523
Symbolic Math Toolbox™ 5 MuPAD® Tutorial

Mupad Tutorial

Embed Size (px)

Citation preview

Page 1: Mupad Tutorial

Symbolic Math Toolbox™ 5MuPAD® Tutorial

Page 2: Mupad Tutorial

How to Contact The MathWorks

www.mathworks.com Webcomp.soft-sys.matlab Newsgroupwww.mathworks.com/contact_TS.html Technical Support

[email protected] Product enhancement [email protected] Bug [email protected] Documentation error [email protected] Order status, license renewals, [email protected] Sales, pricing, and general information

508-647-7000 (Phone)

508-647-7001 (Fax)

The MathWorks, Inc.3 Apple Hill DriveNatick, MA 01760-2098For contact information about worldwide offices, see the MathWorks Web site.

Symbolic Math Toolbox™ MuPAD® Tutorial

© COPYRIGHT 1997–2008 by SciFace Software GmbH & Co. KG.The software described in this document is furnished under a license agreement. The software may be usedor copied only under the terms of the license agreement. No part of this manual may be photocopied orreproduced in any form without prior written consent from The MathWorks, Inc.

FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentationby, for, or through the federal government of the United States. By accepting delivery of the Program orDocumentation, the government hereby agrees that this software or documentation qualifies as commercialcomputer software or commercial computer software documentation as such terms are used or definedin FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014. Accordingly, the terms and conditions ofthis Agreement and only those rights specified in this Agreement, shall pertain to and govern the use,modification, reproduction, release, performance, display, and disclosure of the Program and Documentationby the federal government (or other entity acquiring for or through the federal government) and shallsupersede any conflicting contractual terms or conditions. If this License fails to meet the government’sneeds or is inconsistent in any respect with federal procurement law, the government agrees to return theProgram and Documentation, unused, to The MathWorks, Inc.

Trademarks

MuPAD is a registered trademark of SciFace Software GmbH & Co. KG. MATLAB and Simulinkare registered trademarks of The MathWorks, Inc. See www.mathworks.com/trademarks for a listof additional trademarks. Other product or brand names may be trademarks or registeredtrademarks of their respective holders. Patents

The MathWorks products are protected by one or more U.S. patents. Please seewww.mathworks.com/patents for more information.

Page 3: Mupad Tutorial

Hj]^Y[]

7KLV�ERRN�H[SODLQV�WKH�EDVLF�XVH�RI�WKH�0X3$'� FRPSXWHU�DOJHEUD�V\VWHP� DQGJLYHV�VRPH�LQVLJKW�LQWR�WKH�SRZHU�RI�WKH�V\VWHP� 0X3$' LV�DYDLODEOH�DV�SDUW�RI�WKH6\PEROLF�0DWK�7RROER[��LQ�0$7/$%��

7KLV�LQWURGXFWLRQ�DGGUHVVHV�PDWKHPDWLFLDQV� HQJLQHHUV� FRPSXWHU�VFLHQWLVWV�QDWXUDO�VFLHQWLVWV�DQG� PRUH�JHQHUDOO\� DOO�WKRVH�LQ�QHHG�RI�PDWKHPDWLFDOFRPSXWDWLRQV�IRU�WKHLU�HGXFDWLRQ�RU�WKHLU�SURIHVVLRQ� *HQHUDOO\�VSHDNLQJ� WKLVERRN�DGGUHVVHV�DQ\ERG\�ZKR�ZDQWV�WR�XVH�WKH�SRZHU�RI�D�PRGHUQ�FRPSXWHUDOJHEUD�SDFNDJH�

7KHUH�DUH�WZR�ZD\V�WR�XVH�D�FRPSXWHU�DOJHEUD�V\VWHP� 2Q�WKH�RQH�KDQG� \RX�PD\XVH�WKH�PDWKHPDWLFDO�NQRZOHGJH�LW�LQFRUSRUDWHV�E\�FDOOLQJ�V\VWHP�IXQFWLRQVLQWHUDFWLYHO\� )RU�H[DPSOH� \RX�FDQ�FRPSXWH�V\PEROLF�LQWHJUDOV�RU�JHQHUDWH�DQGLQYHUW�PDWULFHV�E\�FDOOLQJ�DSSURSULDWH�IXQFWLRQV� 7KH\�FRPSULVH�WKH�V\VWHP¶VPDWKHPDWLFDO�LQWHOOLJHQFH�DQG�PD\�LPSOHPHQW�VRSKLVWLFDWHG�DOJRULWKPV�&KDSWHUV � WKURXJK �� GLVFXVV�WKLV�ZD\�RI�XVLQJ�WKH�0X3$' HQJLQH�

2Q�WKH�RWKHU�KDQG� ZLWK�WKH�KHOS�RI�WKH�0X3$' SURJUDPPLQJ�ODQJXDJH� \RX�FDQHDVLO\�DGG�IXQFWLRQDOLW\�WR�WKH�V\VWHP�E\�LPSOHPHQWLQJ�\RXU�RZQ�DOJRULWKPV�DV0X3$' SURFHGXUHV� 7KLV�LV�XVHIXO�IRU�VSHFLDO�SXUSRVH�DSSOLFDWLRQV�LI�QRDSSURSULDWH�V\VWHP�IXQFWLRQV�H[LVW� &KDSWHUV �� WKURXJK �� DUH�DQ�LQWURGXFWLRQ�WR0X3$' SURJUDPPLQJ�

<RX�FDQ�UHDG�WKLV�ERRN�LQ�WKH�VWDQGDUG�ZD\�³OLQHDUO\´�IURP�WKH�ILUVW�WR�WKH�ODVWSDJH� +RZHYHU� WKHUH�DUH�UHDVRQV�WR�SURFHHG�RWKHUZLVH� 7KLV�PD\�EH�WKH�FDVH� H�J��LI�\RX�DUH�LQWHUHVWHG�LQ�D�SDUWLFXODU�SUREOHP� RU�LI�\RX�DOUHDG\�NQRZ�VRPHWKLQJDERXW�0X3$'�

)RU�EHJLQQHUV� ZH�UHFRPPHQG�WR�VWDUW�UHDGLQJ�&KDSWHU �� ZKLFK�JLYHV�D�ILUVWVXUYH\�RI�0X3$'��7KH�GHVFULSWLRQ�RI�WKH�RQOLQH�KHOS�V\VWHP�LQ�6HFWLRQ ��� LVSUREDEO\�WKH�PRVW�LPSRUWDQW�SDUW�RI�WKLV�FKDSWHU� 7KH�KHOS�V\VWHP�SURYLGHVLQIRUPDWLRQ�DERXW�GHWDLOV�RI�V\VWHP�IXQFWLRQV� WKHLU�V\QWD[� WKHLU�FDOOLQJSDUDPHWHUV� HWF� ,W�LV�DYDLODEOH�RQOLQH�ZKHQHYHU�WKH�0X3$' QRWHERRN�LQWHUIDFH�LVUXQQLQJ� ,Q�WKH�EHJLQQLQJ� UHTXHVWLQJ�D�KHOS�SDJH�LV�SUREDEO\�\RXU�PRVW�IUHTXHQWTXHU\�WR�WKH�V\VWHP� $IWHU�\RX�KDYH�JURZQ�IDPLOLDU�ZLWK�WKH�KHOS�V\VWHP� \RX�PD\VWDUW�WR�H[SHULPHQW�ZLWK�0X3$'��&KDSWHU � GHPRQVWUDWHV�VRPH�RI�WKH�PRVWLPSRUWDQW�V\VWHP�IXQFWLRQV�³DW�ZRUN�´ <RX�ZLOO�ILQG�IXUWKHU�GHWDLOV�DERXW�WKHVHIXQFWLRQV�LQ�ODWHU�SDUWV�RI�WKH�ERRN�RU�LQ�WKH�KHOS�SDJHV� )RU�D�GHHSHU

Page 4: Mupad Tutorial

Ik^_Z\^

XQGHUVWDQGLQJ�RI�WKH�GDWD�VWUXFWXUHV�LQYROYHG� \RX�PD\�FRQVXOW�WKH�FRUUHVSRQGLQJVHFWLRQV�LQ�&KDSWHU ��

&KDSWHU � GLVFXVVHV�WKH�0X3$' OLEUDULHV�DQG�WKHLU�XVH� 7KH\�FRQWDLQ�PDQ\IXQFWLRQV�DQG�DOJRULWKPV�IRU�SDUWLFXODU�PDWKHPDWLFDO�WRSLFV�

7KH�EDVLF�GDWD�W\SHV�DQG�WKH�PRVW�LPSRUWDQW�V\VWHP�IXQFWLRQV�IRU�WKHLUPDQLSXODWLRQ�DUH�LQWURGXFHG�LQ�&KDSWHU �� ,W�LV�QRW�QHFHVVDU\�WR�VWXG\�DOO�RI�WKHPLQ�WKH�VDPH�GHSWK� 'HSHQGLQJ�RQ�\RXU�LQWHQGHG�DSSOLFDWLRQ� \RX�PD\�VHOHFWLYHO\UHDG�RQO\�WKH�SDVVDJHV�DERXW�WKH�UHOHYDQW�WRSLFV�

&KDSWHU � H[SODLQV�KRZ�0X3$' HYDOXDWHV�REMHFWV� ZH�VWURQJO\�UHFRPPHQG�WR�UHDGWKLV�FKDSWHU�

&KDSWHUV � WKURXJK �� GLVFXVV�WKH�XVH�RI�VRPH�SDUWLFXODUO\�LPSRUWDQW�V\VWHPIXQFWLRQV� VXEVWLWXWLRQ� GLIIHUHQWLDWLRQ� V\PEROLF�LQWHJUDWLRQ� HTXDWLRQ�VROYLQJ�UDQGRP�QXPEHU�JHQHUDWLRQ� DQG�JUDSKLF�FRPPDQGV�

6HYHUDO�XVHIXO�IHDWXUHV�VXFK�DV�WKH�KLVWRU\�PHFKDQLVP� LQSXW�DQG�RXWSXW�URXWLQHV�RU�WKH�GHILQLWLRQ�RI�XVHU�SUHIHUHQFHV�DUH�GHVFULEHG�LQ�&KDSWHUV ���� WKURXJK ���3UHIHUHQFHV�FDQ�EH�XVHG�WR�FRQILJXUH�WKH�V\VWHP¶V�LQWHUDFWLYH�EHKDYLRU�DIWHU�WKHXVHU¶V�IDQF\�WR�D�FHUWDLQ�H[WHQW�

&KDSWHUV �� WKURXJK �� JLYH�DQ�LQWURGXFWLRQ�WR�WKH�EDVLF�FRQFHSWV�RI�WKH�0X3$'SURJUDPPLQJ�ODQJXDJH�

0X3$' SURYLGHV�DOJRULWKPV�WKDW�FDQ�KDQGOH�D�ODUJH�FODVV�RI�PDWKHPDWLFDO�REMHFWVDQG�FRPSXWDWLRQDO�WDVNV�UHODWHG�WR�WKHP� 8SRQ�UHDGLQJ�WKLV�LQWURGXFWLRQ� LW�LVSRVVLEOH�WKDW�\RX�HQFRXQWHU�XQNQRZQ�PDWKHPDWLFDO�QRWLRQV�VXFK�DV�ULQJV�RUILHOGV� 7KLV�LQWURGXFWLRQ�LV�QRW�LQWHQGHG�WR�H[SODLQ�WKH�PDWKHPDWLFDO�EDFNJURXQGIRU�VXFK�REMHFWV� %DVLF�PDWKHPDWLFDO�NQRZOHGJH�LV�KHOSIXO�EXW�QRW�PDQGDWRU\�WRXQGHUVWDQG�WKH�WH[W� 6RPHWLPHV�\RX�PD\�DVN�ZKDW�DOJRULWKP�0X3$' XVHV�WRVROYH�D�SDUWLFXODU�SUREOHP� 7KH�LQWHUQDO�PRGH�RI�RSHUDWLRQ�RI�WKH�0X3$'SURFHGXUHV�LV�QRW�DGGUHVVHG�KHUH� ZH�GR�QRW�LQWHQG�WR�JLYH�D�JHQHUDO�LQWURGXFWLRQWR�FRPSXWHU�DOJHEUD�DQG�LWV�DOJRULWKPV� 7KH�LQWHUHVWHG�UHDGHU�PD\�FRQVXOW�WH[WERRNV�VXFK�DV� H�J�� >*&/��@�RU�>**��@�

7KLV�ERRN�JLYHV�DQ HOHPHQWDU\ LQWURGXFWLRQ�WR�0X3$'��6RPHZKDW�PRUH�DEVWUDFWPDWKHPDWLFDO�REMHFWV�VXFK�DV� H�J�� ILHOG�H[WHQVLRQV� DUH�HDV\�WR�GHVFULEH�DQG�WRKDQGOH�LQ�0X3$'��+RZHYHU� VXFK�DGYDQFHG�DVSHFWV�RI�WKH�V\VWHP�DUH�QRWGLVFXVVHG�KHUH� 7KH�PDWKHPDWLFDO�DSSOLFDWLRQV�WKDW�DUH�PHQWLRQHG�LQ�WKH�WH[W�DUH

LL

Page 5: Mupad Tutorial

Ik^_Z\^

LQWHQWLRQDOO\�NHSW�RQ�D�UDWKHU�HOHPHQWDU\�OHYHO� 7KLV�LV�WR�NHHS�WKLV�WH[W�SODLQ�IRUUHDGHUV�ZLWK�OLWWOH�PDWKHPDWLFDO�EDFNJURXQG�DQG�WR�PDNH�LW�DSSOLFDEOH�DW�VFKRROOHYHO�

:H�FDQQRW�H[SODLQ�WKH�FRPSOHWH�IXQFWLRQDOLW\�RI�0X3$' LQ�WKLV�LQWURGXFWLRQ�6RPH�SDUWV�RI�WKH�V\VWHP�DUH�PHQWLRQHG�RQO\�EULHIO\� ,W�LV�EH\RQG�WKH�VFRSH�RI�WKLVWXWRULDO�WR�JR�LQWR�WKH�GHWDLOV�RI�WKH�IXOO�SRZHU�RI�WKH�0X3$' SURJUDPPLQJODQJXDJH� <RX�ILQG�WKHVH�LQ�WKH�0X3$' KHOS�V\VWHP� DYDLODEOH�RQOLQH�GXULQJ�D0X3$' VHVVLRQ�

7KLV�WXWRULDO�UHIHUV�WR�0X3$' YHUVLRQ���DQG�ODWHU� 6LQFH�WKH�GHYHORSPHQW�RI�WKHV\VWHP�DGYDQFHV�FRQWLQXRXVO\� VRPH�RI�WKH�GHWDLOV�GHVFULEHG�PD\�FKDQJH�LQ�WKHIXWXUH� )XWXUH�YHUVLRQV�ZLOO�GHILQLWHO\�SURYLGH�DGGLWLRQDO�IXQFWLRQDOLW\�WKURXJKQHZ�V\VWHP�IXQFWLRQV�DQG�DSSOLFDWLRQ�SDFNDJHV� ,Q�WKLV�WXWRULDO� ZH�PDLQO\SUHVHQW�WKH�EDVLF�WRROV�DQG�WKHLU�XVH� ZKLFK�ZLOO�SUREDEO\�UHPDLQ�HVVHQWLDOO\XQFKDQJHG� :H�WU\�WR�ZRUG�DOO�VWDWHPHQWV�LQ�WKH�WH[W�LQ�VXFK�D�ZD\�WKDW�WKH\�VWD\EDVLFDOO\�YDOLG�IRU�IXWXUH�0X3$' YHUVLRQV�

LLL

Page 6: Mupad Tutorial
Page 7: Mupad Tutorial

;gfl]flk

3UHIDFH

b

,QWURGXFWLRQ

*1XPHULFDO�&RPSXWDWLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � ���&RPSXWHU�$OJHEUD � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���&KDUDFWHULVWLFV�RI�&RPSXWHU�$OJHEUD�6\VWHPV � � � � � � � � � � � � � � � ���0X3$'� 6RIWZDUH � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

)LUVW�6WHSV�LQ�0X3$'�

+1RWHERRN�LQWHUIDFH � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���([SODQDWLRQV�DQG�+HOS � � � � � � � � � � � � � � � � � � � � � � � � � � � ���&RPSXWLQJ�ZLWK�1XPEHUV � � � � � � � � � � � � � � � � � � � � � � � � � � ���

([DFW�&RPSXWDWLRQV � � � � � � � � � � � � � � � � � � � � � � � � � ���1XPHULFDO�$SSUR[LPDWLRQV � � � � � � � � � � � � � � � � � � � � � ���&RPSOH[�1XPEHUV � � � � � � � � � � � � � � � � � � � � � � � � � � ���

6\PEROLF�&RPSXWDWLRQ � � � � � � � � � � � � � � � � � � � � � � � � � � � ����,QWURGXFWRU\�([DPSOHV � � � � � � � � � � � � � � � � � � � � � � � � ����&XUYH�6NHWFKLQJ � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����(OHPHQWDU\�1XPEHU�7KHRU\ � � � � � � � � � � � � � � � � � � � � � ����

Page 8: Mupad Tutorial

<hgm^gml

7KH�0X3$'� /LEUDULHV

,,QIRUPDWLRQ�$ERXW�D�3DUWLFXODU�/LEUDU\ � � � � � � � � � � � � � � � � � � ���([SRUWLQJ�/LEUDULHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���7KH�6WDQGDUG�/LEUDU\ � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

0X3$'� 2EMHFWV

-2SHUDQGV� WKH�)XQFWLRQV RS DQG QRSV � � � � � � � � � � � � � � � � � � � ���1XPEHUV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���,GHQWLILHUV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���6\PEROLF�([SUHVVLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����

2SHUDWRUV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����([SUHVVLRQ�7UHHV � � � � � � � � � � � � � � � � � � � � � � � � � � � ����2SHUDQGV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����

6HTXHQFHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����/LVWV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����6HWV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����7DEOHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����$UUD\V � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����%RROHDQ�([SUHVVLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����6WULQJV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����)XQFWLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����6HULHV�([SDQVLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����$OJHEUDLF�6WUXFWXUHV� )LHOGV� 5LQJV� HWF� � � � � � � � � � � � � � � � � � � ����9HFWRUV�DQG�0DWULFHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����

'HILQLWLRQ�RI�0DWULFHV�DQG�9HFWRUV � � � � � � � � � � � � � � � � � ����&RPSXWLQJ�ZLWK�0DWULFHV � � � � � � � � � � � � � � � � � � � � � � ����6SHFLDO�0HWKRGV�IRU�0DWULFHV � � � � � � � � � � � � � � � � � � � � ����7KH�/LEUDULHV OLQDOJ DQG QXPHULF � � � � � � � � � � � � � � � � � ����6SDUVH�0DWULFHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����$Q�$SSOLFDWLRQ � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����

3RO\QRPLDOV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����

YL

Page 9: Mupad Tutorial

<hgm^gml

'HILQLWLRQ�RI�3RO\QRPLDOV � � � � � � � � � � � � � � � � � � � � � � ����&RPSXWLQJ�ZLWK�3RO\QRPLDOV � � � � � � � � � � � � � � � � � � � � ����

+DUGZDUH�)ORDW�$UUD\V � � � � � � � � � � � � � � � � � � � � � � � � � � � ����,QWHUYDO�$ULWKPHWLF � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����1XOO�2EMHFWV� QXOO��� 1,/� )$,/� XQGHILQHG � � � � � � � � � � � � � � � � ����

(YDOXDWLRQ�DQG�6LPSOLILFDWLRQ

.,GHQWLILHUV�DQG�7KHLU�9DOXHV � � � � � � � � � � � � � � � � � � � � � � � � � ���&RPSOHWH� ,QFRPSOHWH� DQG�(QIRUFHG�(YDOXDWLRQ � � � � � � � � � � � � � ���$XWRPDWLF�6LPSOLILFDWLRQ � � � � � � � � � � � � � � � � � � � � � � � � � � ����(YDOXDWLRQ�DW�D�3RLQW � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����

6XEVWLWXWLRQ� VXEV� VXEVH[� DQG VXEVRS/

'LIIHUHQWLDWLRQ�DQG�,QWHJUDWLRQ

0'LIIHUHQWLDWLRQ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���,QWHJUDWLRQ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

6ROYLQJ�(TXDWLRQV� VROYH1

3RO\QRPLDO�(TXDWLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���*HQHUDO�(TXDWLRQV�DQG�,QHTXDOLWLHV � � � � � � � � � � � � � � � � � � � � ���'LIIHUHQWLDO�(TXDWLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����

YLL

Page 10: Mupad Tutorial

<hgm^gml

5HFXUUHQFH�(TXDWLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����

0DQLSXODWLQJ�([SUHVVLRQV

27UDQVIRUPLQJ�([SUHVVLRQV � � � � � � � � � � � � � � � � � � � � � � � � � ���6LPSOLI\LQJ�([SUHVVLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � ����$VVXPSWLRQV�DERXW�0DWKHPDWLFDO�3URSHUWLHV � � � � � � � � � � � � � � � ����

&KDQFH�DQG�3UREDELOLW\

*)

*UDSKLFV

**,QWURGXFWLRQ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����(DV\�3ORWWLQJ� *UDSKV�RI�)XQFWLRQV � � � � � � � � � � � � � � � � � � � � ����

�' )XQFWLRQ�*UDSKV � � � � � � � � � � � � � � � � � � � � � � � � � �����' )XQFWLRQ�*UDSKV � � � � � � � � � � � � � � � � � � � � � � � � � �����

$GYDQFHG�3ORWWLQJ� 3ULQFLSOHV�DQG�)LUVW�([DPSOHV � � � � � � � � � � � � �����*HQHUDO�3ULQFLSOHV � � � � � � � � � � � � � � � � � � � � � � � � � � �����6RPH�([DPSOHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����

7KH�)XOO�3LFWXUH� *UDSKLFDO�7UHHV � � � � � � � � � � � � � � � � � � � � � �����9LHZHU� %URZVHU� DQG�,QVSHFWRU� ,QWHUDFWLYH�0DQLSXODWLRQ � � � � � � � �����3ULPLWLYHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����$WWULEXWHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����

'HIDXOW�9DOXHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����,QKHULWDQFH�RI�$WWULEXWHV � � � � � � � � � � � � � � � � � � � � � � � �����3ULPLWLYHV�5HTXHVWLQJ�6FHQH�$WWULEXWHV� ³+LQWV´ � � � � � � � � � �����7KH�+HOS�3DJHV�RI�$WWULEXWHV � � � � � � � � � � � � � � � � � � � � � �����

YLLL

Page 11: Mupad Tutorial

<hgm^gml

&RORUV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����5*% &RORUV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����+69 &RORUV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����

$QLPDWLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����*HQHUDWLQJ�6LPSOH�$QLPDWLRQV � � � � � � � � � � � � � � � � � � � �����3OD\LQJ�$QLPDWLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � �����7KH�1XPEHU�RI�)UDPHV�DQG�WKH�7LPH�5DQJH � � � � � � � � � � � � �����:KDW�&DQ�%H�$QLPDWHG" � � � � � � � � � � � � � � � � � � � � � � � �����$GYDQFHG�$QLPDWLRQV� 7KH�6\QFKURQL]DWLRQ�0RGHO � � � � � � � � �����)UDPH�E\�)UDPH�$QLPDWLRQV � � � � � � � � � � � � � � � � � � � � � �����([DPSOHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����

*URXSV�RI�3ULPLWLYHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����7UDQVIRUPDWLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����/HJHQGV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ������)RQWV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ������6DYLQJ�DQG�([SRUWLQJ�3LFWXUHV � � � � � � � � � � � � � � � � � � � � � � � ������

,QWHUDFWLYH�6DYLQJ�DQG�([SRUWLQJ � � � � � � � � � � � � � � � � � � ������%DWFK�0RGH � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ������

,PSRUWLQJ�3LFWXUHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ������&DPHUDV�LQ��' � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ������6WUDQJH�(IIHFWV�LQ��'"�$FFHOHUDWHG�2SHQ*/� OLEUDU\" � � � � � � � � � � ������

,QSXW�DQG�2XWSXW

*+2XWSXW�RI�([SUHVVLRQV � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����

3ULQWLQJ�([SUHVVLRQV�RQ�WKH�6FUHHQ � � � � � � � � � � � � � � � � � ����0RGLI\LQJ�WKH�2XWSXW�)RUPDW � � � � � � � � � � � � � � � � � � � � ����

5HDGLQJ�DQG�:ULWLQJ�)LOHV � � � � � � � � � � � � � � � � � � � � � � � � � � ����7KH�)XQFWLRQV ZULWH DQG UHDG � � � � � � � � � � � � � � � � � � � � ����6DYLQJ�D�0X3$'� 6HVVLRQ � � � � � � � � � � � � � � � � � � � � � � ����5HDGLQJ�'DWD�IURP�D�7H[W�)LOH � � � � � � � � � � � � � � � � � � � � ����

L[

Page 12: Mupad Tutorial

<hgm^gml

8WLOLWLHV

*,8VHU�'HILQHG�3UHIHUHQFHV � � � � � � � � � � � � � � � � � � � � � � � � � � ����7KH�+LVWRU\�0HFKDQLVP � � � � � � � � � � � � � � � � � � � � � � � � � � � ����,QIRUPDWLRQ�RQ�0X3$'� $OJRULWKPV � � � � � � � � � � � � � � � � � � � ����5HVWDUWLQJ�D�0X3$'� 6HVVLRQ � � � � � � � � � � � � � � � � � � � � � � � �����([HFXWLQJ�&RPPDQGV�RI�WKH�2SHUDWLQJ�6\VWHP � � � � � � � � � � � � � � �����

7\SH�6SHFLILHUV

*-7KH�)XQFWLRQV W\SH DQG WHVWW\SH � � � � � � � � � � � � � � � � � � � � � ����&RPIRUWDEOH�7\SH�&KHFNLQJ� WKH 7\SH /LEUDU\ � � � � � � � � � � � � � � ����

/RRSV

*.

%UDQFKLQJ� LI�WKHQ�HOVH DQG FDVH*/

0X3$'� 3URFHGXUHV

*0'HILQLQJ�3URFHGXUHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ����7KH�5HWXUQ�9DOXH�RI�D�3URFHGXUH � � � � � � � � � � � � � � � � � � � � � � ����5HWXUQLQJ�6\PEROLF�)XQFWLRQ�&DOOV � � � � � � � � � � � � � � � � � � � � � ����/RFDO�DQG�*OREDO�9DULDEOHV � � � � � � � � � � � � � � � � � � � � � � � � � ����

[

Page 13: Mupad Tutorial

<hgm^gml

6XESURFHGXUHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����6FRSH�RI�9DULDEOHV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����7\SH�'HFODUDWLRQ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����3URFHGXUHV�ZLWK�D�9DULDEOH�1XPEHU�RI�$UJXPHQWV � � � � � � � � � � � � �����2SWLRQV� WKH�5HPHPEHU�7DEOH � � � � � � � � � � � � � � � � � � � � � � � �����,QSXW�3DUDPHWHUV � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �����(YDOXDWLRQ�:LWKLQ�3URFHGXUHV � � � � � � � � � � � � � � � � � � � � � � � �����)XQFWLRQ�(QYLURQPHQWV � � � � � � � � � � � � � � � � � � � � � � � � � � � �����$ 3URJUDPPLQJ�([DPSOH� 'LIIHUHQWLDWLRQ � � � � � � � � � � � � � � � � � �����3URJUDPPLQJ�([HUFLVHV � � � � � � � � � � � � � � � � � � � � � � � � � � � �����

6ROXWLRQV�WR�([HUFLVHV

:

'RFXPHQWDWLRQ�DQG�5HIHUHQFHV

;

*UDSKLFV�*DOOHU\

<

&RPPHQWV�RQ�WKH�*UDSKLFV�*DOOHU\

=

[L

Page 14: Mupad Tutorial

<hgm^gml

,QGH[

Bg]^q

[LL

Page 15: Mupad Tutorial

)

,QWURGXFWLRQ

7R�H[SODLQ�WKH�QRWLRQ�RI�FRPSXWHU�DOJHEUD� ZH�FRPSDUH�DOJHEUDLF�DQG�QXPHULFDOFRPSXWDWLRQV� %RWK�NLQGV�DUH�VXSSRUWHG�E\�D�FRPSXWHU� EXW�WKHUH�DUHIXQGDPHQWDO�GLIIHUHQFHV� ZKLFK�DUH�GLVFXVVHG�LQ�ZKDW�IROORZV�

Page 16: Mupad Tutorial

* Bgmkh]n\mbhg

Fme]ja[Yd�;gehmlYlagfk

0DQ\�D�PDWKHPDWLFDO�SUREOHP�FDQ�EH�VROYHG�DSSUR[LPDWHO\�E\�QXPHULFDOFRPSXWDWLRQV� 7KH�FRPSXWDWLRQ�VWHSV�RSHUDWH�RQ QXPEHUV� ZKLFK�DUH�VWRUHGLQWHUQDOO\�LQ IORDWLQJ�SRLQW�UHSUHVHQWDWLRQ� 7KLV�UHSUHVHQWDWLRQ�KDV�WKH�GUDZEDFNWKDW�QHLWKHU�FRPSXWDWLRQV�QRU�VROXWLRQV�DUH�H[DFW�GXH�WR�URXQGLQJ�HUURUV� ,QJHQHUDO� QXPHULFDO�DOJRULWKPV�ILQG�DSSUR[LPDWH�VROXWLRQV�DV�IDVW�DV�SRVVLEOH�2IWHQ�VXFK�VROXWLRQV�DUH�WKH�RQO\�ZD\�WR�KDQGOH�D�PDWKHPDWLFDO�SUREOHPFRPSXWDWLRQDOO\� LQ�SDUWLFXODU�LI�WKHUH�LV�QR�³FORVHG�IRUP´�VROXWLRQ�NQRZQ� �7KHPRVW�SRSXODU�H[DPSOH�IRU�WKLV�VLWXDWLRQ�DUH�URRWV�RI�SRO\QRPLDOV�RI�KLJK�GHJUHHV��0RUHRYHU� DSSUR[LPDWH�VROXWLRQV�DUH�XVHIXO�LI�H[DFW�UHVXOWV�DUH�XQQHFHVVDU\��H�J��LQ�YLVXDOL]DWLRQ��

���

Page 17: Mupad Tutorial

<hfinm^k�:e`^[kZ

;gehml]j�9d_]ZjY

,Q�FRQWUDVW�WR�QXPHULFDO�FRPSXWDWLRQV� WKHUH�DUH V\PEROLF FRPSXWDWLRQV�LQFRPSXWHU�DOJHEUD� >+HF ��@�GHILQHV�WKHP�DV ³FRPSXWDWLRQV�ZLWK�V\PEROVUHSUHVHQWLQJ�PDWKHPDWLFDO�REMHFWV�´ +HUH� DQ REMHFW PD\�EH�D�QXPEHU� EXW�DOVR�DSRO\QRPLDO� DQ�HTXDWLRQ� DQ�H[SUHVVLRQ�RU�D�IRUPXOD� D�IXQFWLRQ� D�JURXS� D�ULQJ�RU�DQ\�RWKHU�PDWKHPDWLFDO�REMHFW� 6\PEROLF�FRPSXWDWLRQV�ZLWK�QXPEHUV�DUHFDUULHG�RXW H[DFWO\� ,QWHUQDOO\� QXPEHUV�DUH�UHSUHVHQWHG�DV�TXRWLHQWV�RI�LQWHJHUVRI�DUELWUDU\�OHQJWK��OLPLWHG�E\�WKH�DPRXQW�RI�VWRUDJH�DYDLODEOH� RI�FRXUVH�� 7KHVHNLQGV�RI�FRPSXWDWLRQV�DUH�FDOOHG V\PEROLF RU DOJHEUDLF� >+HF ��@�JLYHV�WKHIROORZLQJ�GHILQLWLRQV�

�� ³6\PEROLF´�HPSKDVL]HV�WKDW�LQ�PDQ\�FDVHV�WKH�XOWLPDWH�JRDO�RI�PDWKHPDWLFDOSUREOHP�VROYLQJ�LV�H[SUHVVLQJ�WKH�DQVZHU�LQ�D�FORVHG�IRUPXOD�RU�ILQGLQJ�DV\PEROLF�DSSUR[LPDWLRQ�

�� ³$OJHEUDLF´�PHDQV�WKDW�FRPSXWDWLRQV�DUH�FDUULHG�RXW�H[DFWO\� DFFRUGLQJ�WRWKH�UXOHV�RI�DOJHEUD� LQVWHDG�RI�XVLQJ�DSSUR[LPDWH�IORDWLQJ�SRLQW�DULWKPHWLF�

6RPHWLPHV�³V\PEROLF�PDQLSXODWLRQ´�RU�³IRUPXOD�PDQLSXODWLRQ´�DUH�XVHG�DVV\QRQ\PV�IRU�FRPSXWHU�DOJHEUD� VLQFH�FRPSXWDWLRQV�RSHUDWH�RQ�V\PEROV�DQGIRUPXODH� ([DPSOHV�DUH�V\PEROLF�LQWHJUDWLRQ�RU�GLIIHUHQWLDWLRQ�VXFK�DV

∫xGx =

x2

2,

∫ 4

1xGx =

152,

GGx

OQ OQx =1x OQx

RU�V\PEROLF�VROXWLRQV�RI�HTXDWLRQV� )RU�H[DPSOH� ZH�FRQVLGHU�WKH�HTXDWLRQx4 + p x2 + 1 = 0 LQ x ZLWK�RQH�SDUDPHWHU p� ,WV�VROXWLRQ�VHW�LV

⎧⎨

⎩±

√2√−p−

√p2 − 4

2, ±

√2√−p+

√p2 − 4

2

⎫⎬

⎭ .

7KH�V\PEROLF�FRPSXWDWLRQ�RI�DQ�H[DFW�VROXWLRQ�XVXDOO\�UHTXLUHV�PRUH�FRPSXWLQJWLPH�DQG�PRUH�VWRUDJH�WKDQ�WKH�QXPHULF�FRPSXWDWLRQ�RI�DQ�DSSUR[LPDWH�VROXWLRQ�+RZHYHU� D�V\PEROLF�VROXWLRQ�LV�H[DFW� PRUH�JHQHUDO� DQG�RIWHQ�SURYLGHV�PRUHLQIRUPDWLRQ�DERXW�WKH�SUREOHP�DQG�LWV�VROXWLRQ� 7KH�DERYH�IRUPXOD�H[SUHVVHV�WKHVROXWLRQV�RI�WKH�HTXDWLRQ�LQ�WHUPV�RI�WKH�SDUDPHWHU p� ,W�VKRZV�KRZ�WKH�VROXWLRQV

���

Page 18: Mupad Tutorial

* Bgmkh]n\mbhg

GHSHQG�IXQFWLRQDOO\�RQ p� 7KLV�LQIRUPDWLRQ�FDQ�EH�XVHG� IRU�H[DPSOH� WR�H[DPLQHKRZ�VHQVLWLYH�WKH�VROXWLRQV�EHKDYH�ZKHQ�WKH�SDUDPHWHU�FKDQJHV�

&RPELQDWLRQV�RI�V\PEROLF�DQG�QXPHULF�PHWKRGV�DUH�XVHIXO�IRU�VSHFLDODSSOLFDWLRQV� )RU�H[DPSOH� WKHUH�DUH�DOJRULWKPV�LQ�FRPSXWHU�DOJHEUD�WKDW�EHQHILWIURP�HIILFLHQW�KDUGZDUH�IORDWLQJ�SRLQW�DULWKPHWLF� 2Q�WKH�RWKHU�KDQG� LW�PD\�EHXVHIXO�WR�VLPSOLI\�D�SUREOHP�IURP�QXPHULFDO�DQDO\VLV�V\PEROLFDOO\�EHIRUH�DSSO\LQJWKH�DFWXDO�DSSUR[LPDWLRQ�DOJRULWKP�

���

Page 19: Mupad Tutorial

<aZkZ\m^kblmb\l�h_�<hfinm^k�:e`^[kZ�Lrlm^fl

;`YjY[l]jakla[k�g^�;gehml]j�9d_]ZjY�Kqkl]ek

0RVW�RI�WKH�FXUUHQW�FRPSXWHU�DOJHEUD�V\VWHPV�FDQ�EH�XVHG�LQWHUDFWLYHO\� 7KH�XVHUHQWHUV�VRPH�IRUPXODH�DQG�FRPPDQGV� DQG�WKH�V\VWHP HYDOXDWHV WKHP� 7KHQ�LWUHWXUQV�DQ�DQVZHU� ZKLFK�FDQ�EH�PDQLSXODWHG�IXUWKHU�LI�QHFHVVDU\�

,Q�DGGLWLRQ�WR�H[DFW�V\PEROLF�FRPSXWDWLRQV� PRVW�FRPSXWHU�DOJHEUD�SDFNDJHV�FDQDSSUR[LPDWH�VROXWLRQV�QXPHULFDOO\�DV�ZHOO� 7KH�XVHU�FDQ�VHW�WKH�SUHFLVLRQ�WR�WKHGHVLUHG�QXPEHU�RI�GLJLWV� ,Q�0X3$'�� WKH�JOREDO�YDULDEOH ',*,76 KDQGOHV�WKLV�)RU�H[DPSOH� LI�\RX�HQWHU�WKH�VLPSOH�FRPPDQG ',*,76� ���� WKHQ�0X3$'SHUIRUPV�DOO�IORDWLQJ�SRLQW�FDOFXODWLRQV�ZLWK�D�SUHFLVLRQ�RI 100 GHFLPDO�GLJLWV� 2IFRXUVH� VXFK�FRPSXWDWLRQV�QHHG�PRUH�FRPSXWLQJ�WLPH�DQG�PRUH�VWRUDJH�WKDQ�WKHXVH�RI�KDUGZDUH�IORDWLQJ�SRLQW�DULWKPHWLF�

0RUHRYHU� PRGHUQ�FRPSXWHU�DOJHEUD�V\VWHPV�SURYLGH�D�SRZHUIXO�SURJUDPPLQJODQJXDJH� DQG�WRROV�IRU�YLVXDOL]DWLRQ�DQG�DQLPDWLRQ�RI�PDWKHPDWLFDO�GDWD� $OVR�PDQ\�V\VWHPV�FDQ�SURGXFH�OD\RXWHG�GRFXPHQWV��NQRZQ�DV QRWHERRNV RUZRUNVKHHWV�� 7KH�0X3$' V\VWHP�KDV�VXFK�D�QRWHERRN�FRQFHSW� EXW�ZH�ZLOO�RQO\VNHWFK�LW�EULHIO\�LQ�WKLV�WXWRULDO� 3OHDVH�FKHFN�WKH�XVHU�LQWHUIDFH�GRFXPHQWDWLRQLQVWHDG�IRU�PRUH�GHWDLOV� 7KH�JRDO�RI�WKLV�ERRN�LV�WR�JLYH�DQ�LQWURGXFWLRQ�WR�WKHPDWKHPDWLFDO SRZHU�RI�WKH�0X3$' ODQJXDJH�

�7KH�0X3$' SURJUDPPLQJ�ODQJXDJH�LV�VWUXFWXUHG�VLPLODUO\�WR 3DVFDO� ZLWK�H[WHQVLRQV�VXFK�DV�ODQ�JXDJH�FRQVWUXFWV�IRU�REMHFW�RULHQWHG�SURJUDPPLQJ�

���

Page 20: Mupad Tutorial

* Bgmkh]n\mbhg

EmH9<� Kg^loYj]

,Q�FRPSDULVRQ�WR�RWKHU�FRPSXWHU�DOJHEUD�V\VWHPV� 0X3$'� KDV�WKH�IROORZLQJQRWHZRUWK\�IHDWXUHV� ZKLFK�DUH�QRW�GLVFXVVHG�LQ�GHWDLO�LQ�WKLV�ERRN�

� 7KHUH�DUH�0X3$' ODQJXDJH�FRQVWUXFWV�IRU�REMHFW�RULHQWHG�SURJUDPPLQJ�<RX FDQ�GHILQH�\RXU�RZQ�GDWD�W\SHV� H�J�� WR�UHSUHVHQW�PDWKHPDWLFDOVWUXFWXUHV� $OPRVW�DOO�H[LVWLQJ�RSHUDWRUV�DQG�IXQFWLRQV�FDQ�EH�RYHUORDGHG�

� 7KHUH�LV�D�KLJKO\�LQWHUDFWLYH�0X3$' VRXUFH�FRGH�GHEXJJHU�

� 3URJUDPV�ZULWWHQ�LQ�& RU�&���FDQ�EH�DGGHG�WR�WKH�NHUQHO�E\�WKH�0X3$'G\QDPLF�PRGXOH�FRQFHSW�

7KH�KHDUW�RI�WKH�0X3$' HQJLQH�LV�LWV NHUQHO� ZKLFK�LV�LPSOHPHQWHG�LQ�&��� ,WFRPSULVHV�WKH�IROORZLQJ�PDLQ�FRPSRQHQWV�

± 7KH SDUVHU UHDGV�WKH�LQSXW�WR�WKH�V\VWHP�DQG�SHUIRUPV D�V\QWD[�FKHFN� ,I�QRHUURUV�DUH�IRXQG� LW�FRQYHUWV�WKH�LQSXW�WR�D�0X3$' GDWD�W\SH�

± 7KH HYDOXDWRU SURFHVVHV�DQG�VLPSOLILHV�WKH�LQSXW� ,WV PRGH�RI�RSHUDWLRQ�LVGLVFXVVHG�ODWHU�

± 7KHPHPRU\�PDQDJHPHQW LV�UHVSRQVLEOH�IRU�WKH HIILFLHQW�VWRUDJH�RI�0X3$'REMHFWV�

± 6RPH�IUHTXHQWO\�XVHG�DOJRULWKPV�VXFK�DV� H�J�� DULWKPHWLFDO IXQFWLRQV�DUHLPSOHPHQWHG�DV NHUQHO�IXQFWLRQV LQ�&�

3DUVHU�DQG�HYDOXDWRU�GHILQH�WKH�0X3$' ODQJXDJH�DV�GHVFULEHG�LQ�WKLV�ERRN� 7KH0X3$' OLEUDULHV� ZKLFK�FRQWDLQ�PRVW�RI�WKH�PDWKHPDWLFDO�NQRZOHGJH�RI�WKHV\VWHP� DUH�LPSOHPHQWHG�LQ�WKLV�ODQJXDJH�

,Q�DGGLWLRQ� 0X3$' RIIHUV�FRPIRUWDEOH�XVHU�LQWHUIDFHV�IRU�JHQHUDWLQJ QRWHERRNVRU�JUDSKLFV� RU�WR�GHEXJ�SURJUDPV�ZULWWHQ�LQ�WKH�0X3$' ODQJXDJH� 7KH�0X3$'KHOS�V\VWHP�KDV�K\SHUWH[W�IXQFWLRQDOLW\� <RX�FDQ�QDYLJDWH�ZLWKLQ�GRFXPHQWV�DQGH[HFXWH�H[DPSOHV�E\�D�PRXVH�FOLFN�

6\PEROLF�0DWK�7RROER[��DOVR�RIIHUV�D�ZD\�RI�XVLQJ�WKH�0X3$' HQJLQH�WR�VROYHSUREOHPV�GLUHFWO\�IURP�WKH�0$7/$%� HQYLURQPHQW� XVLQJ�HLWKHU�FRPPDQGV�DV

���

Page 21: Mupad Tutorial

FnI:=� Lh_mpZk^

SUHVHQWHG�KHUH�RU� IRU�D�VHOHFWHG�VXEVHW�RI�FRPPDQGV� DOWHUQDWLYHV�ILWWLQJ�PRUHFORVHO\�LQWR�D�0$7/$% ODQJXDJH�SURJUDP� 7KLV�ERRN�GRHV�QRW�GLVFXVV�WKHVHRSWLRQV� EXW�OLPLWV�LWVHOI�WR�XVLQJ�WKH�0X3$' HQJLQH�GLUHFWO\� DORQJ�ZLWK�WKH0X3$' *8,V�

���

Page 22: Mupad Tutorial
Page 23: Mupad Tutorial

*

)LUVW�6WHSV�LQ�0X3$'�

&RPSXWHU�DOJHEUD�V\VWHPV�VXFK�DV�0X3$'� DUH�RIWHQ�XVHG�LQWHUDFWLYHO\� )RUH[DPSOH� \RX�FDQ�HQWHU�DQ�LQVWUXFWLRQ�WR�PXOWLSO\�WZR�QXPEHUV�DQG�KDYH�0X3$'FRPSXWH�WKH�UHVXOW�DQG�SULQW�LW�RQ�WKH�VFUHHQ�

7KH�IROORZLQJ�VHFWLRQ�ZLOO�JLYH�D�VKRUW�GHVFULSWLRQ�RI�WKH�XVHU�LQWHUIDFH� 7KH�QH[WRQH�DIWHU�WKDW�GHVFULEHV�WKH�KHOS�V\VWHP� 5HTXHVWLQJ�D�KHOS�SDJH�LV�SUREDEO\�WKHPRVW�IUHTXHQWO\�XVHG�FRPPDQG�IRU�WKH�EHJLQQHU� 7KH�VHFWLRQ�DIWHU�WKDW�LV�DERXWXVLQJ�0X3$' DV�DQ�³LQWHOOLJHQW�SRFNHW�FDOFXODWRU´� FDOFXODWLQJ�ZLWK�QXPEHUV�7KLV�LV�WKH�HDVLHVW�DQG�WKH�PRVW�LQWXLWLYH�SDUW�RI�WKLV�WXWRULDO� $IWHUZDUGV�ZHLQWURGXFH�VRPH�V\VWHP�IXQFWLRQV�IRU�V\PEROLF�FRPSXWDWLRQV� 7KH�FRUUHVSRQGLQJVHFWLRQ�LV�ZULWWHQ�TXLWH�LQIRUPDOO\�DQG�JLYHV�D�ILUVW�LQVLJKW�LQWR�WKH�V\PEROLFIHDWXUHV�RI�WKH�V\VWHP�

Page 24: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

Fgl]Zggc�afl]j^Y[]

$IWHU�\RX�FDOO�WKH�0X3$'� SURJUDP� \RX�ZLOO�VHH�WKH�ZHOFRPH�GLDORJ� )URP�WKHUH�\RX�FDQ�HQWHU�WKH�KHOS�V\VWHP��ZKLFK�LV� RI�FRXUVH� DOVR�DYDLODEOH�ODWHU�DQG�ZLOO�EHGHVFULEHG�RQ�SDJH ����� RSHQ�DQ�H[LVWLQJ�ILOH� FUHDWH�D�QHZ�0X3$' VRXUFH�ILOH�RURSHQ�D�QHZ�QRWHERRN� 8VXDOO\� \RX�ZLOO�ZLVK�WR�RSHQ�DQ�H[LVWLQJ�RU�D�QHZQRWHERRN�

$IWHU�VWDUWLQJ�WKH�SURJUDP� \RX�FDQ�HQWHU�FRPPDQGV�LQ�WKH�0X3$' ODQJXDJH�7KHVH�FRPPDQGV�VKRXOG�EH�W\SHG�LQWR�³LQSXW�UHJLRQV�´ ZKLFK�ORRN�OLNH�WKLV�

,I�\RX�SUHVV�WKH �5HWXUQ! NH\� WKLV�ILQLVKHV�\RXU�LQSXW�DQG�0X3$' HYDOXDWHV�WKHFRPPDQG�WKDW�\RX�KDYH�HQWHUHG� +ROGLQJ �6KLIW! ZKLOH�SUHVVLQJ �5HWXUQ!SURYRNHV�D�OLQHIHHG� ZKLFK�\RX�FDQ�XVH�WR�IRUPDW�\RXU�LQSXW�

$IWHU�LQYRNLQJ�WKH�FRPPDQG� \RX�ZLOO�VHH�LWV�UHVXOW�SULQWHG�LQ�WKH�VDPH�EUDFNHW�EHORZ�WKH�LQSXW� DQG�LI�\RX�ZHUH�DW�WKH�HQG�RI�WKH�QRWHERRN� D�QHZ�LQSXW�UHJLRQZLOO�EH�RSHQHG�EHORZ�

VLQ�������0.0005926535551

7KH�V\VWHP�HYDOXDWHG�WKH�XVXDO�VLQH�IXQFWLRQ�DW�WKH�SRLQW 3.141 DQG�UHWXUQHG�DIORDWLQJ�SRLQW�DSSUR[LPDWLRQ�RI�WKH�YDOXH� VLPLODU�WR�WKH�RXWSXW�RI�D�SRFNHWFDOFXODWRU�

<RX�FDQ�DW�DQ\�WLPH�JR�EDFN�WR�ROGHU�LQSXWV�DQG�HGLW�WKHP� $OVR� \RX�FDQ�FOLFNEHWZHHQ�WKH�EUDFNHWV�DQG�DGG�WH[W� (GLWLQJ�RXWSXW�LV�QRW�SRVVLEOH� EXW�\RX�FDQFRS\�IRUPXODV�IURP�WKH�RXWSXW�DQG�SDVWH�WKHP�VRPHSODFH�HOVH� �:KHQ�FRS\LQJ�DIRUPXOD�WR�DQ�LQSXW�UHJLRQ� LW�ZLOO�EH�DXWRPDWLFDOO\�WUDQVODWHG�LQWR�DQ�HTXLYDOHQWFRPPDQG�LQ�WH[WXDO�IRUP��

,I�\RX�WHUPLQDWH�\RXU�FRPPDQG�ZLWK�D�FRORQ� WKHQ�0X3$' H[HFXWHV�WKH�FRPPDQGZLWKRXW�SULQWLQJ�LWV�UHVXOW�RQ�WKH�VFUHHQ� 7KLV�HQDEOHV�\RX�WR�VXSSUHVV�WKH�RXWSXWRI�LUUHOHYDQW�LQWHUPHGLDWH�UHVXOWV�

���

Page 25: Mupad Tutorial

Ghm^[hhd�bgm^k_Z\^

<RX�FDQ�HQWHU�PRUH�WKDQ�RQH�FRPPDQG�LQ�RQH�OLQH� 7ZR�VXEVHTXHQW�FRPPDQGVKDYH�WR�EH�VHSDUDWHG�E\�D�VHPLFRORQ�RU�D�FRORQ� LI�WKH�UHVXOW�RI�WKH�ILUVW�FRPPDQGLV�WR�EH�SULQWHG�RU�QRW� UHVSHFWLYHO\�

GLII�VLQ�[A��� [�� LQW��� [�2x FRV

(x2)

VLQ(x2)

+HUH [A� GHQRWHV�WKH�VTXDUH�RI [� DQG�WKH�0X3$' IXQFWLRQV GLII DQG LQWSHUIRUP�WKH�RSHUDWLRQV�³GLIIHUHQWLDWH´�DQG�³LQWHJUDWH´��&KDSWHU ��� 7KH�FKDUDFWHU� UHWXUQV�WKH�SUHYLRXV�H[SUHVVLRQ��LQ�WKH�H[DPSOH� WKLV�LV�WKH�GHULYDWLYH�RI VLQ(x2)��7KH�FRQFHSW�XQGHUO\LQJ � LV�GLVFXVVHG�LQ�&KDSWHU �����

,Q�WKH�IROORZLQJ�H[DPSOH� WKH�RXWSXW�RI�WKH�ILUVW�FRPPDQG�LV�VXSSUHVVHG�E\�WKHFRORQ� DQG�RQO\�WKH�UHVXOW�RI�WKH�VHFRQG�FRPPDQG�DSSHDUV�RQ�WKH�VFUHHQ�

HTXDWLRQV � ^[ � \ �� [ � \ �`�VROYH�HTXDWLRQV�{[x = 1, y = 0]}

,Q�WKH�SUHYLRXV�H[DPSOH� D�VHW�RI�WZR�HTXDWLRQV�LV�DVVLJQHG�WR�WKH�LGHQWLILHUHTXDWLRQV� 7KH�FRPPDQG VROYH�HTXDWLRQV� FRPSXWHV�WKH�VROXWLRQ� &KDSWHU �GLVFXVVHV�WKH�VROYHU�LQ�PRUH�GHWDLO�

7R�TXLW�D�0X3$' VHVVLRQ� XVH�WKH�FRUUHVSRQGLQJ�HQWU\�LQ�WKH�³)LOH´�PHQX�

���

Page 26: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

=phdYfYlagfk�Yf\�@]dh

,I�\RX�GR�QRW�NQRZ�WKH�FRUUHFW�V\QWD[�RI�D�0X3$'� FRPPDQG� \RX�FDQ�REWDLQ�WKLVLQIRUPDWLRQ�GLUHFWO\�IURP�WKH�RQOLQH�KHOS�V\VWHP� )RU�D�EULHI�H[SODQDWLRQ� VLPSO\W\SH�WKH�FRPPDQG�QDPH� SRLQW�\RXU�PRXVH�FXUVRU�DW�WKH�FRPPDQG�DQG�OHW�LW�UHVWWKHUH�IRU�D�PRPHQW� <RX�ZLOO�JHW�D�WRROWLS�ZLWK�D�EULHI�H[SODQDWLRQ�

7KH KHOS�SDJH RI�WKH�FRUUHVSRQGLQJ�IXQFWLRQ�SURYLGHV�PRUH�GHWDLOHG�LQIRUPDWLRQ�<RX�FDQ�UHTXHVW�LW�E\�HQWHULQJ KHOS��QDPH��� ZKHUH QDPH LV�WKH�QDPH�RI�WKHIXQFWLRQ� 7KH�IXQFWLRQ KHOS H[SHFWV�LWV�DUJXPHQW�WR�EH�D�VWULQJ� ZKLFK�LVJHQHUDWHG�E\�GRXEOH�TXRWHV � �6HFWLRQ ������ 7KH�RSHUDWRU " LV�D�VKRUW�IRUP�IRUKHOS� ,W�LV�XVHG�ZLWKRXW�SDUHQWKHVLV�RU�TXRWHV�

"VROYH

7KH�KHOS�V\VWHP�LV�D�K\SHUWH[W�V\VWHP� L�H�� VLPLODU�WR�EURZVLQJ�WKH�ZHE� $FWLYHNH\ZRUGV�DUH�XQGHUOLQHG� ,I�\RX�FOLFN�RQ�WKHP� \RX�REWDLQ�IXUWKHU�LQIRUPDWLRQDERXW�WKH�FRUUHVSRQGLQJ�QRWLRQ� 7KH�WRROWLSV�PHQWLRQHG�DERYH�DOVR�ZRUN�LQ�WKHKHOS�V\VWHP� <RX�FDQ�HGLW�DQG�H[HFXWH�WKH�H[DPSOHV�LQ�WKH�KHOS�V\VWHP�RU�FRS\DQG�SDVWH�WKHP�WR�D�QRWHERRN�

([HUFLVH����� )LQG�RXW�KRZ�WR�XVH�WKH�0X3$' GLIIHUHQWLDWRU GLII� DQG�FRPSXWHWKH�ILIWK�GHULYDWLYH�RI VLQ(x2)�

���

Page 27: Mupad Tutorial

<hfinmbg`�pbma�Gnf[^kl

;gehmlaf_�oal`�FmeZ]jk

7R�FRPSXWH�ZLWK�QXPEHUV� \RX�FDQ�XVH�0X3$'� OLNH�D�SRFNHW�FDOFXODWRU� 7KHUHVXOW�RI�WKH�IROORZLQJ�LQSXW�LV�D�UDWLRQDO�QXPEHU�

� � ���72

<RX�VHH�WKDW�WKH�UHWXUQV�DUH�H[DFW�UHVXOWV��DQG�QRW�URXQGHG�IORDWLQJ�SRLQWQXPEHUV��ZKHQ�FRPSXWLQJ�ZLWK�LQWHJHUV�DQG�UDWLRQDO�QXPEHUV�

�� � ���� �������� � ����A�674736728

7KH�V\PERO A UHSUHVHQWV�H[SRQHQWLDWLRQ� 0X3$' FDQ�FRPSXWH�ELJ�QXPEHUVHIILFLHQWO\� 7KH�OHQJWK�RI�D�QXPEHU�WKDW�\RX�PD\�FRPSXWH�LV�RQO\�OLPLWHG�E\�WKHDYDLODEOH�PDLQ�VWRUDJH� )RU�H[DPSOH� WKH 123UG�SRZHU�RI 1234 LV�D�IDLUO\�ELJLQWHJHU��

����A��������������������������������������������������������?�����������������������������������������������������?�����������������������������������������������������?�����������������������������������������������������?�����������������������������������������������������?�����������������������������������������������������?�����������������������������������������������������?����������

%HVLGHV�WKH�EDVLF�DULWKPHWLF�IXQFWLRQV� 0X3$' SURYLGHV�D�YDULHW\�RI�IXQFWLRQVRSHUDWLQJ�RQ�QXPEHUV�

�,Q�WKLV�SULQWRXW� WKH�³EDFNVODVK´ \ DW�WKH�HQG�RI�D�OLQH�LQGLFDWHV�WKDW�WKH�UHVXOW�LV�FRQWLQXHG�RQ�WKHQH[W�OLQH� 7R�EUHDN�ORQJ�OLQHV�LQ�WKLV�ZD\� \RX�QHHG�WR�VZLWFK�RII�W\SHVHWWLQJ�LQ�WKH�³1RWHERRN´�PHQX�

���

Page 28: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

$ VLPSOH�H[DPSOH�LV�WKH�IDFWRULDO n! = 1 · 2 · · ·n RI�D�QRQQHJDWLYH�LQWHJHU� ZKLFKFDQ�EH�HQWHUHG�LQ�PDWKHPDWLFDO�QRWDWLRQ�

���������������������������������������������������������?�����������������������������������������������������?����������������������������������������������������

7KH�IXQFWLRQ LVSULPH FKHFNV�ZKHWKHU�D�SRVLWLYH�LQWHJHU�LV�SULPH� ,W�UHWXUQV�HLWKHU758( RU )$/6(�LVSULPH�����������

)$/6(

8VLQJ LIDFWRU� \RX�FDQ�REWDLQ�WKH�SULPH�IDFWRUL]DWLRQ�

LIDFWRU�����������32 · 3607 · 3803

([DFW�&RPSXWDWLRQV

1RZ�VXSSRVH�WKDW�ZH�ZDQW�WR�³FRPSXWH´�WKH�QXPEHU√

56� 7KH�SUREOHP�LV�WKDW�WKHYDOXH�RI�WKLV�LUUDWLRQDO�QXPEHU�FDQQRW�EH�H[SUHVVHG�DV�D�TXRWLHQWQXPHUDWRU�GHQRPLQDWRU�RI�WZR�LQWHJHUV�H[DFWO\� 7KXV�³FRPSXWDWLRQ´�FDQ�RQO\PHDQ�WR�ILQG�DQ�H[DFW�UHSUHVHQWDWLRQ�WKDW�LV DV�VLPSOH�DV�SRVVLEOH� :KHQ�\RXLQSXW

√56 YLD VTUW� 0X3$' UHWXUQV�WKH�IROORZLQJ�

VTUW����

2√

14

7KH�UHVXOW�RI�WKH�VLPSOLILFDWLRQ�RI√

56 LV�WKH�H[DFW�YDOXH 2 ·√

14� ,Q�0X3$'�√

14�RU� VRPHWLPHV� ��A�������UHSUHVHQWV�WKH�SRVLWLYH�VROXWLRQ�RI�WKH�HTXDWLRQx2 = 14� ,QGHHG� WKLV�LV�SUREDEO\�WKH�PRVW�VLPSOH�UHSUHVHQWDWLRQ�RI�WKH�UHVXOW� :HVWUHVV�WKDW

√14 LV�D�JHQXLQH�REMHFW�ZLWK�FHUWDLQ�SURSHUWLHV��H�J�� WKDW�LWV�VTXDUH�FDQ

EH�VLPSOLILHG�WR 14�� 7KH�V\VWHP�DSSOLHV�WKHP�DXWRPDWLFDOO\�ZKHQ�FRPSXWLQJ�ZLWKVXFK�REMHFWV� )RU�H[DPSOH�

VTUW����A�196

���

Page 29: Mupad Tutorial

<hfinmbg`�pbma�Gnf[^kl

$V�DQRWKHU�H[DPSOH�IRU�H[DFW�FRPSXWDWLRQ� OHW�XV�GHWHUPLQH�WKH�OLPLW

e = OLPn→∞

(1 +

1n

)n.

:H�XVH�WKH�IXQFWLRQ OLPLW DQG�WKH�V\PERO LQILQLW\�

OLPLW��� � ��Q�AQ� Q LQILQLW\�H

7R�HQWHU�WKLV�QXPEHU�LQ�0X3$'��\RX�KDYH�WR�XVH ( RU H[S���� ZKHUH H[SUHSUHVHQWV�WKH�H[SRQHQWLDO�IXQFWLRQ� 0X3$' NQRZV�H[DFW�UXOHV�RI�PDQLSXODWLRQIRU�WKLV�REMHFW� )RU�H[DPSOH� XVLQJ�WKH�QDWXUDO�ORJDULWKP OQ ZH�ILQG�

OQ���H[S����−1

:H�ZLOO�HQFRXQWHU�PRUH�H[DFW�FRPSXWDWLRQV�ODWHU�LQ�WKLV�WXWRULDO�

1XPHULFDO�$SSUR[LPDWLRQV

%HVLGHV�H[DFW�FRPSXWDWLRQV� 0X3$' FDQ�DOVR�SHUIRUP�QXPHULFDO�DSSUR[LPDWLRQV�)RU�H[DPSOH� \RX�FDQ�XVH�WKH�IXQFWLRQ IORDW WR�ILQG�D�GHFLPDO�DSSUR[LPDWLRQ�WR√

56� 7KLV�IXQFWLRQ�FRPSXWHV�WKH�YDOXH�RI�LWV�DUJXPHQW�LQ IORDWLQJ�SRLQWUHSUHVHQWDWLRQ�

IORDW�VTUW�����7.483314774

7KH�SUHFLVLRQ�RI�WKH�DSSUR[LPDWLRQ�GHSHQGV�RQ�WKH�YDOXH�RI�WKH�JOREDO�YDULDEOH',*,76� ZKLFK�GHWHUPLQHV�WKH�QXPEHU�RI�GHFLPDO�GLJLWV�IRU�QXPHULFDOFRPSXWDWLRQV� ,WV�GHIDXOW�YDOXH�LV 10�

',*,76� IORDW������������10

10.02868609

���

Page 30: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

*OREDO�YDULDEOHV�VXFK�DV ',*,76 DIIHFW�WKH�EHKDYLRU�RI�0X3$'��DQG�DUH�DOVR�FDOOHGHQYLURQPHQW�YDULDEOHV�� <RX�ILQG�D�FRPSOHWH�OLVW�RI�DOO�HQYLURQPHQW�YDULDEOHV�LQ6HFWLRQ ³(QYLURQPHQW�9DULDEOHV´�RI�WKH�0X3$' 4XLFN�5HIHUHQFH�LQ�WKH�RQOLQHGRFXPHQWDWLRQ� 7KH�YDULDEOH ',*,76 FDQ�DVVXPH�DQ\�LQWHJUDO�YDOXH�EHWZHHQ 1 DQG232 − 1� DOWKRXJK 1000 FDQ�DOUHDG\�EH�FRQVLGHUHG�YHU\�ODUJH�DQG�LW�LV�XQOLNHO\�WKDWXVLQJ�PRUH�WKDQ 10 000 GLJLWV�LV�UHDVRQDEOH�IRU�DQ\WKLQJ�EXW�WKH�PRVW�XQXVXDOFDOFXODWLRQV�

',*,76 � ���� IORDW������������� ',*,76 � ��������������������������������������������������������?������������������������������������������������

:H�KDYH�UHVHW�WKH�YDOXH�RI ',*,76 WR 10 IRU�WKH�IROORZLQJ�FRPSXWDWLRQV� 7KLV�FDQDOVR�EH�DFKLHYHG�YLD�WKH�FRPPDQG GHOHWH ',*,76� )RU�DULWKPHWLF�RSHUDWLRQV�ZLWKQXPEHUV� 0X3$' DXWRPDWLFDOO\�XVHV�DSSUR[LPDWH�FRPSXWDWLRQ�DV�VRRQ�DV DWOHDVW�RQH RI�WKH�QXPEHUV�LQYROYHG�LV�D�IORDWLQJ�SRLQW�YDOXH�

���� � ���� �������� � ����A�10.02868609

3OHDVH�QRWH�WKDW�QRQH�RI�WKH�WZR�IROORZLQJ�FDOOV

��� VLQ���� ������������ VLQ���

UHVXOWV�LQ�DQ�DSSUR[LPDWH�FRPSXWDWLRQ�RI VLQ(2)� VLQFH� WHFKQLFDOO\� VLQ��� LV�DQH[SUHVVLRQ�UHSUHVHQWLQJ�WKH��H[DFW��YDOXH�RI VLQ(2) DQG QRW D�QXPEHU�

��� VLQ���� ������������ VLQ���2 VLQ(2)

3, 0.6666666666 VLQ(2)

�7KH�VHSDUDWLRQ�RI�ERWK�YDOXHV�E\�D�FRPPD�JHQHUDWHV�D�VSHFLDO�GDWD�W\SH� QDPHO\�DVHTXHQFH� ZKLFK�LV�GHVFULEHG�LQ�6HFWLRQ �����

�<RX�VKRXOG�EH�SDUWLFXODUO\�FDXWLRXV�ZKHQ�WKH�VDPH�FRPSXWDWLRQ�LV�SHUIRUPHG�ZLWK�GLIIHUHQW�YDOXHVRI ',*,76��6RPH�RI�WKH�PRUH�LQWULFDWH�QXPHULFDO�DOJRULWKPV�LQ�0X3$' HPSOR\�WKH�RSWLRQ�³UHPHPEHU�´7KLV�LPSOLHV�WKDW�WKH\�VWRUH�SUHYLRXVO\�FRPSXWHG�YDOXHV�WR�EH�XVHG�DJDLQ��6HFWLRQ ������ ZKLFK�FDQ�OHDGWR�LQDFFXUDWH�QXPHULFDO�UHVXOWV�LI�WKH�UHPHPEHUHG�YDOXHV�ZHUH�FRPSXWHG�ZLWK�ORZHU�SUHFLVLRQ� 7R�EHVDIH� \RX�VKRXOG�UHVWDUW�WKH�0X3$' VHVVLRQ�XVLQJ UHVHW�� EHIRUH�LQFUHDVLQJ�WKH�YDOXH�RI ',*,76��7KLVFRPPDQG�FOHDUV�PHPRU\�DQG�UHVHWV�DOO�HQYLURQPHQW�YDULDEOHV�WR�WKHLU�GHIDXOW�YDOXHV��6HFWLRQ ������

���

Page 31: Mupad Tutorial

<hfinmbg`�pbma�Gnf[^kl

<RX�KDYH�WR�XVH�WKH�IXQFWLRQ IORDW WR�FRPSXWH�IORDWLQJ�SRLQW�DSSUR[LPDWLRQV�RIWKH�DERYH�H[SUHVVLRQV��

IORDW���� VLQ����� ������������ IORDW�VLQ����0.6061982846, 0.6061982845

0RVW�DULWKPHWLF�IXQFWLRQV�VXFK�DV�WKH�VTXDUH�URRW� WKH�WULJRQRPHWULF�IXQFWLRQV�WKH�H[SRQHQWLDO�IXQFWLRQ� RU�WKH�ORJDULWKP� DXWRPDWLFDOO\�UHWXUQ�DSSUR[LPDWHYDOXHV�ZKHQ�WKHLU�DUJXPHQW�LV�D�IORDWLQJ�SRLQW�QXPEHU�

VTUW������� VLQ������7.483314774, 0.001592652916

7KH�FRQVWDQWV π DQG e DUH�GHQRWHG�E\ 3, DQG ( H[S���� UHVSHFWLYHO\� 0X3$' FDQSHUIRUP�H[DFW�FRPSXWDWLRQV�ZLWK�WKHP�

FRV�3,�� OQ�(�−1, 1

,I�GHVLUHG� \RX�FDQ�REWDLQ�QXPHULFDO�DSSUR[LPDWLRQV�RI�WKHVH�FRQVWDQWV�E\DSSO\LQJ IORDW�

',*,76 � ���� IORDW�3,�� IORDW�(�� GHOHWH ',*,76������������������������������������������������������?������������������������������������������������

�����������������������������������������������������?������������������������������������������������

([HUFLVH����� &RPSXWH√

27− 2√

3 DQG FRV(π/8) H[DFWO\� 'HWHUPLQH�QXPHULFDODSSUR[LPDWLRQV�WR�D�SUHFLVLRQ�RI���GLJLWV�

&RPSOH[�1XPEHUV

7KH�LPDJLQDU\�XQLW√−1 LV�UHSUHVHQWHG�LQ�0X3$' E\�WKH�V\PERO , LQ�WKH�LQSXW

DQG�DQ�XSULJKW L LQ�WKH�W\SHVHW�RXWSXW�

�7DNH�D�ORRN�DW�WKH�ODVW�GLJLWV� 7KH�VHFRQG�FRPPDQG�\LHOGV�D�VOLJKWO\�OHVV�DFFXUDWH�UHVXOW� VLQFH0.666 . . . LV�DOUHDG\�DQ�DSSUR[LPDWLRQ�RI 2/3 DQG�WKH�URXQGLQJ�HUURU�LV�SURSDJDWHG�WR�WKH�ILQDO�UHVXOW�

���

Page 32: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

VTUW����� ,A�L, −1

<RX�FDQ�LQSXW�FRPSOH[�QXPEHUV�LQ�WKH�XVXDO�PDWKHPDWLFDO�QRWDWLRQ x+ y L� %RWKWKH�UHDO�SDUW x DQG�WKH�LPDJLQDU\�SDUW y PD\�EH�LQWHJHUV� UDWLRQDO�QXPEHUV� RUIORDWLQJ�SRLQW�QXPEHUV�

�� � � ,� �� � ,�� ���� � ,� ���� � ,���A�2 + 9 L, 0.073− 0.129 L

,I�\RX�XVH�V\PEROLF�H[SUHVVLRQV�VXFK�DV� H�J�� VTUW���� 0X3$' PD\�QRW�UHWXUQ�WKHUHVXOW�RI�D�FDOFXODWLRQ�LQ�&DUWHVLDQ�FRRUGLQDWHV�

���VTUW��� � ,�1√

2 + L

7KH�IXQFWLRQ UHFWIRUP �VKRUW�IRU� UHFWDQJXODU�IRUP��HQVXUHV�WKDW�WKH�UHVXOW�LV�VSOLWLQWR�LWV�UHDO�DQG�LPDJLQDU\�SDUWV�

UHFWIRUP����VTUW��� � ,��√

23− L

3

7KH�IXQFWLRQV 5H DQG ,P UHWXUQ�WKH�UHDO�SDUW x DQG�WKH�LPDJLQDU\�SDUW y�UHVSHFWLYHO\� RI�D�FRPSOH[�QXPEHU x+ y L� 7KH�IXQFWLRQV FRQMXJDWH� DEV� DQG DUJFRPSXWH�WKH�FRPSOH[�FRQMXJDWH x− y L� WKH�DEVROXWH�YDOXH |x+ y L| =

√x2 + y2�

DQG�WKH�SRODU�DQJOH� UHVSHFWLYHO\�

5H����VTUW��� � ,��� ,P����VTUW��� � ,���FRQMXJDWH����VTUW��� � ,���DEV����VTUW��� � ,��� DUJ����VTUW��� � ,���UHFWIRUP�FRQMXJDWH����VTUW��� � ,���√

23, −1

3,

1√2− L,

√3

3, − DUFWDQ

(√2

2

),

√2

3+L3

����

Page 33: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

KqeZgda[�;gehmlYlagf

7KLV�VHFWLRQ�FRPSULVHV�VRPH�H[DPSOHV�RI�0X3$'� VHVVLRQV�WKDW�LOOXVWUDWH�D�VPDOOVHOHFWLRQ�RI�WKH�V\VWHP¶V�SRZHU�IRU�V\PEROLF�PDQLSXODWLRQ� 7KH�PDWKHPDWLFDONQRZOHGJH�LV�FRQWDLQHG�HVVHQWLDOO\�LQ�WKH�0X3$' IXQFWLRQV�IRU�GLIIHUHQWLDWLRQ�LQWHJUDWLRQ� VLPSOLILFDWLRQ�RI�H[SUHVVLRQV�HWF� 7KLV�GHPRQVWUDWLRQ�GRHV�QRWSURFHHG�LQ�D�SDUWLFXODUO\�V\VWHPDWLF�PDQQHU� ZH�DSSO\�WKH�V\VWHP�IXQFWLRQV�WRREMHFWV�RI�YDULRXV�W\SHV� VXFK�DV�VHTXHQFHV� VHWV� OLVWV� H[SUHVVLRQV�HWF� 7KH\�DUHH[SODLQHG�LQ�GHWDLO�RQH�E\�RQH�LQ�&KDSWHU ��

,QWURGXFWRU\�([DPSOHV

$ V\PEROLF�H[SUHVVLRQ�PD\�FRQWDLQ�XQGHWHUPLQHG�TXDQWLWLHV��LGHQWLILHUV�� 7KHIROORZLQJ�H[SUHVVLRQ�FRQWDLQV�WZR�XQNQRZQV x DQG y�

I � \A� � � [ � � [A� � � [A� � [A�x4 + 4x3 + 6x2 + 4x+ y2

8VLQJ�WKH�DVVLJQPHQW�RSHUDWRU � � ZH�KDYH�DVVLJQHG�WKH�H[SUHVVLRQ�WR�DQLGHQWLILHU I� ZKLFK�FDQ�QRZ�EH�XVHG�DV�DQ�DEEUHYLDWLRQ�IRU�WKH�H[SUHVVLRQ� :H�VD\WKDW�WKH�ODWWHU�LV�WKH YDOXH RI�WKH�LGHQWLILHU I� :H�QRWH�WKDW�0X3$' KDV�H[FKDQJHGWKH�RUGHU�RI�WKH�WHUPV��

0X3$' RIIHUV�WKH�V\VWHP�IXQFWLRQ GLII IRU�GLIIHUHQWLDWLQJ�H[SUHVVLRQV�

GLII�I� [�� GLII�I� \�4x3 + 12x2 + 12x+ 4, 2 y

+HUH� ZH�KDYH�FRPSXWHG�ERWK�WKH�GHULYDWLYH�ZLWK�UHVSHFW�WR x DQG�WR y� <RX�PD\REWDLQ�KLJKHU�GHULYDWLYHV�HLWKHU�E\�QHVWHG�FDOOV�RI GLII� RU�E\�D�VLQJOH�FDOO�

GLII�GLII�GLII�I� [�� [�� [� GLII�I� [� [� [�24x+ 24 = 24x+ 24

�,QWHUQDOO\� V\PEROLF�VXPV�DUH�RUGHUHG�DFFRUGLQJ�WR�FHUWDLQ�UXOHV�WKDW�HQDEOH�WKH�V\VWHP�WR�DFFHVVWKH�WHUPV�IDVWHU� 2I�FRXUVH� VXFK�D�UHRUGHULQJ�RI�WKH�LQSXW�KDSSHQV�RQO\�IRU�FRPPXWDWLYH�RSHUDWLRQV�VXFKDV� H�J�� DGGLWLRQ�RU�PXOWLSOLFDWLRQ� ZKHUH�FKDQJLQJ�WKH�RUGHU�RI�WKH�RSHUDQGV�\LHOGV�D�PDWKHPDWLFDOO\HTXLYDOHQW�REMHFW�

����

Page 34: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

$OWHUQDWLYHO\� \RX�FDQ�XVH�WKH�GLIIHUHQWLDO�RSHUDWRU � ZKLFK�PDSV�D�IXQFWLRQ�WR�LWVGHULYDWLYH�

VLQ� VLQ�[�FRV, FRV(x)

7KH�V\PERO IRU�WKH�GHULYDWLYH�LV�D�VKRUW�IRUP�RI�WKH�GLIIHUHQWLDO�RSHUDWRU '� 7KHFDOO '�IXQFWLRQ� UHWXUQV�WKH�GHULYDWLYH�'�VLQ�� '�VLQ��[�

FRV, FRV(x)

1RWH� 0X3$' XVHV�D�PDWKHPDWLFDOO\�VWULFW�QRWDWLRQ�IRU�WKH�GLIIHUHQWLDO�RSHUDWRU� '�RU� HTXLYDOHQWO\� WKH RSHUDWRU��GLIIHUHQWLDWHV�IXQFWLRQV� ZKLOH GLII GLIIHUHQWLDWHVH[SUHVVLRQV� ,Q�WKH�H[DPSOH� WKH PDSV�WKH��QDPH�RI�WKH��IXQFWLRQ�WR�WKH��QDPHRI�WKH��IXQFWLRQ�UHSUHVHQWLQJ�WKH�GHULYDWLYH� <RX�RIWHQ�ILQG�D�VORSS\�QRWDWLRQ�VXFKDV� H�J�� (x+ x2)′ IRU�WKH�GHULYDWLYH�RI�WKH�IXQFWLRQ F : x #→ x+ x2� 7KLV�QRWDWLRQFRQIXVHV�WKH�PDS F DQG�WKH�LPDJH�SRLQW f = F (x) DW�D�SRLQW x� 0X3$' KDV�DVWULFW�GLVWLQFWLRQ�EHWZHHQ�WKH IXQFWLRQ F DQG�WKH H[SUHVVLRQ f = F (x)� ZKLFK�DUHUHDOL]HG�DV�GLIIHUHQW�GDWD�W\SHV� 7KH�PDS�FRUUHVSRQGLQJ�WR f FDQ�EH�GHILQHG�E\

) � [ �! [ � [A��

7KHQ

GLII�I� [� )�[�2x+ 1 = 2x+ 1

DUH�HTXLYDOHQW�ZD\V�RI�REWDLQLQJ�WKH�GHULYDWLYH�DV�H[SUHVVLRQV� 7KH�FDOOI� [�[A�� I� GRHV�QRW�PDNH�VHQVH�LQ�0X3$'�

<RX�FDQ�FRPSXWH�LQWHJUDOV�E\�XVLQJ LQW� 7KH�IROORZLQJ�FRPPDQG�FRPSXWHV�DGHILQLWH�LQWHJUDO�RQ�WKH�UHDO�LQWHUYDO�EHWZHHQ 0 DQG 1�

LQW�I� [ �����

y2 +265

����

Page 35: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

7KH�QH[W�FRPPDQG�GHWHUPLQHV�DQ�LQGHILQLWH�LQWHJUDO�DQG�UHWXUQV�DQ�H[SUHVVLRQFRQWDLQLQJ�WKH�LQWHJUDWLRQ�YDULDEOH x DQG�D�V\PEROLF�SDUDPHWHU y�

LQW�I� [�x5

5+ x4 + 2x3 + 2x2 + x y2

1RWH�WKDW LQW UHWXUQV�D�VSHFLDO�DQWLGHULYDWLYH� QRW�D�JHQHUDO�RQH��ZLWK�DGGLWLYHFRQVWDQW��

,I�\RX�WU\�WR�FRPSXWH�WKH�LQGHILQLWH�LQWHJUDO�RI�DQ�H[SUHVVLRQ�DQG�LW�FDQQRW�EHUHSUHVHQWHG�E\�HOHPHQWDU\�IXQFWLRQV� WKHQ LQW UHWXUQV�WKH�FDOO�V\PEROLFDOO\�

LQWHJUDO � LQW����H[S�[A�� � ��� [�∫

1Hx2 + 1

Gx

1HYHUWKHOHVV� WKLV�REMHFW�KDV�PDWKHPDWLFDO�SURSHUWLHV� 7KH�GLIIHUHQWLDWRUUHFRJQL]HV�WKDW�LWV�GHULYDWLYH�LV�WKH�LQWHJUDQG�

GLII�LQWHJUDO� [�1

Hx2 + 1

'HILQLWH�LQWHJUDOV�PD\�DOVR�EH�UHWXUQHG�V\PEROLFDOO\�E\ LQW�

LQW����H[S�[A�� � ��� [ �����∫ 1

0

1Hx2 + 1

Gx

7KH�FRUUHVSRQGLQJ�PDWKHPDWLFDO�REMHFW�LV�D�UHDO�QXPEHU� DQG�WKH�RXWSXW�LV�DQH[DFW�UHSUHVHQWDWLRQ�RI�WKLV�QXPEHU� ZKLFK�0X3$' ZDV�XQDEOH�WR�VLPSOLI\�IXUWKHU�$V�XVXDO� \RX�FDQ�REWDLQ�D�IORDWLQJ�SRLQW�DSSUR[LPDWLRQ�E\�DSSO\LQJ IORDW�

IORDW���0.41946648

$V�DOUHDG\�QRWHG� WKH�V\PERO � LV�DQ�DEEUHYLDWLRQ�IRU�WKH�SUHYLRXVO\�FRPSXWHGH[SUHVVLRQ��&KDSWHU ������

0X3$' NQRZV�WKH�PRVW�LPSRUWDQW�PDWKHPDWLFDO�IXQFWLRQV�VXFK�DV�WKH�VTXDUH

����

Page 36: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

URRW VTUW� WKH�H[SRQHQWLDO�IXQFWLRQ H[S� WKH�WULJRQRPHWULF�IXQFWLRQV VLQ� FRV� WDQ�WKH�K\SHUEROLF�IXQFWLRQV VLQK� FRVK� WDQK� WKH�FRUUHVSRQGLQJ�LQYHUVH�IXQFWLRQV OQ�DUFVLQ� DUFFRV� DUFWDQ� DUFVLQK� DUFFRVK� DUFWDQK� DV�ZHOO�DV�D�YDULHW\�RI�RWKHUVSHFLDO�IXQFWLRQV�VXFK�DV� H�J�� WKH�JDPPD�IXQFWLRQ� WKH�HUURU�IXQFWLRQ HUI� %HVVHOIXQFWLRQV��EHVVHO,� EHVVHO-��«�� HWF���6HFWLRQ�³6SHFLDO�0DWKHPDWLFDO�)XQFWLRQV´RI�WKH�0X3$' 4XLFN�5HIHUHQFH�JLYHV�D�VXUYH\�� ,Q�SDUWLFXODU� 0X3$' NQRZV�WKHUXOHV�RI�PDQLSXODWLRQ�IRU�WKHVH�IXQFWLRQV��H�J�� WKH�DGGLWLRQ�WKHRUHPV�IRU�WKHWULJRQRPHWULF�IXQFWLRQV��DQG�DSSOLHV�WKHP� ,W�FDQ�FRPSXWH�IORDWLQJ�SRLQWDSSUR[LPDWLRQV�VXFK�DV� H�J�� IORDW�H[S���� ��������� DQG�NQRZV�VSHFLDOYDOXHV��H�J�� VLQ�3,� ���

,I�\RX�FDOO�WKHVH�IXQFWLRQV� WKH\�RIWHQ�UHWXUQ�WKHPVHOYHV�V\PEROLFDOO\� VLQFH�WKLV�LVWKH�PRVW�VLPSOH�H[DFW�UHSUHVHQWDWLRQ�RI�WKH�FRUUHVSRQGLQJ�YDOXH�

VTUW���� H[S���� VLQ�[ � \�√

2, H, VLQ(x+ y)

)RU�PDQ\�XVHUV� WKH�PDLQ�IHDWXUH�RI�WKH�V\VWHP�LV�WR�VLPSOLI\�RU�WUDQVIRUP�VXFKH[SUHVVLRQV�XVLQJ�WKH�UXOHV�IRU�FRPSXWDWLRQ� )RU�H[DPSOH� WKH�V\VWHP�IXQFWLRQH[SDQG ³H[SDQGV´�IXQFWLRQV�VXFK�DV H[S� VLQ� HWF��E\�PHDQV�RI�WKH�DGGLWLRQWKHRUHPV LI�WKHLU�DUJXPHQW�LV�D�V\PEROLF�VXP�

H[SDQG�H[S�[ � \��� H[SDQG�VLQ�[ � \���H[SDQG�WDQ�[ � � 3,����

Hx Hy, FRV(x) VLQ(y) + FRV(y) VLQ(x) , − 1WDQ(x)

*HQHUDOO\�VSHDNLQJ� RQH�RI�WKH�PDLQ�WDVNV�RI�D�FRPSXWHU�DOJHEUD�V\VWHP�LV�WRPDQLSXODWH�DQG�WR�VLPSOLI\�H[SUHVVLRQV� %HVLGHV H[SDQG� 0X3$' SURYLGHV�WKHIXQFWLRQV FROOHFW� FRPELQH� IDFWRU� QRUPDO� SDUWIUDF� UDGVLPS� UHZULWH�VLPSOLI\� DQG 6LPSOLI\ IRU�PDQLSXODWLRQ� 7KH\�DUH�SUHVHQWHG�LQ�JUHDWHU�GHWDLO�LQ&KDSWHU �� :H�EULHIO\�PHQWLRQ�VRPH�RI�WKHP�LQ�ZKDW�IROORZV�

7KH�IXQFWLRQ QRUPDO ILQGV�D�FRPPRQ�GHQRPLQDWRU�IRU�UDWLRQDO�H[SUHVVLRQV�

I � [��� � [� � ���� � [�� J � QRUPDO�I�x2 + x+ 2x2 − 1

����

Page 37: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

0RUHRYHU� QRUPDO DXWRPDWLFDOO\�FDQFHOV�FRPPRQ�IDFWRUV�LQ�WKH�QXPHUDWRU�DQGWKH�GHQRPLQDWRU�

QRUPDO�[A���[ � \� � \A���[ � \��x− y

&RQYHUVHO\� SDUWIUDF �VKRUW�IRU�³SDUWLDO�IUDFWLRQ´��GHFRPSRVHV�D�UDWLRQDOH[SUHVVLRQ�LQWR�D�VXP�RI�UDWLRQDO�WHUPV�ZLWK�VLPSOH�GHQRPLQDWRUV�

SDUWIUDF�J� [�2x− 1

− 1x+ 1

+ 1

7KH�IXQFWLRQV VLPSOLI\ DQG 6LPSOLI\ DUH�XQLYHUVDO�VLPSOLILHUV�DQG�WU\�WR�ILQG�DUHSUHVHQWDWLRQ�WKDW�LV�DV�³VLPSOH´�DV�SRVVLEOH�

VLPSOLI\��H[S�[� � ����H[S�[��� � ���Hx2 − 1

<RX�PD\�FRQWURO�WKH�VLPSOLILFDWLRQ�E\�VXSSO\LQJ VLPSOLI\ RU 6LPSOLI\ ZLWKDGGLWLRQDO�DUJXPHQWV��VHH�WKH�RQOLQH�GRFXPHQWDWLRQ�IRU�GHWDLOV��

7KH�IXQFWLRQ UDGVLPS VLPSOLILHV�DULWKPHWLFDO�H[SUHVVLRQV�FRQWDLQLQJ�UDGLFDOV�URRWV��

I � VTUW�� � � VTUW����� I UDGVLPS�I�√

2√√

3 + 2 =√

3 + 1

+HUH� ZH�KDYH�JHQHUDWHG�DQ�HTXDWLRQ� ZKLFK�LV�D�JHQXLQH�0X3$' REMHFW�

$QRWKHU�LPSRUWDQW�IXQFWLRQ�LV IDFWRU� ZKLFK�GHFRPSRVHV�DQ�H[SUHVVLRQ�LQWR�DSURGXFW�RI�VLPSOHU�RQHV�

IDFWRU�[A� � � [A� � � [ � ���IDFWRU�� [ \ � � [ � � \ � [A� � \A���IDFWRU�[A���[ � \� � ]A���[ � \��

(x+ 1)3, (x+ y − 2) · (x+ y) ,(x− z) · (x+ z)

(x+ y)

����

Page 38: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

7KH�IXQFWLRQ OLPLW GRHV�ZKDW�LWV�QDPH�VXJJHVWV� )RU�H[DPSOH� WKH�IXQFWLRQVLQ(x)/x KDV�D�UHPRYDEOH�SROH�DW x = 0� ,WV�OLPLW�IRU x→ 0 LV 1�

OLPLW�VLQ�[��[� [ ��1

,Q�D�0X3$' VHVVLRQ� \RX�FDQ�GHILQH�IXQFWLRQV�RI�\RXU�RZQ�LQ�VHYHUDO�ZD\V� $VLPSOH�DQG�LQWXLWLYH�PHWKRG�LV�WR�XVH�WKH�DUURZ�RSHUDWRU �! �WKH�PLQXV�V\PEROIROORZHG�E\�WKH�³JUHDWHU�WKDQ´�V\PERO��

) � [ �! �[A��� )�[�� )�\�� )�D � E�� )�[�

x2, y2, (a+ b)2, 2x

,Q�&KDSWHU ��� ZH�GLVFXVV�0X3$' SURJUDPPLQJ�IHDWXUHV�DQG�GHVFULEH�KRZ�WRLPSOHPHQW�PRUH�FRPSOH[�DOJRULWKPV�DV�0X3$' SURFHGXUHV�

<RX�FDQ�DOVR�XVH�WKH�JUDSKLFV�IDFLOLWLHV�WR�YLVXDOL]H�PDWKHPDWLFDO�REMHFWVLPPHGLDWHO\� 7KH�UHOHYDQW�0X3$' IXQFWLRQV�IRU�JHQHUDWLQJ�JUDSKLFV�DUH�WKH SORWFRPPDQG�DQG�WKH�URXWLQHV�IURP�WKH�JUDSKLFV�OLEUDU\ SORW� +HUH� ZH�SUHVHQWVLPSOH�FDOOV�RQO\� SOHDVH�FRQVXOW�&KDSWHU �� IRU�LQ�GHSWK�LQIRUPDWLRQ�

SORW�VLQ�[A��� [ ������

����

Page 39: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

SORW�VLQ�[A� � \A��� [ ���3,� \ ���3,� ��'�

6ROYLQJ�HTXDWLRQV�RU�V\VWHPV�RI�HTXDWLRQV�LV�FHUWDLQO\�DQ�LPSRUWDQW�WDVN�IRU�DFRPSXWHU�DOJHEUD�V\VWHP� 7KLV�LV�GRQH�YLD VROYH�

HTXDWLRQV � ^[ � \ D� [ � D \ E`�XQNQRZQV � ^[� \`�VROYH�HTXDWLRQV� XQNQRZQV� ,JQRUH6SHFLDO&DVHV�{[x =a2 + ba+ 1

, y =a− ba+ 1

]}

+HUH� ZH�KDYH�JHQHUDWHG�D�VHW�RI�WZR�HTXDWLRQV�DQG�D�VHW�RI�XQNQRZQV�ZKLFK�ZHZLVK�WR�VROYH�IRU� VROYH UHWXUQV�WKH�UHVXOW�LQ�WHUPV�RI�VLPSOLILHG�HTXDWLRQV� IURPZKLFK�\RX�FDQ�UHDG�RII�WKH�VROXWLRQ� ,Q�WKH�DERYH�H[DPSOH� WKHUH�DUH�WZR�PRUHV\PEROLF�SDUDPHWHUV a DQG b� 7KLV�LV�ZK\�ZH�KDYH�WROG VROYH ZKLFK�RI�WKH�V\PEROVLW�VKRXOG�H[SUHVV�LQ�WHUPV�RI�WKH�RWKHUV� 7KH�³RSWLRQ´ ,JQRUH6SHFLDO&DVHV WHOOV0X3$'�WR�LJQRUH�WKH�SRVVLELOLW\�WKDW D FRXOG�EH −1� ZKHUH�WKH�DERYH�VROXWLRQZRXOG�EH�LQFRUUHFW� :LWKRXW�WKLV�RSWLRQ� 0X3$'�UHWXUQV�D�FRPSOHWH�VROXWLRQ�ZLWKWKUHH�EUDQFKHV�

����

Page 40: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

VROYH�HTXDWLRQV� XQNQRZQV�⎧⎪⎪⎨

⎪⎪⎩

{[x = a2+b

a+1 , y = a−ba+1

]}LI a ̸= −1

{[x = −z − 1, y = z]} LI a = −1 ∧ b = −1∅ LI a = −1 ∧ b ̸= −1

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�KDYH�RQO\�RQH�HTXDWLRQ�LQ�RQH�XQNQRZQ� 0X3$'DXWRPDWLFDOO\�UHFRJQL]HV�WKH�XQNQRZQ�DQG�VROYHV�IRU�LW�

VROYH�[A� � � [ � � ��{[x = 1− L] , [x = 1 + L]}

,I�ZH�VXSSO\�WKH�XQNQRZQ [ WR�VROYH�IRU� WKH�IRUPDW�RI�WKH�RXWSXW�FKDQJHV�

VROYH�[A� � � [ � � �� [�{1− L, 1 + L}

7KH�UHVXOW�LV�D�VHW�FRQWDLQLQJ�WKH�WZR��FRPSOH[��VROXWLRQV�RI�WKH�TXDGUDWLFHTXDWLRQ� <RX�ILQG�D�GHWDLOHG�GHVFULSWLRQ�RI VROYH LQ�&KDSWHU ��

7KH�IXQFWLRQV VXP DQG SURGXFW KDQGOH�V\PEROLF�VXPV�DQG�SURGXFWV� )RU�H[DPSOH�WKH�ZHOO�NQRZQ�VXP 1 + 2 + · · ·+ n LV�VXP�L� L ���Q�n (n+ 1)

2

7KH�SURGXFW 1 · 2 · · ·n LV�NQRZQ�DV�IDFWRULDO n!�

SURGXFW�LA�� L ���Q�n!3

����

Page 41: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

7KHUH�H[LVW�VHYHUDO�GDWD�VWUXFWXUHV�IRU�YHFWRUV�DQG�PDWULFHV� ,Q�SULQFLSOH� \RX�PD\XVH�DUUD\V��6HFWLRQ �����WR�UHSUHVHQW�VXFK�REMHFWV� +RZHYHU� LW�LV�IDU�PRUH�LQWXLWLYHWR�ZRUN�ZLWK�WKH�GDWD�W\SH�³PDWUL[�´ <RX�FDQ�JHQHUDWH�PDWULFHV�E\�XVLQJ�WKHV\VWHP�IXQFWLRQ PDWUL[�

$ � PDWUL[�>>�� �@� >D� �@@�(

1 2a 4

)

0DWUL[�REMHFWV�FRQVWUXFWHG�WKLV�ZD\�KDYH�WKH�FRQYHQLHQW�SURSHUW\�WKDW�WKH�EDVLFDULWKPHWLF�RSHUDWLRQV �� � HWF��DUH�VSHFLDOL]HG��³RYHUORDGHG´��DFFRUGLQJ�WR�WKHDSSURSULDWH�PDWKHPDWLFDO�FRQWH[W� )RU�H[DPSOH� \RX�PD\�XVH � RU WR�DGG�RUPXOWLSO\�PDWULFHV� UHVSHFWLYHO\��LI�WKH�GLPHQVLRQV�PDWFK��

% � PDWUL[�>>\� �@� >]� �@@��$� %� $ � %� $ %(

1 2a 4

),

(y 3z 5

),

(y + 1 5a+ z 9

),

(y + 2 z 13

4 z + a y 3 a+ 20

)

7KH�SRZHU $A����� HTXLYDOHQW�WR ��$��GHQRWHV�WKH�LQYHUVH�RI�WKH�PDWUL[�

$A����(− 2a−2

1a−2

a2 a−4 − 1

2 a−4

)

7KH�IXQFWLRQ OLQDOJ��GHW� IURP�WKH�0X3$' OLQDOJ OLEUDU\�IRU�OLQHDU�DOJHEUD�6HFWLRQ ������ FRPSXWHV�WKH�GHWHUPLQDQW�

OLQDOJ��GHW�$�4− 2 a

&ROXPQ�YHFWRUV RI�GLPHQVLRQ n FDQ�EH�LQWHUSUHWHG�DV n× 1PDWULFHV�

E � PDWUL[�>�� [@�(

1x

)

����

Page 42: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

<RX�FDQ�FRPIRUWDEO\�GHWHUPLQH�WKH�VROXWLRQ A−1⃗b RI�WKH�V\VWHP�RI�OLQHDUHTXDWLRQV Ax⃗ = b⃗� ZLWK�WKH�DERYH�FRHIILFLHQW�PDWUL[ A DQG�WKH�SUHYLRXVO\�GHILQHG b⃗RQ�WKH�ULJKW�KDQG�VLGH�

VROXWLRQ9HFWRU � $A���� E(

xa−2 −

2a−2

a2 a−4 −

x2 a−4

)

1RZ�\RX�FDQ�DSSO\�WKH�IXQFWLRQ QRUPDO WR�HDFK�FRPSRQHQW�RI�WKH�YHFWRU�E\�PHDQVRI�WKH�V\VWHP�IXQFWLRQ PDS� WKXV�VLPSOLI\LQJ�WKH�UHSUHVHQWDWLRQ�

PDS��� QRUPDO�(

x−2a−2a−x2 a−4

)

7R�YHULI\�WKH�FRPSXWDWLRQ� \RX�PD\�PXOWLSO\�WKH�VROXWLRQ�YHFWRU�E\�WKH�PDWUL[ $�

$ �(

x−2a−2 + 2 (a−x)

2 a−44 (a−x)2 a−4 + a (x−2)

a−2

)

$IWHU�VLPSOLILFDWLRQ� \RX�FDQ�FKHFN�WKDW�WKH�UHVXOW�HTXDOV E�

PDS��� QRUPDO�(

1x

)

6HFWLRQ ���� SURYLGHV�PRUH�LQIRUPDWLRQ�RQ�KDQGOLQJ�PDWULFHV�DQG�YHFWRUV�

([HUFLVH����� &RPSXWH�DQ�H[SDQGHG�IRUP�RI�WKH�H[SUHVVLRQ (x2 + y)5�

([HUFLVH����� 8VH�0X3$' WR�FKHFN�WKDWx2 − 1x+ 1

= x− 1 KROGV�

([HUFLVH����� *HQHUDWH�D�SORW�RI�WKH�IXQFWLRQ 1/ VLQ(x) IRU 1 ≤ x ≤ 10 �

����

Page 43: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

([HUFLVH����� 2EWDLQ�GHWDLOHG�LQIRUPDWLRQ�DERXW�WKH�IXQFWLRQ OLPLW� 8VH0X3$' WR�YHULI\�WKH�IROORZLQJ�OLPLWV�

OLPx→0

VLQ(x)x

= 1 , OLPx→0

1− FRV(x)x

= 0 , OLPx→0+

OQ(x) = −∞ ,

OLPx→0xVLQ(x) = 1 , OLP

x→∞

(1 +

1x

)x= e , OLP

x→∞

OQ(x)ex

= 0 ,

OLPx→0+

xOQ(x) =∞ , OLPx→∞

(1 +π

x

)x= eπ , OLP

x→0−

21 + e−1/x

= 0 .

7KH�OLPLW OLPx→0eFRW(x) GRHV�QRW�H[LVW� +RZ�GRHV�0X3$' UHDFW"

([HUFLVH����� 2EWDLQ�GHWDLOHG�LQIRUPDWLRQ�DERXW�WKH�IXQFWLRQ VXP� 7KH�FDOOVXP�I�N��N D��E� FRPSXWHV�D FORVHG�IRUP RI�D�ILQLWH�RU�LQILQLWH�VXP� LI�SRVVLEOH�8VH�0X3$' WR�YHULI\�WKH�IROORZLQJ�LGHQWLW\�

n∑

k=1

(k2 + k + 1) =n (n2 + 3n+ 5)

3.

'HWHUPLQH�WKH�YDOXHV�RI�WKH�IROORZLQJ�VHULHV�

∞∑

k=0

2 k − 3(k + 1) (k + 2) (k + 3)

,∞∑

k=2

k

(k − 1)2 (k + 1)2.

([HUFLVH����� &RPSXWH 2 · (A+B)� A ·B� DQG (A−B)−1 IRU�WKH�IROORZLQJPDWULFHV�

A =

⎝1 2 34 5 67 8 0

⎠ , B =

⎝1 1 00 0 10 1 0

⎠ .

&XUYH�6NHWFKLQJ

,Q�WKH�IROORZLQJ�VDPSOH�VHVVLRQ� ZH�XVH�VRPH�RI�WKH�V\VWHP�IXQFWLRQV�IURP�WKHSUHYLRXV�VHFWLRQ�WR�VNHWFK�DQG�GLVFXVV�WKH�FXUYH�JLYHQ�E\�WKH�UDWLRQDO�IXQFWLRQ

f : x #→ (x− 1)2

x− 2+ a

����

Page 44: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

ZLWK�D�SDUDPHWHU a� )LUVW� ZH�GHWHUPLQH�VRPH�FKDUDFWHULVWLFV�RI�WKLV�IXQFWLRQ�GLVFRQWLQXLWLHV� H[WUHPDO�YDOXHV� DQG�EHKDYLRU�IRU�ODUJH x� :H�DVVXPH�WKDW a LV�DUHDO�QXPEHU�

DVVXPH�D LQ 5B��I � [ �! �[ � ��A���[ � �� � D�VLQJXODULWLHV � GLVFRQW�I�[�� [�{2}

7KH�IXQFWLRQ GLVFRQW GHWHUPLQHV�WKH�GLVFRQWLQXLWLHV�RI�WKH�IXQFWLRQ f ZLWKUHVSHFW�WR�WKH�YDULDEOH x� ,W�UHWXUQV�D�VHW�RI�VXFK�SRLQWV� 7KXV� WKH�DERYH f LVGHILQHG�DQG�FRQWLQXRXV�IRU�DOO x ̸= 2� 2EYLRXVO\� x = 2 LV�D�SROH� ,QGHHG� 0X3$'�

ILQGV�WKH�OLPLW ∓∞ ZKHQ�\RX�DSSURDFK�WKLV�SRLQW�IURP�WKH�OHIW�RU�IURP�WKH�ULJKW�UHVSHFWLYHO\�

OLPLW�I�[�� [ �� /HIW�� OLPLW�I�[�� [ �� 5LJKW�−∞, ∞

<RX�ILQG�WKH�URRWV�RI f E\�VROYLQJ�WKH�HTXDWLRQ f(x) = 0�

URRWV � VROYH�I�[� �� [�{

1−√a2 + 4 a

2− a

2,

√a2 + 4 a

2− a

2+ 1

}

'HSHQGLQJ�RQ a� HLWKHU�ERWK�RU�QRQH�RI�WKH�WZR�URRWV�DUH�UHDO� 1RZ� ZH�ZDQW�WRILQG�WKH�ORFDO�H[WUHPD�RI f � 7R�WKLV�HQG� ZH�GHWHUPLQH�WKH�URRWV�RI�WKH�ILUVWGHULYDWLYH f ′�

I�[�

2x− 2x− 2

− (x− 1)2

(x− 2)2

H[WUHPD � VROYH�I�[� �� [�{1, 3}

7KHVH�DUH�WKH�FDQGLGDWHV�IRU�ORFDO�H[WUHPD� +RZHYHU� VRPH�RI�WKHP�PLJKW�EHVDGGOH�SRLQWV� ,I�WKH�VHFRQG�GHULYDWLYH f ′′ GRHV�QRW�YDQLVK�DW�WKHVH�SRLQWV� WKHQERWK�DUH�UHDOO\�H[WUHPD� :H�FKHFN�

����

Page 45: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

I���� I���−2, 2

2XU�UHVXOWV�LPSO\�WKDW f KDV�WKH�IROORZLQJ�SURSHUWLHV� IRU�DQ\�FKRLFH�RI�WKHSDUDPHWHU a� WKHUH�LV�D�ORFDO�PD[LPXP�DW x = 1� D�SROH�DW x = 2� DQG�D�ORFDOPLQLPXP�DW x = 3� 7KH�FRUUHVSRQGLQJ�YDOXHV�RI f DW�WKHVH�SRLQWV�DUH

PD[YDOXH � I���� PLQYDOXH � I���a

a+ 4

f WHQGV�WR ∓∞ IRU x→ ∓∞�

OLPLW�I�[�� [ �LQILQLW\�� OLPLW�I�[�� [ LQILQLW\�−∞, ∞

:H�FDQ�VSHFLI\�WKH�EHKDYLRU�RI f PRUH�SUHFLVHO\�IRU�ODUJH�YDOXHV�RI x� ,WDV\PSWRWLFDOO\�DSSURDFKHV�WKH�OLQHDU�IXQFWLRQ x #→ x+ a�

VHULHV�I�[�� [ LQILQLW\�

x+ a+1x

+2x2

+4x3

+8x4

+O(

1x5

)

+HUH�ZH�KDYH�HPSOR\HG�WKH�IXQFWLRQ VHULHV WR�FRPSXWH�DQ�DV\PSWRWLF�H[SDQVLRQRI f �6HFWLRQ ������ :H�FDQ�HDVLO\�FKHFN�RXU�UHVXOWV�YLVXDOO\�E\�SORWWLQJ�WKH�JUDSKRI f IRU�VHYHUDO�YDOXHV�RI a�

) � I�[� _ D ��� * � I�[� _ D �� + � I�[� _ D ��)� *� +

(x− 1)2

x− 2− 4,

(x− 1)2

x− 2,

(x− 1)2

x− 2+ 4

7KH�RSHUDWRU _ �&KDSWHU ���HYDOXDWHV�DQ�H[SUHVVLRQ�DW�VRPH�SRLQW� LQ�WKH�H[DPSOH�ZH�KDYH�VXEVWLWXWHG�WKH�FRQFUHWH�YDOXHV −4� 0 DQG 4� UHVSHFWLYHO\� IRU a� :H�QRZFDQ�SORW�WKH�WKUHH�IXQFWLRQV�WRJHWKHU�LQ�RQH�SLFWXUH�

����

Page 46: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

SORW�)� *� +� [ ������

(OHPHQWDU\�1XPEHU�7KHRU\

0X3$'� SURYLGHV�D�ORW�RI�HOHPHQWDU\�QXPEHU�WKHRUHWLF�IXQFWLRQV� IRU�H[DPSOH�

� LVSULPH�Q� WHVWV�ZKHWKHU n ∈ N LV�D�SULPH�QXPEHU�

� LWKSULPH�Q� UHWXUQV�WKH n�WK�SULPH�QXPEHU�

� QH[WSULPH�Q� ILQGV�WKH�OHDVW�SULPH�QXPEHU ≥ n�

� SUHYSULPH�Q� ILQGV�WKH�ODUJHVW�SULPH�QXPEHU ≤ n�

� LIDFWRU�Q� FRPSXWHV�WKH�SULPH�IDFWRUL]DWLRQ�RI n�

7KHVH�URXWLQHV�DUH�TXLWH�IDVW� +RZHYHU� VLQFH�WKH\�HPSOR\�SUREDELOLVWLF�SULPDOLW\WHVWV� WKH\�PD\�UHWXUQ�ZURQJ�UHVXOWV�ZLWK�YHU\�VPDOO�SUREDELOLW\�� ,QVWHDG�RILVSULPH� \RX�FDQ�XVH�WKH��VORZHU��IXQFWLRQ QXPOLE��SURYHSULPH DV�DQ�HUURU�IUHHSULPDOLW\�WHVW�

�,Q�SUDFWLFH� \RX�QHHG�QRW�ZRUU\�DERXW�WKLV�EHFDXVH�WKH�FKDQFHV�RI�D�ZURQJ�DQVZHU�DUH�QHJOLJLEOH�WKH�SUREDELOLW\�RI�D�KDUGZDUH�IDLOXUH�LV�PXFK�KLJKHU�WKDQ�WKH�SUREDELOLW\�WKDW�WKH�UDQGRPL]HG�WHVW�UHWXUQVWKH�ZURQJ�DQVZHU�RQ�FRUUHFWO\�ZRUNLQJ�KDUGZDUH�

����

Page 47: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

/HW�XV�JHQHUDWH�D�OLVW�RI�DOO�SULPHV�XS�WR 10 000� +HUH�LV�RQH�RI�PDQ\�ZD\V�WR�GRWKLV�

SULPHV � VHOHFW�>� ��������@� LVSULPH�>�� �� �� �� ��� ��� ��� ���� ����� ����� ����@

)LUVW� ZH�KDYH�JHQHUDWHG�WKH�VHTXHQFH�RI�DOO�SRVLWLYH�LQWHJHUV�XS�WR 10 000 E\PHDQV�RI�WKH�VHTXHQFH�JHQHUDWRU � �6HFWLRQ ����� 7KH�VTXDUH�EUDFNHWV >@ FRQYHUWWKLV�WR�D�0X3$' OLVW� 7KHQ VHOHFW �6HFWLRQ �����HOLPLQDWHV�DOO�WKRVH�OLVW�HOHPHQWVIRU�ZKLFK�WKH�IXQFWLRQ LVSULPH� VXSSOLHG�DV�VHFRQG�DUJXPHQW� UHWXUQV )$/6(��7KHQXPEHU�RI�WKHVH�SULPHV�HTXDOV�WKH�QXPEHU�RI�OLVW�HOHPHQWV� ZKLFK�ZH�FDQ�REWDLQYLD QRSV �³QXPEHU�RI�RSHUDQGV�´ 6HFWLRQ �����

QRSV�SULPHV�1229

$OWHUQDWLYHO\� ZH�PD\�JHQHUDWH�WKH�VDPH�SULPH�OLVW�E\

SULPHV � >LWKSULPH�L� � L �������@�

+HUH�ZH�KDYH�XVHG�WKH�IDFW�WKDW�ZH�DOUHDG\�NQRZ�WKH�QXPEHU�RI�SULPHV�XS�WR10 000� $QRWKHU�SRVVLELOLW\�LV�WR�JHQHUDWH�D�ODUJH�OLVW�RI�SULPHV�DQG�GLVFDUG�WKHRQHV�JUHDWHU�WKDQ 10 000�

SULPHV � VHOHFW�>LWKSULPH�L� � L �������@�[ �! �[� ��������

+HUH� WKH�REMHFW [ �!�[� ������ UHSUHVHQWV�WKH�IXQFWLRQ�WKDW�PDSV�HDFK [ WR�WKHLQHTXDOLW\ [� ������ 7KH VHOHFW FRPPDQG�WKHQ�NHHSV�RQO\�WKRVH�OLVW�HOHPHQWVIRU�ZKLFK�WKH�LQHTXDOLW\�HYDOXDWHV�WR 758(�

,Q�WKH�QH[W�H[DPSOH� ZH�XVH�D UHSHDW ORRS��&KDSWHU ����WR�JHQHUDWH�WKH�OLVW�RISULPHV� :LWK�WKH�KHOS�RI�WKH�FRQFDWHQDWLRQ�RSHUDWRU � �6HFWLRQ ����� ZHVXFFHVVLYHO\�DSSHQG�SULPHV i WR�D�OLVW�XQWLO QH[WSULPH�L���� WKH�QH[W�SULPH�JUHDWHUWKDQ i� H[FHHGV 10 000� :H�VWDUW�ZLWK�WKH�HPSW\�OLVW�DQG�WKH�ILUVW�SULPH i = 2�

SULPHV � >@� L � ��UHSHDW

SULPHV � SULPHV � >L@�L � QH[WSULPH�L � ��

XQWLO L ! ����� HQGBUHSHDW�

����

Page 48: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

1RZ� ZH�FRQVLGHU�*ROGEDFK¶V�IDPRXV�FRQMHFWXUH�

³(YHU\�HYHQ�LQWHJHU�JUHDWHU�WKDQ 2 LV�WKH�VXP�RI�WZR�SULPHV�´

:H�ZDQW�WR�YHULI\�WKLV�FRQMHFWXUH�IRU�DOO�HYHQ�QXPEHUV�XS�WR 10 000� )LUVW� ZHJHQHUDWH�WKH�OLVW�RI�HYHQ�LQWHJHUV >����«������@�

OLVW � >� L � L �������@�QRSV�OLVW�

4999

1RZ� ZH�VHOHFW�WKRVH�QXPEHUV�IURP�WKH�OLVW�WKDW�FDQQRW�EH�ZULWWHQ�LQ�WKH�IRUP³SULPH�����´ 7KLV�LV�GRQH�E\�WHVWLQJ�IRU�HDFK i LQ�WKH�OLVW�ZKHWKHU i− 2 LV�D�SULPH�

OLVW � VHOHFW�OLVW� L �! �QRW LVSULPH�L � �����QRSV�OLVW�

4998

7KH�RQO\�LQWHJHU�WKDW�KDV�EHHQ�HOLPLQDWHG�LV 4 �VLQFH�IRU�DOO�RWKHU�HYHQ�SRVLWLYHLQWHJHUV i− 2 LV�HYHQ�DQG�JUHDWHU�WKDQ 2� DQG�KHQFH�QRW�SULPH�� 1RZ�ZH�GLVFDUG�DOOQXPEHUV�RI�WKH�IRUP�³SULPH����´�

OLVW � VHOHFW�OLVW� L �! �QRW LVSULPH�L � �����QRSV�OLVW�

3770

7KH�UHPDLQLQJ 3770 LQWHJHUV�DUH�QHLWKHU�RI�WKH�IRUP�³SULPH����´�QRU�RI�WKH�IRUP³SULPH�����´ :H�QRZ�FRQWLQXH�WKLV�SURFHGXUH�E\�PHDQV�RI�D ZKLOH ORRS�&KDSWHU ���� ,Q�WKH�ORRS� j VXFFHVVLYHO\�UXQV�WKURXJK�DOO�SULPHV > 3� DQG�WKHQXPEHUV�RI�WKH�IRUP�³SULPH�� j´�DUH�HOLPLQDWHG� $ SULQW FRPPDQG��6HFWLRQ �����RXWSXWV�WKH�QXPEHU�RI�UHPDLQLQJ�LQWHJHUV�LQ�HDFK�VWHS� 7KH�ORRS�HQGV�DV�VRRQ�DVWKH�OLVW�LV�HPSW\�

M � ��

����

Page 49: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

ZKLOH OLVW �! >@ GRM � QH[WSULPH�M � ���OLVW � VHOHFW�OLVW� L �! �QRW LVSULPH�L � M����SULQW�M� QRSV�OLVW���

HQGBZKLOH��� ����

�� ����

��� ����

���

���� �

���� �

���� �

7KXV�ZH�KDYH�FRQILUPHG�WKDW�*ROGEDFK¶V�FRQMHFWXUH�KROGV�WUXH�IRU�DOO�HYHQSRVLWLYH�LQWHJHUV�XS�WR 10 000� :H�KDYH�HYHQ�VKRZQ�WKDW�DOO�WKRVH�QXPEHUV�FDQ�EHZULWWHQ�DV�D�VXP�RI�D�SULPH�OHVV�RU�HTXDO�WR 173 DQG�DQRWKHU�SULPH�

,Q�WKH�QH[W�H[DPSOH� ZH�JHQHUDWH�D�OLVW�RI�GLVWDQFHV�EHWZHHQ�WZR�VXFFHVVLYH�SULPHVXS�WR 500�

SULPHV � VHOHFW�>� ������@� LVSULPH��GLVWDQFHV � >SULPHV>L@ � SULPHV>L � �@

� L ���QRSV�SULPHV�@>�� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ��

�� �� �� �� �� �� �� �� �� �� �� �� ��� �� �� ��

��� �� �� �� �� �� �� �� ��� �� �� �� ��� ��� ��

�� �� �� �� ��� �� �� �� �� �� �� �� ��� ��� �� ��

�� ��� �� ��� �� �� �� �� �� �� �� �� �� �� �� ���

�� ��� �� �� �� �� �� �� �� �� ��� �� �� �@

7KH�LQGH[HG�FDOO SULPHV>L@ UHWXUQV�WKH iWK�HOHPHQW�LQ�WKH�OLVW�

����

Page 50: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

7KH�IXQFWLRQ ]LS �6HFWLRQ �����SURYLGHV�DQ�DOWHUQDWLYH�PHWKRG� 7KH�FDOO]LS�D�E�I� FRPELQHV�WZR�OLVWV a = [a1, a2, . . . ] DQG b = [b1, b2, . . . ]FRPSRQHQWZLVH�E\�PHDQV�RI�WKH�IXQFWLRQ f � WKH�UHVXOWLQJ�OLVW�LV

[f(a1, b1), f(a2, b2), . . . ]

DQG�KDV�DV�PDQ\�HOHPHQWV�DV�WKH�VKRUWHU�RI�WKH�WZR�OLVWV� ,Q�RXU�H[DPSOH� ZH�DSSO\WKLV�WR�WKH�SULPH�OLVW a = [a1, . . . , an]� WKH�³VKLIWHG´�SULPH�OLVW b = [a2, . . . , an]� DQGWKH�IXQFWLRQ (x, y) #→ y − x� :H�ILUVW�JHQHUDWH�D�VKLIWHG�FRS\�RI�WKH�SULPH�OLVW�E\GHOHWLQJ�WKH�ILUVW�HOHPHQW� WKXV�VKRUWHQLQJ�WKH�OLVW�

E � SULPHV� GHOHWH E>�@�

7KH�IROORZLQJ�FRPPDQG�UHWXUQV�WKH�VDPH�UHVXOW�DV�DERYH�

GLVWDQFHV � ]LS�SULPHV� E� �[� \� �! �\ � [���

:H�KDYH�SUHVHQWHG�DQRWKHU�XVHIXO�IXQFWLRQ�LQ�6HFWLRQ ���� WKH�URXWLQH LIDFWRU IRUIDFWRULQJ�DQ�LQWHJHU�LQWR�SULPHV� 7KH�FDOO LIDFWRU�Q� UHWXUQV�DQ�REMHFW�RI�WKHVDPH�W\SH�DV IDFWRU� LW�LV�D�VSHFLDO�GDWD�W\SH�FDOOHG )DFWRUHG� 2EMHFWV�RI�WKLV�W\SHDUH�SULQWHG�RQ�WKH�VFUHHQ�LQ�D�IRUP�WKDW�LV�HDVLO\�UHDGDEOH�

LIDFWRU������������−32 · 3607 · 3803

,QWHUQDOO\� WKH�SULPH�IDFWRUV�DQG�WKH�H[SRQHQWV�DUH�VWRUHG�LQ�IRUP�RI�D�OLVW� DQG\RX�FDQ�H[WUDFW�WKHP�E\�XVLQJ RS RU�E\�DQ�LQGH[HG�DFFHVV� &RQVXOW�WKH�KHOS�SDJHVRI LIDFWRU DQG )DFWRUHG IRU�GHWDLOV� 7KH�LQWHUQDO�OLVW�KDV�WKH�IRUPDW

[s, p1, e1, . . . , pk, ek]

ZLWK�SULPHV p1, . . . , pk� WKHLU�H[SRQHQWV e1, . . . , ek� DQG�WKH�VLJQ s = ±1� VXFK�WKDWn = s · pe11 · p

e22 · · · p

ekk �

RS���−1, 3, 2, 3607, 1, 3803, 1

����

Page 51: Mupad Tutorial

Lrf[heb\�<hfinmZmbhg

:H�QRZ�HPSOR\�WKH�IXQFWLRQ LIDFWRU WR�ILQG�RXW�KRZ�PDQ\�LQWHJHUV�EHWZHHQ 2DQG 10 000 DUH�GLYLVLEOH�E\�H[DFWO\�WZR�GLVWLQFW�SULPH�QXPEHUV� :H�QRWH�WKDW�WKHREMHFW�UHWXUQHG�E\ LIDFWRU�Q� KDV 2m+ 1 HOHPHQWV�LQ�LWV�OLVW�UHSUHVHQWDWLRQ�ZKHUHm LV�WKH�QXPEHU�RI�GLVWLQFW�SULPH�GLYLVRUV�RI n� 7KXV� WKH�IXQFWLRQ

P � Q �! �QRSV�LIDFWRU�Q�� � �����

UHWXUQV�WKH�QXPEHU�RI�GLVWLQFW�SULPH�IDFWRUV� :H�FRQVWUXFW�WKH�OLVW�RI�YDOXHV m(k)IRU k = 2, . . . , 10000�

OLVW � >P�N� � N ��������@�

7KH�IROORZLQJ IRU ORRS��6HFWLRQ ����GLVSOD\V�WKH�QXPEHU�RI�LQWHJHUV�ZLWK�SUHFLVHO\i = 1, 2, . . . , 6 GLVWLQFW�SULPH�GLYLVRUV�

IRU L IURP � WR � GRSULQW�L� QRSV�VHOHFW�OLVW� [ �! �[ L����

HQGBIRU��� ����

�� ����

�� ����

�� ���

�� ��

�� �

7KXV�WKHUH�DUH 1280 LQWHJHUV�ZLWK�H[DFWO\�RQH�SULPH�GLYLVRU�LQ�WKH�VFDQQHGLQWHUYDO�� 4097 LQWHJHUV�ZLWK�SUHFLVHO\�WZR�GLVWLQFW�SULPH�IDFWRUV� DQG�VR�RQ� ,W�LVHDV\�WR�VHH�ZK\�WKH�LQWHUYDO�FRQWDLQV�QR�LQWHJHU�ZLWK�VL[�RU�PRUH�SULPH�GLYLVRUV�WKH�VPDOOHVW�VXFK�QXPEHU 2 · 3 · 5 · 7 · 11 · 13 = 30 030 H[FHHGV 10 000�

7KH QXPOLE OLEUDU\�FRPSULVHV�YDULRXV�QXPEHU�WKHRUHWLF�IXQFWLRQV� $PRQJ�RWKHUV�LW�FRQWDLQV�WKH�URXWLQH QXPOLE��QXPSULPHGLYLVRUV� HTXLYDOHQW�WR�WKH�DERYH P� IRUFRPSXWLQJ�WKH�QXPEHU�RI�SULPH�GLYLVRUV� :H�UHIHU�WR�&KDSWHU � IRU�D�GHVFULSWLRQRI�WKH�0X3$' OLEUDULHV�

�:H�KDYH�DOUHDG\�VHHQ�WKDW�WKH�LQWHUYDO�FRQWDLQV 1229 SULPH�QXPEHUV� &DQ�\RX�H[SODLQ�WKH�GLI�IHUHQFH"

����

Page 52: Mupad Tutorial

+ ?bklm�Lm^il�bg�FnI:=�

([HUFLVH����� 3ULPHV�RI�WKH�IRUP 2n ± 1 DOZD\V�KDYH�SURGXFHG�SDUWLFXODULQWHUHVW�

D� 3ULPHV�RI�WKH�IRUP 2p − 1� ZKHUH p LV�D�SULPH� DUH�FDOOHG0HUVHQQH�SULPHV�)LQG�DOO�0HUVHQQH�SULPHV�IRU 1 < p ≤ 1000�

E� )RU�D�SRVLWLYH�LQWHJHU n� WKH n�WK�)HUPDW�QXPEHU LV 2(2n) + 1� 5HIXWH)HUPDW¶V�FRQMHFWXUH�WKDW�DOO�WKRVH�QXPEHUV�DUH�SULPHV�

����

Page 53: Mupad Tutorial

+

7KH�0X3$'� /LEUDULHV

0RVW�RI�0X3$'�¶V�PDWKHPDWLFDO�NQRZOHGJH�LV�RUJDQL]HG�LQ�OLEUDULHV� 6XFK�DOLEUDU\�FRPSULVHV�D�FROOHFWLRQ�RI�IXQFWLRQV�IRU�VROYLQJ�SUREOHPV�LQ�D�SDUWLFXODUDUHD� VXFK�DV�OLQHDU�DOJHEUD� QXPEHU�WKHRU\� QXPHULFDO�DQDO\VLV� HWF� /LEUDU\IXQFWLRQV�DUH�ZULWWHQ�LQ�WKH�0X3$' SURJUDPPLQJ�ODQJXDJH� <RX�FDQ�XVH�WKHP�LQWKH�VDPH�ZD\�DV�NHUQHO�IXQFWLRQV� ZLWKRXW�NQRZLQJ�DQ\WKLQJ�DERXW�WKHSURJUDPPLQJ�ODQJXDJH�

6HFWLRQ ³/LEUDULHV�DQG�0RGXOHV´�RI�WKH�0X3$' 4XLFN�5HIHUHQFH�FRQWDLQV�D�VXUYH\RI�DOO�OLEUDULHV� $OWHUQDWLYHO\� WKH�KHOS�YLHZHU�FRQWDLQV�D�OLVW�RI�OLEUDULHV��DQGWKLUG�SDUW\�SDFNDJHV�LQVWDOOHG��LQ�LWV�FRQWHQWV�SDQH�RQ�WKH�OHIW�

7KH�OLEUDULHV�DUH�GHYHORSHG�FRQWLQXRXVO\� DQG�IXWXUH�YHUVLRQV�RI�0X3$' ZLOOSURYLGH�DGGLWLRQDO�IXQFWLRQDOLW\�

,Q�WKLV�FKDSWHU� ZH�GR�QRW�DGGUHVV�WKH�PDWKHPDWLFDO�IXQFWLRQDOLW\�RI�OLEUDULHV�EXWUDWKHU�GLVFXVV�WKHLU�JHQHUDO�XVH�

Page 54: Mupad Tutorial

, Ma^�FnI:=� Eb[kZkb^l

Af^gjeYlagf�9Zgml�Y�HYjla[mdYj�DaZjYjq

<RX�FDQ�REWDLQ�LQIRUPDWLRQ�DQG�KHOS�IRU�OLEUDULHV�E\�FDOOLQJ�WKH�IXQFWLRQV LQIRDQG KHOS� 7KH�IXQFWLRQ LQIR JLYHV�D�OLVW�RI�DOO�IXQFWLRQV�LQVWDOOHG�LQ�WKH�OLEUDU\�7KH QXPOLE OLEUDU\�LV�D�FROOHFWLRQ�RI�QXPEHU�WKHRUHWLF�IXQFWLRQV�

LQIR�QXPOLE�/LEUDU\ QXPOLE� WKH SDFNDJH IRU HOHPHQWDU\QXPEHU WKHRU\

�� ,QWHUIDFH�QXPOLE��/DPEGD� QXPOLE��2PHJD�QXPOLE��FRQWIUDF� QXPOLE��FRUQDFFKLD�QXPOLE��GHFLPDO� QXPOLE��GLYLVRUV�QXPOLE��HFP� QXPOLE��IDFWRU*DXVV,QW�QXPOLE��ILERQDFFL� QXPOLE��IURP$VFLL����

7KH�FRPPDQGV KHOS RU " VXSSO\�D�PRUH�GHWDLOHG�GHVFULSWLRQ�RI�WKH�OLEUDU\�

7KH�IXQFWLRQ QXPOLE��GHFLPDO RI�WKH QXPOLE OLEUDU\�FRPSXWHV�WKH�GHFLPDOH[SDQVLRQ�RI�D�UDWLRQDO�QXPEHU��

QXPOLE��GHFLPDO�������17, [5, 7, 1, 4, 2, 8]

$V�IRU�RWKHU�V\VWHP�IXQFWLRQV� \RX�ZLOO�VHH�LQIRUPDWLRQ�ZKHQ�KRYHULQJ�\RXU�PRXVHSRLQWHU�RYHU�WKH�QDPH�RI�D�OLEUDU\�IXQFWLRQ�DQG�\RX�PD\�UHTXHVW�LQIRUPDWLRQ�RQWKH�IXQFWLRQ�E\�PHDQV�RI KHOS RU "�

"QXPOLE��GHFLPDO

�7KH�UHVXOW�LV�WR�EH�LQWHUSUHWHG�DV� 123/7 = 17.571428 = 17.571428 571428 . . .

���

Page 55: Mupad Tutorial

Bg_hkfZmbhg�:[hnm�Z�IZkmb\neZk�Eb[kZkr

<RX�FDQ�KDYH�D�ORRN�DW�WKH�LPSOHPHQWDWLRQ�RI�D�OLEUDU\�IXQFWLRQ E\�XVLQJ H[SRVH�

H[SRVH�QXPOLE��GHFLPDO�SURF�D�

QDPH QXPOLE��GHFLPDO�ORFDO S� T� V� O� L�

EHJLQLI QRW WHVWW\SH�D� 7\SH��1XPHULF� WKHQ

���HQGBSURF

���

Page 56: Mupad Tutorial

, Ma^�FnI:=� Eb[kZkb^l

=phgjlaf_�DaZjYja]k

,Q�WKH�SUHYLRXV�VHFWLRQ� \RX�KDYH�VHHQ�WKDW�WKH�FDOOLQJ�V\QWD[�IRU�D�OLEUDU\�IXQFWLRQLV OLEUDU\��IXQFWLRQ� ZKHUH OLEUDU\ DQG IXQFWLRQ DUH�WKH�QDPHV�RI�WKH�OLEUDU\DQG�WKH�IXQFWLRQ� UHVSHFWLYHO\� )RU�H[DPSOH� WKH�OLEUDU\ QXPHULF IRU�QXPHULFDOFRPSXWDWLRQV�FRQWDLQV�WKH�IXQFWLRQ QXPHULF��IVROYH� ,W�LPSOHPHQWV�D�PRGLILHGYHUVLRQ�RI�WKH�ZHOO�NQRZQ�1HZWRQ�PHWKRG�IRU�QXPHULFDO�URRW�ILQGLQJ� ,Q�WKHIROORZLQJ�H[DPSOH� ZH�DSSUR[LPDWH�D�URRW�RI�WKH�VLQH�IXQFWLRQ�LQ�WKH�LQWHUYDO [2, 4]�

QXPHULF��IVROYH�VLQ�[�� [ �����[x = 3.141592654]

7KH�IXQFWLRQ XVH� PDNHV�IXQFWLRQV�RI�D�OLEUDU\�³JOREDOO\�NQRZQ�´ VR�WKDW�\RX�FDQXVH�WKHP�ZLWKRXW�VSHFLI\LQJ�WKH�OLEUDU\�QDPH�

XVH�QXPHULF� IVROYH�� IVROYH�VLQ�[�� [ �����[x = 3.141592654]

,I�\RX�DOUHDG\�KDYH�DVVLJQHG�D�YDOXH�WR�WKH�QDPH�RI�WKH�IXQFWLRQ�WR�EH�H[SRUWHG�XVH UHWXUQV�D�ZDUQLQJ�DQG�WKH�IXQFWLRQ�LV�QRW�H[SRUWHG� ,Q�WKH�IROORZLQJ� WKHQXPHULFDO�LQWHJUDWRU QXPHULF��TXDGUDWXUH FDQQRW�EH�H[SRUWHG�WR�WKH�QDPHTXDGUDWXUH� EHFDXVH�WKLV�LGHQWLILHU�KDV�D�YDOXH�TXDGUDWXUH � �� XVH�QXPHULF� TXDGUDWXUH�

:DUQLQJ� TXDGUDWXUH DOUHDG\ KDV D YDOXH� QRW H[SRUWHG�

$IWHU�GHOHWLRQ�RI�WKH�YDOXH� WKH�QDPH�RI�WKH�IXQFWLRQ�LV�DYDLODEOH�DQG�WKHFRUUHVSRQGLQJ�OLEUDU\�IXQFWLRQ�FDQ�EH�H[SRUWHG�VXFFHVVIXOO\� 2QH�FDQ�H[SRUWVHYHUDO�IXQFWLRQV�DW�RQFH�

GHOHWH TXDGUDWXUH�XVH�QXPHULF� UHDOURRWV� TXDGUDWXUH��

1RZ�\RX�FDQ�XVH UHDOURRWV �WR�ILQG DOO UHDO�URRWV�RI�DQ�H[SUHVVLRQ�LQ�DQ�LQWHUYDO�DQG TXDGUDWXUH �IRU�QXPHULFDO�LQWHJUDWLRQ��GLUHFWO\� 3OHDVH�UHIHU�WR�WKHFRUUHVSRQGLQJ�KHOS�SDJHV�IRU�WKH�PHDQLQJ�RI�WKH�LQSXW�SDUDPHWHUV�DQG�WKHUHWXUQHG�RXWSXW� 3OHDVH�QRWH�WKDW�QHLWKHU�WRROWLSV�QRU�WKH KHOS FRPPDQG�ZLOO

�7KLV�IXQFWLRQ�XVHG�WR�EH�FDOOHG H[SRUW� EXW�WKDW�QDPH�LV�PRUH�DSSURSULDWHO\�XVHG�IRU�WKH�OLEUDU\H[SRUWLQJ�GDWD�LQ�IRUPDWV�IRU�RWKHU�SURJUDPV�

���

Page 57: Mupad Tutorial

>qihkmbg`�Eb[kZkb^l

UHDFW�WR�H[SRUWHG�QDPHV� <RX�ZLOO�QHHG�WR�XVH�WKH�IXOO�QDPH��RU�WKH�VHDUFKIXQFWLRQDOLW\�RI�WKH�KHOS�EURZVHU��LQ�WKHVH�FDVHV�

UHDOURRWV�[A� � [A� � � [A� � �� [ � ��[ �������� ������

[[−3.623046875,−3.62109375] , [0.8217773438, 0.822265625]]

TXDGUDWXUH�H[S�[� � �� [ �����2.718281828

,I�\RX�FDOO XVH ZLWK�RQO\�RQH�DUJXPHQW� QDPHO\�WKH�QDPH�RI�WKH�OLEUDU\� WKHQ�DOOIXQFWLRQV�LQ�WKDW�OLEUDU\�DUH�H[SRUWHG� ,I�WKHUH�DUH�QDPH�FRQIOLFWV�ZLWK�DOUHDG\H[LVWLQJ�LGHQWLILHUV� WKHQ XVH LVVXHV�ZDUQLQJV�

HLJHQYDOXHV � �� XVH�QXPHULF�,QIR� QXPHULF��TXDGUDWXUH DOUHDG\ LV H[SRUWHG�,QIR� QXPHULF��UHDOURRWV DOUHDG\ LV H[SRUWHG�:DUQLQJ� LQGHWV DOUHDG\ KDV D YDOXH� QRW H[SRUWHG�,QIR� QXPHULF��IVROYH DOUHDG\ LV H[SRUWHG�:DUQLQJ� UDWLRQDOL]H DOUHDG\ KDV D YDOXH� QRW

H[SRUWHG�:DUQLQJ� OLQVROYH DOUHDG\ KDV D YDOXH� QRW H[SRUWHG�:DUQLQJ� VXP DOUHDG\ KDV D YDOXH� QRW H[SRUWHG�:DUQLQJ� LQW DOUHDG\ KDV D YDOXH� QRW H[SRUWHG�:DUQLQJ� VROYH DOUHDG\ KDV D YDOXH� QRW H[SRUWHG�:DUQLQJ� VRUW DOUHDG\ KDV D YDOXH� QRW H[SRUWHG�:DUQLQJ� HLJHQYDOXHV DOUHDG\ KDV D YDOXH� QRW

H[SRUWHG�

$IWHU�GHOHWLQJ�WKH�YDOXH�RI�WKH�LGHQWLILHU HLJHQYDOXHV� WKH�OLEUDU\�IXQFWLRQ�ZLWK�WKHVDPH�QDPH�FDQ�EH�H[SRUWHG�VXFFHVVIXOO\�

GHOHWH HLJHQYDOXHV� XVH�QXPHULF� HLJHQYDOXHV��

+RZHYHU� WKH�RWKHU�QDPH�FRQIOLFWV LQW� VROYH HWF� FDQQRW�EH�UHVROYHG� 7KHLPSRUWDQW�V\PEROLF�V\VWHP�IXQFWLRQV LQW� VROYH HWF� VKRXOG�QRW�EH�UHSODFHG�E\WKHLU�QXPHULFDO�FRXQWHUSDUWV QXPHULF��LQW� QXPHULF��VROYH HWF�

���

Page 58: Mupad Tutorial

, Ma^�FnI:=� Eb[kZkb^l

L`]�KlYf\Yj\�DaZjYjq

7KH�VWDQGDUG�OLEUDU\�LV�WKH�PRVW�LPSRUWDQW�0X3$'� OLEUDU\� ,W�FRQWDLQV�WKH�PRVWIUHTXHQWO\�XVHG�IXQFWLRQV�VXFK�DV GLII� VLPSOLI\� HWF� �)RU�DQ�HYHQ�PRUHUHVWULFWHG�VHOHFWLRQ�RI�IUHTXHQWO\�XVHG�FRPPDQGV� FKHFN�WKH�FRPPDQG�EDU�DW�WKHULJKW�KDQG�VLGH�RI�WKH�QRWHERRN�LQWHUIDFH�DQG�WKH�0X3$' VRXUFH�FRGH�HGLWRU��7KH�IXQFWLRQV�RI�WKH�VWDQGDUG�OLEUDU\�GR�QRW�KDYH�D�SUHIL[�VHSDUDWHG�ZLWK ��� DVRWKHU�IXQFWLRQV�GR��XQOHVV�H[SRUWHG�� ,Q�WKLV�UHVSHFW� WKHUH�LV�QR�QRWDEOHGLIIHUHQFH�EHWZHHQ�WKH�0X3$' NHUQHO�IXQFWLRQV� ZKLFK�DUH�ZULWWHQ�LQ�WKH�&��SURJUDPPLQJ�ODQJXDJH� DQG�WKH�RWKHU�IXQFWLRQV�IURP�WKH�VWDQGDUG�OLEUDU\� ZKLFKDUH�LPSOHPHQWHG�LQ�WKH�0X3$' SURJUDPPLQJ�ODQJXDJH�

<RX�REWDLQ�PRUH�LQIRUPDWLRQ�DERXW�WKH�DYDLODEOH�IXQFWLRQV�RI�WKH�VWDQGDUG�OLEUDU\E\�QDYLJDWLQJ�WR�WKH�UHOHYDQW�VHFWLRQ�LQ�WKH�KHOS�V\VWHP� ZKLFK�LV�WKH�ILUVWUHIHUHQFH�VHFWLRQ� 7KH�0X3$' 4XLFN�5HIHUHQFH�OLVWV�DOO�IXQFWLRQV�RI�WKH�VWDQGDUGOLEUDU\�LQ�0X3$' YHUVLRQ ��

0DQ\�RI�WKHVH�IXQFWLRQV�DUH�LPSOHPHQWHG�DV�IXQFWLRQ�HQYLURQPHQWV�6HFWLRQ ������� <RX�FDQ�YLHZ�WKH�VRXUFH�FRGH�YLD H[SRVH�QDPH��H[SRVH�H[S�

SURF�[�QDPH H[S�ORFDO \� OQ\� F�RSWLRQ QR'HEXJ�

EHJLQLI DUJV��� � WKHQ

HUURU��H[SHFWLQJ RQH DUJXPHQW��HOVH

LI [��GRP��H[S �! )$,/ WKHQUHWXUQ�[��GRP��H[S�DUJV����

HOVHLI DUJV��� �! � WKHQ

HUURU��H[SHFWLQJ RQH DUJXPHQW��HQGBLI

HQGBLIHQGBLI����

HQGBSURF

���

Page 59: Mupad Tutorial

,

0X3$'� 2EMHFWV

,Q�&KDSWHU �� ZH�LQWURGXFHG�0X3$'� REMHFWV�VXFK�DV�QXPEHUV� V\PEROLFH[SUHVVLRQV� PDSV� RU�PDWULFHV� 1RZ� ZH�GLVFXVV�WKHVH�REMHFWV�PRUHV\VWHPDWLFDOO\�

7KH�REMHFWV�VHQW�WR�WKH�NHUQHO�IRU�HYDOXDWLRQ�FDQ�EH�RI�YDULRXV�IRUPV� DULWKPHWLFH[SUHVVLRQV�ZLWK�QXPEHUV�VXFK�DV �����,���� DULWKPHWLF�H[SUHVVLRQV�ZLWKV\PEROLF�REMHFWV�VXFK�DV [��\�,���� OLVWV� VHWV� HTXDWLRQV� LQHTXDOLWLHV� PDSV�DUUD\V� DEVWUDFW�PDWKHPDWLFDO�REMHFWV� DQG�PRUH� (YHU\�0X3$' REMHFW�EHORQJV�WRVRPH�GDWD�W\SH� FDOOHG�WKH GRPDLQ�W\SH� ,W�FRUUHVSRQGV�WR�D�FHUWDLQ�LQWHUQDOUHSUHVHQWDWLRQ�RI�WKH�REMHFW� ,Q�ZKDW�IROORZV� ZH�GLVFXVV�WKH�IROORZLQJIXQGDPHQWDO�GRPDLQ�W\SHV� $V�D�FRQYHQWLRQ� WKH�QDPHV�RI�GRPDLQV�SURYLGHG�E\WKH�0X3$'�NHUQHO�FRQVLVW�RI�FDSLWDO�OHWWHUV� VXFK�DV '20B,17� '20B5$7 HWF�� ZKLOHGRPDLQV�LPSOHPHQWHG�LQ�WKH�0X3$' ODQJXDJH�VXFK�DV 6HULHV��3XLVHX[ RU'RP��0DWUL[�5� LQYROYH�VPDOO�OHWWHUV�✞

GRPDLQ�W\SH PHDQLQJ

'20B,17 LQWHJHUV� H�J�� �����A�'20B5$7 UDWLRQDO�QXPEHUV� H�J�� ����'20B)/2$7 IORDWLQJ�SRLQW�QXPEHUV� H�J�� �����'20B&203/(; FRPSOH[�QXPEHUV� H�J�� ����� ,'20B,17(59$/ IORDWLQJ�SRLQW�LQWHUYDOV� H�J�� ���������'20B,'(17 V\PEROLF�LGHQWLILHUV� H�J�� [�\�I'20B(;35 V\PEROLF�H[SUHVVLRQV� H�J�� [�\6HULHV��3XLVHX[ V\PEROLF�VHULHV�H[SDQVLRQV� H�J��

��[�[A��2�[A��'20B/,67 OLVWV� H�J�� >�����@'20B6(7 VHWV� H�J�� ^�����`'20B$55$< DUUD\V

Page 60: Mupad Tutorial

- FnI:=� H[c^\ml

GRPDLQ�W\SH PHDQLQJ

'20B7$%/( WDEOHV'20B%22/ %RROHDQ�YDOXHV� 758(� )$/6(� 81.12:1'20B675,1* VWULQJV� H�J�� �, DP�D�VWULQJ�'RP��0DWUL[�5� PDWULFHV�DQG�YHFWRUV�RYHU�WKH�ULQJ 5'20B32/< SRO\QRPLDOV� H�J�� SRO\�[A��[���>[@�'20B352& IXQFWLRQV�DQG�SURFHGXUHV

0RUHRYHU� \RX�FDQ�GHILQH�\RXU�RZQ�GDWD�W\SHV� EXW�ZH�GR�QRW�GLVFXVV�WKLV�KHUH�7KH�V\VWHP�IXQFWLRQ GRPW\SH UHWXUQV�WKH�GRPDLQ�W\SH�RI�D�0X3$' REMHFW�

,Q�WKH�IROORZLQJ�VHFWLRQ� ZH�ILUVW�SUHVHQW�WKH�LPSRUWDQW�RSHUDQG�IXQFWLRQ RS�ZKLFK�HQDEOHV�\RX�WR�GHFRPSRVH�D�0X3$' REMHFW�LQWR�LWV�EXLOGLQJ�EORFNV� 7KHIROORZLQJ�VHFWLRQV�GLVFXVV�WKH�DERYH�GDWD�W\SHV�DQG�VRPH�RI�WKH�PDLQ�V\VWHPIXQFWLRQV�IRU�KDQGOLQJ�WKHP�

���

Page 61: Mupad Tutorial

Hi^kZg]l3 ma^�?ng\mbhgl RS Zg] QRSV

Gh]jYf\k2 l`]�>mf[lagfk RS Yf\ QRSV

,W�LV�RIWHQ�QHFHVVDU\�WR�GHFRPSRVH�D�0X3$'� REMHFW�LQWR�LWV�FRPSRQHQWV�LQ�RUGHUWR�SURFHVV�WKHP�LQGLYLGXDOO\� 7KH�EXLOGLQJ�EORFNV�RI�DQ�REMHFW�DUH�FDOOHG RSHUDQGV�7KH�V\VWHP�IXQFWLRQV�IRU�DFFHVVLQJ�WKHP�DUH RS DQG QRSV �VKRUW�IRU� QXPEHU�RIRSHUDQGV��✞

QRSV�REMHFW� � WKH�QXPEHU�RI�RSHUDQGV�

RS�REMHFW�L� � WKH i�WK�RSHUDQG� 0 ≤ i ≤ QRSV�REMHFW��RS�REMHFW�L��M� � WKH�VHTXHQFH�RI�RSHUDQGV i WKURXJK j�

ZKHUH 0 ≤ i ≤ j ≤ QRSV�REMHFW��RS�REMHFW� � WKH�VHTXHQFH RS�REMHFW���� RS�REMHFW���� ���

RI�DOO�RSHUDQGV�

7KH�PHDQLQJ�RI�DQ�RSHUDQG�GHSHQGV�RQ�WKH�GDWD�W\SH�RI�WKH�REMHFW� :H�GLVFXVV�WKLVIRU�HDFK�GDWD�W\SH�LQ�GHWDLO�LQ�WKH�IROORZLQJ�VHFWLRQV� )RU�H[DPSOH� WKH�RSHUDQGV�RID�UDWLRQDO�QXPEHU�DUH�WKH�QXPHUDWRU�DQG�WKH�GHQRPLQDWRU� WKH�RSHUDQGV�RI�D�OLVWRU�VHW�DUH�WKH�HOHPHQWV� DQG�WKH�RSHUDQGV�RI�D�IXQFWLRQ�FDOO�DUH�WKH�DUJXPHQWV�+RZHYHU� WKHUH�DUH�DOVR�REMHFWV�ZKHUH�WKH�GHFRPSRVLWLRQ�LQWR�RSHUDQGV�LV�OHVVLQWXLWLYH� VXFK�DV�VHULHV�H[SDQVLRQV�DV�JHQHUDWHG�E\�WKH�V\VWHP�IXQFWLRQV WD\ORURU VHULHV �6HFWLRQ ������ +HUH�LV�DQ�H[DPSOH�ZLWK�D�OLVW��6HFWLRQ �����

OLVW � >D� E� F� G� VLQ�[�@� QRSV�OLVW�5

RS�OLVW� ��b

RS�OLVW� �����c, d, VLQ(x)

RS�OLVW�a, b, c, d, VLQ(x)

%\�UHSHDWHGO\�FDOOLQJ�WKH RS IXQFWLRQ� \RX�FDQ�GHFRPSRVH�DUELWUDU\�0X3$'H[SUHVVLRQV�LQWR�³DWRPLF´�RQHV� ,Q�WKLV�PRGHO� D�0X3$' DWRP LV�DQ�H[SUHVVLRQ

���

Page 62: Mupad Tutorial

- FnI:=� H[c^\ml

WKDW�FDQQRW�EH�IXUWKHU�GHFRPSRVHG�E\ RS� VXFK�WKDW RS�DWRP� DWRP KROGV�� 7KLVLV�HVVHQWLDOO\�WKH�FDVH�IRU�LQWHJHUV� IORDWLQJ�SRLQW�QXPEHUV� LGHQWLILHUV�WKDW�KDYHQRW�EHHQ�DVVLJQHG�D�YDOXH� DQG�IRU�VWULQJV�

RS����� RS��������� RS�D�� RS��, DP D WH[W��−2, 0.1234, a, ´, DP�D�WH[W´

,Q�WKH�IROORZLQJ�H[DPSOH� D�QHVWHG�OLVW�LV�GHFRPSRVHG�FRPSOHWHO\�LQWR�LWV�DWRPVD���D���D���[���OLVW � >>D��� D��@� >D��� [A�@@

7KH�RSHUDQGV�DQG�VXERSHUDQGV�DUH�✞

RS�OLVW� �� � >D��� D��@RS�OLVW� �� � >D��� [A�@RS�RS�OLVW� ��� �� � D��RS�RS�OLVW� ��� �� � D��RS�RS�OLVW� ��� �� � D��RS�RS�OLVW� ��� �� � [A�RS�RS�RS�OLVW� ��� ��� �� � [RS�RS�RS�OLVW� ��� ��� �� � �

�7KLV�PRGHO�LV�D�JRRG�DSSUR[LPDWLRQ�WR�0X3$'¶V�LQWHUQDO�PRGH�RI�RSHUDWLRQ� EXW�WKHUH�DUH�VRPHH[FHSWLRQV� )RU�H[DPSOH� \RX�FDQ�GHFRPSRVH�UDWLRQDO�QXPEHUV�YLD RS� EXW�WKH�NHUQHO�UHJDUGV�WKHP�DVDWRPV� 2Q�WKH�RWKHU�KDQG� DOWKRXJK�VWULQJV�DUH�LQGHFRPSRVDEOH�ZLWK�UHVSHFW�WR RS� LW�LV�VWLOO�SRVVLEOH�WRDFFHVV�WKH�FKDUDFWHUV�RI�D�VWULQJ�LQGLYLGXDOO\��6HFWLRQ ������

���

Page 63: Mupad Tutorial

Hi^kZg]l3 ma^�?ng\mbhgl RS Zg] QRSV

,QVWHDG�RI�WKH�DQQR\LQJ�QHVWHG�FDOOV�RI RS� \RX�PD\�DOVR�XVH�WKH�IROORZLQJ�VKRUWIRUP�WR�DFFHVV�VXEH[SUHVVLRQV�✞

RS�OLVW� >�@� � >D��� D��@RS�OLVW� >�@� � >D��� [A�@RS�OLVW� >�� �@� � D��RS�OLVW� >�� �@� � D��RS�OLVW� >�� �@� � D��RS�OLVW� >�� �@� � [A�RS�OLVW� >�� �� �@� � [RS�OLVW� >�� �� �@� � �

([HUFLVH����� 'HWHUPLQH�WKH�RSHUDQGV�RI�WKH�SRZHU DAE� WKH�HTXDWLRQ D E� DQGWKH�V\PEROLF�IXQFWLRQ�FDOO I�D�E��

([HUFLVH����� 7KH�IROORZLQJ�FDOO�RI VROYH �&KDSWHU ���UHWXUQV�D�VHW�

VHW � VROYH�^[ � VLQ��� \ H[S�D��\ � VLQ��� \ H[S��D�`� ^[�\`�

{[x =

Ha VLQ(3)− Ha + VLQ(3)Ha

VLQ(3)− 1, y = − 1

Ha (VLQ(3)− 1)

]}

([WUDFW�WKH�YDOXH�RI�WKH�VROXWLRQ�IRU y DQG�DVVLJQ�LW�WR�WKH�LGHQWLILHU \�

���

Page 64: Mupad Tutorial

- FnI:=� H[c^\ml

FmeZ]jk

:H�KDYH�GHPRQVWUDWHG�LQ�6HFWLRQ ��� KRZ�WR�ZRUN�ZLWK�QXPEHUV� 7KHUH�DUHYDULRXV�GDWD�W\SHV�IRU�QXPEHUV�

GRPW\SH������ GRPW\SH������ GRPW\SH���������GRPW\SH���� � � ,�

'20B,17, '20B5$7, '20B)/2$7, '20B&203/(;

$ UDWLRQDO�QXPEHU�LV�D�FRPSRXQG�REMHFW� WKH�EXLOGLQJ�EORFNV�DUH�WKH�QXPHUDWRUDQG�WKH�GHQRPLQDWRU� 6LPLODUO\� D�FRPSOH[�QXPEHU�FRQVLVWV�RI�WKH�UHDO�DQG�WKHLPDJLQDU\�SDUW� <RX�FDQ�XVH�WKH�RSHUDQG�IXQFWLRQ RS IURP�WKH�SUHYLRXV�VHFWLRQ�WRDFFHVV�WKHVH�FRPSRQHQWV�

RS��������� ��� RS��������� ��111, 223

RS���� � ��� ,� ��� RS���� � ��� ,� ��100, 200

$OWHUQDWLYHO\� \RX�FDQ�XVH�WKH�V\VWHP�IXQFWLRQV QXPHU� GHQRP� 5H� DQG ,P�

QXPHU���������� GHQRP����������5H���� � ��� ,�� ,P���� � ��� ,�

111, 223, 100, 200

%HVLGHV�WKH�FRPPRQ�DULWKPHWLF�RSHUDWLRQV �� �� � DQG �� WKHUH�DUH�WKH�DULWKPHWLFRSHUDWRUV GLY DQG PRG IRU�GLYLVLRQ�RI�DQ�LQWHJHU x E\�D�QRQ�]HUR�LQWHJHU p ZLWKUHPDLQGHU� ,I x = k p+ r KROGV�ZLWK�LQWHJHUV k DQG 0 ≤ r < |p|� WKHQ [ GLY SUHWXUQV�WKH�³LQWHJUDO�TXRWLHQW´ k DQG [ PRG S UHWXUQV�WKH�³UHPDLQGHU´ r�

�� GLY �� �� PRG �6, 1

7DEOH ��� JLYHV�D�FRPSLODWLRQ�RI�WKH�PDLQ�0X3$' IXQFWLRQV�DQG�RSHUDWRUV�IRUKDQGOLQJ�QXPEHUV� :H�UHIHU�WR�WKH�KHOS�V\VWHP��L�H�� "DEV� "FHLO HWF�� IRU�DGHWDLOHG�GHVFULSWLRQ�RI�WKHVH�IXQFWLRQV� :H�VWUHVV�WKDW�ZKLOH�H[SUHVVLRQV�VXFK�DV√

2PDWKHPDWLFDOO\�UHSUHVHQW�QXPEHUV� 0X3$' WUHDWV�WKHP�DV�V\PEROLFH[SUHVVLRQV��6HFWLRQ �����

���

Page 65: Mupad Tutorial

Gnf[^kl

�� �� � �� A � EDVLF�DULWKPHWLF�RSHUDWLRQVDEV � DEVROXWH�YDOXHFHLO � URXQGLQJ�XSGLY � TXRWLHQW�³PRGXOR´IDFW � IDFWRULDOIORDW � DSSUR[LPDWLRQ�E\�IORDWLQJ�SRLQW�QXPEHUVIORRU � URXQGLQJ�GRZQIUDF � IUDFWLRQDO�SDUWLIDFWRU� IDFWRU � SULPH�IDFWRUL]DWLRQLVSULPH � SULPDOLW\�WHVWPRG � UHPDLQGHU�³PRGXOR´URXQG � URXQGLQJ�WR�QHDUHVWVLJQ � VLJQVTUW � VTXDUH�URRWWUXQF � LQWHJUDO�SDUW

7DEOH ���� 0X3$'� IXQFWLRQV�DQG�RSHUDWRUV�IRU�QXPEHUV

GRPW\SH�VTUW����'20B(;35

([HUFLVH����� :KDW�LV�WKH�GLIIHUHQFH�EHWZHHQ 1/3 + 1/3 + 1/3 DQG1.0/3 + 1/3 + 1/3 LQ�0X3$'"

([HUFLVH����� &RPSXWH�WKH�GHFLPDO�H[SDQVLRQV�RI π(ππ) DQG e

13 π√

163 ZLWK�DSUHFLVLRQ�RI 10 DQG 100 GLJLWV� UHVSHFWLYHO\� :KDW�LV�WKH 234�WK�GLJLW�DIWHU�WKHGHFLPDO�SRLQW�RI π"

([HUFLVH����� $IWHU�\RX�H[HFXWH [� ��A������� RQO\�WKH�ILUVW ',*,76 GHFLPDOGLJLWV�RI [ DUH�JXDUDQWHHG�WR�EH�FRUUHFW�

D� 7UXQFDWLQJ�WKH�IUDFWLRQDO�SDUW�YLD WUXQF LV�WKHUHIRUH�TXHVWLRQDEOH� :KDWGRHV�0X3$' GR"

E� :KDW�LV�UHWXUQHG�IRU [ DIWHU�LQFUHDVLQJ ',*,76"

���

Page 66: Mupad Tutorial

- FnI:=� H[c^\ml

A\]flaÚ]jk

,GHQWLILHUV DUH�QDPHV�VXFK�DV [ RU I WKDW�PD\�UHSUHVHQW�YDULDEOHV�DQG�XQNQRZQV�7KH\�PD\�FRQVLVW�RI�DUELWUDU\�FRPELQDWLRQV�RI�OHWWHUV� GLJLWV� WKH�KDVK�PDUN�³��´DQG�WKH�XQGHUVFRUH�³B�´ ZLWK�WKH�RQO\�H[FHSWLRQ�WKDW�WKH�ILUVW�V\PERO�PXVW�QRW�EH�DGLJLW� ,GHQWLILHUV�VWDUWLQJ�ZLWK�D�KDVK�PDUN�FDQ�QHYHU�KDYH�YDOXHV�RU�SURSHUWLHV�0X3$'� GLVWLQJXLVKHV�XSSHUFDVH�DQG�ORZHUFDVH�OHWWHUV� ([DPSOHV�RI�DGPLVVLEOHLGHQWLILHUV�DUH [� B[��� DQG WKHB0X3$'BV\VWHP� ZKLOH�0X3$' ZRXOG�QRW�DFFHSW��[� S��� DQG [!\ DV�LGHQWLILHUV� 0X3$' DOVR�DFFHSWV�DQ\�VHTXHQFH�RI�FKDUDFWHUVVWDUWLQJ�DQG�HQGLQJ�ZLWK�D�µEDFNWLFN¶ C DV�DQ�LGHQWLILHU� VR C[!\C LV� LQ�IDFW� DQLGHQWLILHU� :H�ZLOO�QRW�XVH�VXFK�LGHQWLILHUV�LQ�WKLV�WXWRULDO�

,GHQWLILHUV�WKDW�KDYH�QRW�EHHQ�DVVLJQHG�D�YDOXH�HYDOXDWH�WR�WKHLU�QDPH� ,Q�0X3$'�WKH\�UHSUHVHQW�V\PEROLF�REMHFWV�VXFK�DV�XQNQRZQV�LQ�HTXDWLRQV� 7KHLU�GRPDLQW\SH�LV '20B,'(17�

GRPW\SH�[�'20B,'(17

,GHQWLILHUV�FDQ�FRQWDLQ�VSHFLDO�FKDUDFWHUV� VXFK�DV�DQ α� :H�UHFRPPHQG�XVLQJ�WKH6\PERO OLEUDU\�IRU�JHQHUDWLQJ�WKHVH� 8VLQJ B� A� �� DQG ^`� \RX�FDQ�DOVR�FUHDWHLGHQWLILHUV�ZLWK�VXSHU��DQG�VXEVFULSWV� DV�LQ a11� :H�UHFRPPHQG�XVLQJ�WKH 6\PEROOLEUDU\�IRU�JHQHUDWLQJ�WKHVH� WRR�

<RX�FDQ�DVVLJQ�DQ�DUELWUDU\�REMHFW�WR�DQ�LGHQWLILHU�E\�PHDQV�RI�WKH DVVLJQPHQWRSHUDWRU � � $IWHUZDUGV� WKLV�REMHFW�LV�WKH YDOXH RI�WKH�LGHQWLILHU� )RU�H[DPSOH�DIWHU�WKH�FRPPDQG

[ � � � ,�

WKH�LGHQWLILHU [ KDV�WKH�YDOXH ��,��ZKLFK�LV�D�FRPSOH[�QXPEHU�RI�GRPDLQ�W\SH'20B&203/(;��<RX�VKRXOG�EH�FDUHIXO�WR�GLVWLQJXLVK�EHWZHHQ�DQ�LGHQWLILHU� LWV�YDOXH�DQG�LWV�HYDOXDWLRQ� :H�UHIHU�WR�WKH�LPSRUWDQW�&KDSWHU �� ZKHUH�0X3$'¶VHYDOXDWLRQ�VWUDWHJ\�LV�GHVFULEHG�

,I�DQ�LGHQWLILHU�DOUHDG\�KDV�EHHQ�DVVLJQHG�D�YDOXH� DQRWKHU�DVVLJQPHQW�RYHUZULWHVWKH�SUHYLRXV�YDOXH� 7KH�VWDWHPHQW \� [ GRHV�QRW�DVVLJQ�WKH�LGHQWLILHU [ WR�WKHLGHQWLILHU \� EXW�WKH�FXUUHQW�YDOXH��WKH�HYDOXDWLRQ��RI [�

���

Page 67: Mupad Tutorial

B]^gmbÛ^kl

[ � �� \ � [� [� \1, 1

,I�WKH�YDOXH�RI [ LV�FKDQJHG�ODWHU�RQ� WKLV�GRHV�QRW�DIIHFW \�

[ � �� [� \2, 1

+RZHYHU� LI [ LV�D�V\PEROLF�LGHQWLILHU� ZKLFK�HYDOXDWHV�WR�LWVHOI� WKH�QHZ�LGHQWLILHU \UHIHUV�WR�WKLV�V\PERO�

GHOHWH [� \ � [� [� \� [ � �� [� \x, x

2, 2

+HUH�ZH�KDYH�GHOHWHG�WKH�YDOXH�RI�WKH�LGHQWLILHU [ E\�PHDQV�RI�WKH�IXQFWLRQ GHOHWH�$IWHU�GHOHWLRQ� [ LV�DJDLQ�D�V\PEROLF�LGHQWLILHU�ZLWKRXW�D�YDOXH�

7KH�DVVLJQPHQW�RSHUDWRU � LV�D�VKRUW�IRUP�RI�WKH�V\VWHP�IXQFWLRQ BDVVLJQ� ZKLFKPD\�DOVR�EH�FDOOHG�GLUHFWO\�

BDVVLJQ�[� YDOXH�� [YDOXH

7KLV�IXQFWLRQ�UHWXUQV�LWV�VHFRQG�DUJXPHQW� QDPHO\�WKH�ULJKW�KDQG�VLGH�RI�DQDVVLJQPHQW� 7KLV�H[SODLQV�WKH�VFUHHQ�RXWSXW�DIWHU�DQ�DVVLJQPHQW�

\ � � [2 YDOXH

<RX�FDQ�ZRUN�ZLWK�WKH�UHWXUQHG�YDOXH�LPPHGLDWHO\� )RU�H[DPSOH� WKH�IROORZLQJFRQVWUXFWLRQ�LV�DOORZHG��WKH�DVVLJQPHQW�PXVW�EH�SXW�LQ�SDUHQWKHVHV��

\ � FRV��[ � ���� [� \0, 1

+HUH�WKH�YDOXH 0 LV�DVVLJQHG�WR�WKH�LGHQWLILHU [� 7KH�UHWXUQ�YDOXH�RI�WKH�DVVLJQPHQW�L�H�� 0� LV�IHG�GLUHFWO\�DV�DUJXPHQW�WR�WKH�FRVLQH�IXQFWLRQ� DQG�WKH�UHVXOW FRV(0) = 1LV�DVVLJQHG�WR \� 7KXV� ZH�KDYH�VLPXOWDQHRXVO\�DVVLJQHG�YDOXHV�WR�ERWK [ DQG \�

���

parisakhosravi
Highlight
parisakhosravi
Highlight
Page 68: Mupad Tutorial

- FnI:=� H[c^\ml

$QRWKHU�DVVLJQPHQW�IXQFWLRQ�LV DVVLJQ� ,WV�LQSXW�DUH�VHWV�RU�OLVWV�RI�HTXDWLRQV�ZKLFK�DUH�WUDQVIRUPHG�LQWR�DVVLJQPHQWV�

GHOHWH [� \� DVVLJQ�^[ �� \ �`�� [� \0, 1

7KLV�IXQFWLRQ�LV�SDUWLFXODUO\�XVHIXO�LQ�FRQQHFWLRQ�ZLWK VROYH �6HFWLRQ ��� ZKLFKUHWXUQV�VROXWLRQV�DV�D�VHW�FRQWDLQLQJ�OLVWV�RI�HTXDWLRQV�RI�WKH�IRUPLGHQWLILHU YDOXH ZLWKRXW�DVVLJQLQJ�WKHVH�YDOXHV�

7KHUH�H[LVW�PDQ\�LGHQWLILHUV�ZLWK�SUHGHILQHG�YDOXHV� 7KH\�UHSUHVHQW�PDWKHPDWLFDOIXQFWLRQV��VXFK�DV VLQ� H[S� RU VTUW�� PDWKHPDWLFDO�FRQVWDQWV��VXFK�DV 3,���RU0X3$' DOJRULWKPV��VXFK�DV GLII� LQW� RU OLPLW�� ,I�\RX�WU\�WR�FKDQJH�WKH�YDOXH�RIVXFK�D�SUHGHILQHG�LGHQWLILHU� 0X3$' LVVXHV�D�ZDUQLQJ�RU�DQ�HUURU�PHVVDJH�

VLQ � �(UURU� ,GHQWLILHU VLQ LV SURWHFWHG >BDVVLJQ@

<RX�FDQ�SURWHFW�\RXU�RZQ�LGHQWLILHUV�DJDLQVW�RYHUZULWLQJ�YLD�WKH�FRPPDQGSURWHFW�LGHQWLILHU�� 7KH�ZULWH�SURWHFWLRQ�RI�ERWK�\RXU�RZQ�DQG�RI�V\VWHPLGHQWLILHUV�FDQ�EH�UHPRYHG�E\ XQSURWHFW�LGHQWLILHU�� +RZHYHU� ZH�VWURQJO\UHFRPPHQG�QRW�WR�RYHUZULWH�SUHGHILQHG�LGHQWLILHUV� VLQFH�WKH\�DUH�XVHG�E\�PDQ\V\VWHP�IXQFWLRQV�ZKLFK�ZRXOG�UHWXUQ�XQSUHGLFWDEOH�UHVXOWV�DIWHU�D�UHGHILQLWLRQ�,GHQWLILHUV�VWDUWLQJ�ZLWK�D�KDVK�PDUN�����DUH�DOZD\V�SURWHFWHG�DQG�FDQQRW�EHXQSURWHFWHG� 7KH�FRPPDQG DQDPHV�$OO� OLVWV�DOO�FXUUHQWO\�GHILQHG�LGHQWLILHUV�

<RX�FDQ�XVH�WKH�FRQFDWHQDWLRQ�RSHUDWRU�³�´ WR�JHQHUDWH�QDPHV�RI�LGHQWLILHUVG\QDPLFDOO\� ,I [ DQG L DUH�LGHQWLILHUV� WKHQ [�L JHQHUDWHV�D�QHZ�LGHQWLILHU�E\FRQFDWHQDWLQJ�WKH HYDOXDWLRQV �VHH�&KDSWHU ���RI [ DQG L�

[ � ]� L � �� [�L]�

[�L � YDOXH� ]�YDOXH

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�XVH�D IRU ORRS��&KDSWHU ����WR�DVVLJQ�YDOXHV�WR�WKHLGHQWLILHUV [������[�����GHOHWH [�IRU L IURP � WR ���� GR [�L � LA� HQGBIRU�

����

Page 69: Mupad Tutorial

B]^gmbÛ^kl

'XH�WR�SRVVLEOH�VLGH�HIIHFWV�RU�FRQIOLFWV�ZLWK�DOUHDG\�H[LVWLQJ�LGHQWLILHUV� ZHVWURQJO\�UHFRPPHQG�WR�XVH�WKLV�FRQFHSW�RQO\�LQWHUDFWLYHO\�DQG�QRW�ZLWKLQ�0X3$'SURFHGXUHV�

7KH�IXQFWLRQ JHQLGHQW JHQHUDWHV�D�QHZ�LGHQWLILHU�WKDW�KDV�QRW�EHHQ�XVHG�EHIRUH�LQWKH�FXUUHQW�0X3$' VHVVLRQ�

;� � �;� � �;� � ���� JHQLGHQW��;�

<RX�PD\�XVH�VWULQJV�HQFORVHG�LQ�TXRWDWLRQ�PDUNV � �6HFWLRQ ������WR�JHQHUDWHLGHQWLILHUV�G\QDPLFDOO\�

D � HPDLO� E � ��\RX�� D�EHPDLO�\RX

(YHQ�LI�WKH�VWULQJ�FRQWDLQV�EODQNV�RU�RSHUDWRU�V\PEROV� D�YDOLG�LGHQWLILHU�LVJHQHUDWHG� ZKLFK�0X3$' GLVSOD\V�XVLQJ�WKH�EDFNWLFN�QRWDWLRQ�PHQWLRQHG�DERYH�

D � HPDLO� E � ��\RX � [�� D�EHPDLO�\RX+ [

6WULQJV�DUH�QRW�LGHQWLILHUV�DQG�FDQQRW�EH�DVVLJQHG�D�YDOXH�

�VWULQJ� � �(UURU� ,QYDOLG OHIW�KDQG VLGH LQ DVVLJQPHQW >OLQH �� FRO ��@

([HUFLVH����� :KLFK�RI�WKH�IROORZLQJ�QDPHV [� [�� �[� [BW� GLII� H[S�FDXWLRQ���� [�\� -DFN-LOO� DBYDOLGBLGHQWLILHU� �� DUH�YDOLG�LGHQWLILHUV":KLFKRI�WKHP�FDQ�EH�DVVLJQHG�YDOXHV"

([HUFLVH����� 5HDG�WKH�KHOS�SDJH�IRU VROYH� 6ROYH�WKH�V\VWHP�RI�HTXDWLRQV

x1 + x2 = 1, x2 + x3 = 1, . . . , x19 + x20 = 1, x20 = π

LQ�WKH�XQNQRZQV x1, x2, . . . , x20� 5HDG�WKH�KHOS�SDJH�IRU DVVLJQ DQG�DVVLJQ�WKHYDOXHV�RI�WKH�VROXWLRQ�WR�WKH�XQNQRZQV�

����

Page 70: Mupad Tutorial

- FnI:=� H[c^\ml

KqeZgda[�=phj]kkagfk

:H�VD\�WKDW�DQ�REMHFW�FRQWDLQLQJ�V\PEROLF�WHUPV�VXFK�DV�WKH�HTXDWLRQ

0.3 + VLQ(3) +f(x, y)

5= 0

LV�DQ H[SUHVVLRQ� ([SUHVVLRQV�RI�GRPDLQ�W\SH '20B(;35 DUH�SUREDEO\�WKH�PRVWJHQHUDO�GDWD�W\SH�LQ�0X3$'�� ([SUHVVLRQV�DUH�EXLOW�RI�DWRPLF�FRPSRQHQWV� DV�DOO0X3$' REMHFWV� DQG�DUH�FRPSRVHG�E\�PHDQV�RI RSHUDWRUV� 7KLV�FRPSULVHV�ELQDU\RSHUDWRUV� VXFK�DV�WKH�EDVLF�DULWKPHWLF�RSHUDWLRQV ���� ���A� DQG�IXQFWLRQ�FDOOVVXFK�DV VLQ�·��I�·�� HWF�

2SHUDWRUV

0X3$'� WKURXJKRXW�XVHV�IXQFWLRQV�WR�FRPELQH�RU�PDQLSXODWH�REMHFWV�� ,W�ZRXOGEH�OLWWOH�LQWXLWLYH� KRZHYHU� WR�XVH�IXQFWLRQ�FDOOV�HYHU\ZKHUH�� VD\� BSOXV�D�E� IRUWKH�DGGLWLRQ a+ b� )RU�WKDW�UHDVRQ� D�YDULHW\�RI�LPSRUWDQW�RSHUDWLRQV�LVLPSOHPHQWHG�LQ�VXFK�D�ZD\�WKDW�\RX�FDQ�XVH�WKH�IDPLOLDU�PDWKHPDWLFDO�QRWDWLRQ�³RSHUDWRU�QRWDWLRQ´��IRU�LQSXW� $OVR�WKH�RXWSXW�LV�JLYHQ�LQ�VXFK�D�IRUP� ,Q�WKHIROORZLQJ� ZH�OLVW�WKH�RSHUDWRUV�IRU�EXLOGLQJ�PRUH�FRPSOH[�0X3$' H[SUHVVLRQVIURP�DWRPV�

7KH�RSHUDWRUV ���� �� IRU�WKH�EDVLF�DULWKPHWLF�RSHUDWLRQV�DQG A IRUH[SRQHQWLDWLRQ�DUH�YDOLG�IRU�V\PEROLF�H[SUHVVLRQV�DV�ZHOO�

D � E � F� D � E� �D� D E F� D�E� DAE

a+ b+ c, a− b, −a, a b c, ab, ab

<RX�PD\�LQSXW�WKHVH�RSHUDWRUV�LQ�WKH�IDPLOLDU�PDWKHPDWLFDO�ZD\� EXW�LQWHUQDOO\WKH\�DUH�IXQFWLRQ�FDOOV�

�5HPDUNDEO\� WKH�0X3$' NHUQHO�WUHDWV�QRW�RQO\�JHQXLQH�IXQFWLRQ�FDOOV� VXFK�DV VLQ������ DVVLJQ�PHQWV� RU�DULWKPHWLFDO�RSHUDWLRQV�LQ�D�IXQFWLRQDO�ZD\� EXW�DOVR�ORRSV��&KDSWHU ����DQG�FDVH�GLVWLQFWLRQV�&KDSWHU ����

�/,63 SURJUDPPHUV�PD\�GLVDJUHH�

����

Page 71: Mupad Tutorial

Lrf[heb\�>qik^llbhgl

BSOXV�D�E�F�� BVXEWUDFW�D�E�� BQHJDWH�D��BPXOW�D�E�F�� BGLYLGH�D�E�� BSRZHU�D�E�

a+ b+ c, a− b, −a, a b c, ab, ab

7KH�VDPH�KROGV�IRU�WKH�IDFWRULDO�RI�D�QRQQHJDWLYH�LQWHJHU� <RX�PD\�LQSXW�LW�LQ�WKHPDWKHPDWLFDO�QRWDWLRQ Q�� ,QWHUQDOO\�LW�LV�FRQYHUWHG�WR�D�FDOO�RI�WKH�IXQFWLRQ IDFW�

Q� IDFW�Q�� IDFW����n! = n!, 3628800

7KH�DULWKPHWLF�RSHUDWRUV GLY DQG PRG� ZHUH�SUHVHQWHG�LQ�&KDSWHU ���� 7KH\�PD\DOVR�EH�XVHG�LQ�D�V\PEROLF�FRQWH[W� EXW�WKHQ�UHWXUQ�RQO\�V\PEROLF�UHVXOWV�

[ GLY �� �� PRG Sx GLY 4, 25PRG p

6HYHUDO�0X3$' REMHFWV�VHSDUDWHG�E\�FRPPDV�IRUP�D�VHTXHQFH�

VHTXHQFH � D� E� F � Ga, b, c+ d

7KH�RSHUDWRU � LV�DQ�LPSRUWDQW�WRRO�WR�JHQHUDWH�VXFK�VHTXHQFHV�

LA� � L ����� [AL � L ����4, 9, 16, 25, 36, 49

x, x2, x3, x4, x5

(TXDWLRQV�DQG�LQHTXDOLWLHV DUH�YDOLG�0X3$' REMHFWV� 7KH\�DUH�JHQHUDWHG�E\�WKHHTXDOLW\�VLJQ DQG�E\ �!� UHVSHFWLYHO\�

HTXDWLRQ � [ � \ �� LQHTXDOLW\ � [ �! \x+ y = 2

x ̸= y

�7KH�REMHFW [�PRG�S LV�FRQYHUWHG�WR�WKH�IXQFWLRQ�FDOO BPRG�[�S�� 7KH�IXQFWLRQ BPRG FDQ�EH�UHGHILQHG�H�J�� BPRG� PRGS RU BPRG� PRGV� 7KH�EHKDYLRU�RI PRGS DQG PRGV LV�GRFXPHQWHG�RQ�WKH�FRUUHVSRQGLQJKHOS�SDJHV� $ UHGHILQLWLRQ�RI BPRG DOVR�UHGHILQHV�WKH�RSHUDWRU PRG�

����

Page 72: Mupad Tutorial

- FnI:=� H[c^\ml

7KH�RSHUDWRUV �� � � !� DQG ! FRPSDUH�WKH�PDJQLWXGHV�RI�WKHLU�DUJXPHQWV� 7KHFRUUHVSRQGLQJ�H[SUHVVLRQV�UHSUHVHQW�FRQGLWLRQV�

FRQGLWLRQ � L � �i ≤ 2

,Q�D�FRQFUHWH�FRQWH[W� WKH\�XVXDOO\�FDQ�EH�HYDOXDWHG�WR�RQH�RI�WKH�WUXWK��³%RROHDQ´�YDOXHV 758( RU )$/6(��7\SLFDOO\� WKH\�DUH�XVHG�LQ LI VWDWHPHQWV�RU�DV�WHUPLQDWLRQFRQGLWLRQV�LQ�ORRSV� <RX�PD\�FRPELQH�%RROHDQ�H[SUHVVLRQV�YLD�WKH�ORJLFDORSHUDWRUV DQG DQG RU� RU�QHJDWH�WKHP�YLD QRW�

FRQGLWLRQ� � FRQGLWLRQ� DQG �QRW FRQGLWLRQ��FRQGLWLRQ� ∧ ¬FRQGLWLRQ�

<RX�FDQ�GHILQH�PDSV��IXQFWLRQV��LQ�VHYHUDO�ZD\V�LQ�0X3$'��7KH�VLPSOHVW�PHWKRGLV�WR�XVH�WKH DUURZ�RSHUDWRU �! �WKH�PLQXV�V\PERO�IROORZHG�E\�WKH�³JUHDWHU�WKDQ´V\PERO��

I � [ �! [A�x→ x2

$IWHU�GHILQLQJ�D�IXQFWLRQ�LQ�WKLV�ZD\� \RX�PD\�FDOO�LW�OLNH�D�V\VWHP�IXQFWLRQ�

I���� I�[ � ��� I�\�

16, (x+ 1)2, y2

,I�\RX�ZLVK�WR�GHILQH�D�IXQFWLRQ�WKDW�ILUVW�WUHDWV�WKH�ULJKW�KDQG�VLGH�DV�D�FRPPDQGWR�EH�SHUIRUPHG� XVH�WKH GRXEOH�DUURZ�RSHUDWRU ��!�

J� � [ �! LQW�I�[�� [��J� � [ ��! LQW�I�[�� [�

x→∫f(x) Gx

x→ x3

3

����

Page 73: Mupad Tutorial

Lrf[heb\�>qik^llbhgl

7KH�FRPSRVLWLRQ�RI�IXQFWLRQV�LV�GHILQHG�E\�PHDQV�RI�WKH FRPSRVLWLRQ�RSHUDWRU #�

F � D#E� F�[�a(b(x))

7KH LWHUDWLRQ�RSHUDWRU ## GHQRWHV�LWHUDWHG�FRPSRVLWLRQ�RI�D�IXQFWLRQ�ZLWK�LWVHOI�

I � J##�� I�[�g(g(g(g(x))))

6RPH�V\VWHP�IXQFWLRQV� VXFK�DV�GHILQLWH�LQWHJUDWLRQ�YLD LQW RU�WKH � RSHUDWRU�UHTXLUH�D UDQJH� <RX�JHQHUDWH�D�UDQJH�E\�PHDQV�RI�WKH�RSHUDWRU ���

UDQJH � ����� LQW�[� [ UDQJH�0..1

12

5DQJHV�VKRXOG�QRW�EH�FRQIXVHG�ZLWK�IORDWLQJ�SRLQW�LQWHUYDOV�RI�GRPDLQ�W\SH'20B,17(59$/��ZKLFK�PD\�EH�FUHDWHG�YLD�WKH�RSHUDWRU ��� RU�WKH�IXQFWLRQ KXOO�

3, ��� ����� KXOO�3,� �����3.141592653 . . . 6.666666667, 3.141592653 . . . 6.666666667

7KLV�GDWD�W\SH�LV�H[SODLQHG�LQ�6HFWLRQ �����

0X3$' WUHDWV�DQ\�H[SUHVVLRQ�RI�WKH�IRUP LGHQWLILHU�DUJXPHQW� DV�D�IXQFWLRQFDOO�

GHOHWH I�H[SUHVVLRQ � VLQ�[� � I�[� \� � LQW�J�[�� [ �����

VLQ(x) + f(x, y) +∫ 1

0g(x) Gx

7DEOH ��� OLVWV�DOO�RSHUDWRUV�SUHVHQWHG�DERYH�WRJHWKHU�ZLWK�WKHLU�HTXLYDOHQWIXQFWLRQDO�IRUP� <RX�PD\�XVH�HLWKHU�IRUP�WR�LQSXW�H[SUHVVLRQV�

���� BGLYLGH��� ����17

=17

����

Page 74: Mupad Tutorial

- FnI:=� H[c^\ml

RSHUDWRUV\VWHPIXQFWLRQ PHDQLQJ H[DPSOH

� BSOXV DGGLWLRQ 680� D�E� BVXEWUDFW VXEWUDFWLRQ 'LIIHUHQFH� D�E BPXOW PXOWLSOLFDWLRQ 3URGXFW� D E� BGLYLGH GLYLVLRQ 4XRWLHQW� D�EA BSRZHU H[SRQHQWLDWLRQ 3RZHU� DAE

GLY BGLY TXRWLHQW 4XRWLHQW� D�GLY�SPRGXOR p

PRG BPRG UHPDLQGHU 5HPDLQGHU� D�PRG�SPRGXOR p

� IDFW IDFWRULDO Q�� BVHTJHQ VHTXHQFH 6HTXHQFH� LA��L ����

JHQHUDWLRQ� BH[SUVHT VHTXHQFH 6HT� 6HT��6HT�

FRQFDWHQDWLRQXQLRQ BXQLRQ VHW�XQLRQ 6� 6HW��XQLRQ�6HW�

LQWHUVHFW BLQWHUVHFW VHW�LQWHUVHFWLRQ 6� 6HW��LQWHUVHFW�6HW�PLQXV BPLQXV VHW�GLIIHUHQFH 6� 6HW��PLQXV�6HW�

BHTXDO HTXDWLRQ (TXDWLRQ� [�\ ��! BXQHTXDO LQHTXDOLW\ &RQGLWLRQ� [�!\� BOHVV FRPSDULVRQ &RQGLWLRQ� D�E! FRPSDULVRQ &RQGLWLRQ� D!E� BOHHTXDO FRPSDULVRQ &RQGLWLRQ� D� E! FRPSDULVRQ &RQGLWLRQ� D! EQRW BQRW QHJDWLRQ &RQGLWLRQ�� QRW�&RQGLWLRQ�DQG BDQG ORJLFDO�µDQG¶ &RQGLWLRQ� D�E�DQG�E�FRU BRU ORJLFDO�µRU¶ &RQGLWLRQ� D�E�RU�E�F�! PDSSLQJ 6TXDUH� [��!�[A� ' GLIIHUHQWLDO I�[�

RSHUDWRU# BIFRQFDW FRPSRVLWLRQ K� I#J## BIQHVW LWHUDWLRQ J� I##�� BFRQFDW FRQFDWHQDWLRQ 1HZ1DPH� 1DPH��1DPH��� BUDQJH UDQJH 5DQJH� D��E��� LQWHUYDO LQWHUYDO ,9� ���������

QDPH�� IXQFWLRQ�FDOO VLQ����I�[��UHVHW��

7DEOH ���� 7KH�PDLQ�RSHUDWRUV�IRU�JHQHUDWLQJ�0X3$'� H[SUHVVLRQV

����

Page 75: Mupad Tutorial

Lrf[heb\�>qik^llbhgl

>L � L ����@ >BVHTJHQ�L� L� �����@�[3, 4, 5] = [3, 4, 5]

D � E BOHVV�D� E��(a < b) = (a < b)

�I#J��[� BIFRQFDW�I� J��[�f(g(x)) = f(g(x))

:H�UHPDUN�WKDW�VRPH�RI�WKH�V\VWHP�IXQFWLRQV�VXFK�DV BSOXV� BPXOW� BXQLRQ� RUBFRQFDW DFFHSW�DUELWUDULO\�PDQ\�DUJXPHQWV� ZKLOH�WKH�FRUUHVSRQGLQJ�RSHUDWRUVDUH�RQO\�ELQDU\�

BSOXV�D� E� X� Y�� BFRQFDW�D� E� X� Y�� BXQLRQ��a+ b+ u+ v, DEXY, ∅

,W�LV�RIWHQ�XVHIXO�WR�NQRZ�DQG�WR�XVH�WKH�IXQFWLRQDO�IRUP�RI�WKH�RSHUDWRUV� )RUH[DPSOH� LW�LV�YHU\�HIILFLHQW�WR�IRUP�ORQJHU�VXPV�E\�DSSO\LQJ BSOXV WR�PDQ\DUJXPHQWV� <RX�PD\�JHQHUDWH�WKH�DUJXPHQW�VHTXHQFH�TXLFNO\�E\�PHDQV�RI�WKHVHTXHQFH�JHQHUDWRU ��

BSOXV���L� � L �������� IORDW���2.718281828

7KH�IXQFWLRQ PDS LV�D�XVHIXO�WRRO� ,W�DSSOLHV�D�IXQFWLRQ�WR�DOO�RSHUDQGV�RI�D�0X3$'REMHFW� )RU�H[DPSOH�

BSOXV���L� � L �������� IORDW���[IXQFWLRQ ([�, y, z) , IXQFWLRQ ([�, y, z) , IXQFWLRQ ([�, y, z)]

,I�\RX�ZDQW�WR�DSSO\�RSHUDWRUV�YLD PDS� XVH�WKHLU�IXQFWLRQDO�HTXLYDOHQW�

PDS�>[�� [�� [�@� BSRZHU� ��� PDS�>I� J@� BIQHVW� ��[[�5, [�5, [�5

], [f ◦ f ◦ f ◦ f ◦ f, g ◦ g ◦ g ◦ g ◦ g]

)RU�WKH�PRVW�FRPPRQ�RSHUDWRUV� QDPHO\ �� �� � �� A� � �!� �� !� � � ! � !� DQG� !� WKH�FRUUHVSRQGLQJ�IXQFWLRQV�FDQ�EH�DFFHVVHG�LQ�WKH�IRUP C�C� C�C� C C HWF��

����

Page 76: Mupad Tutorial

- FnI:=� H[c^\ml

PDS�>�� �� �� �@� C C� ��� PDS�>�� �� �� �@� CAC� ��[3, 6, 9, 12] , [1, 4, 9, 16]

1RWH�WKDW C�C LV�QHJDWLRQ� QRW�VXEWUDFWLRQ�

6RPH�RSHUDWLRQV�DUH�LQYDOLG�EHFDXVH�WKH\�GR�QRW�PDNH�VHQVH�PDWKHPDWLFDOO\�

� DQG [(UURU� ,OOHJDO RSHUDQG >BDQG@

7KH�V\VWHP�IXQFWLRQ BDQG UHFRJQL]HV�WKDW�WKH�DUJXPHQW � FDQQRW�UHSUHVHQW�D%RROHDQ�YDOXH�DQG�LVVXHV�DQ�HUURU�PHVVDJH� +RZHYHU� 0X3$' DFFHSWV�D�V\PEROLFH[SUHVVLRQ�VXFK�DV D�DQG�E ZLWK�V\PEROLF�LGHQWLILHUV D�E� $V�VRRQ�DV D DQG E JHW%RROHDQ�YDOXHV� WKH�H[SUHVVLRQ�FDQ�EH�HYDOXDWHG�WR�D�%RROHDQ�YDOXH�DV�ZHOO�

F � D DQG E� D � 758(� E � 758(� F758(

7KH�RSHUDWRUV�KDYH�GLIIHUHQW SULRULWLHV� IRU�H[DPSOH�✞

D�IDFW���PHDQV D��IDFW���� DQG�UHWXUQV D��D��A� PHDQV �D���A� DQG�UHWXUQV D�A��D EAF PHDQV D �EAF��D�E F PHDQV D��E F��D�E PRG F PHDQV �D�E� PRG F�D E PRG F PHDQV D �E PRG F��D�E�� PHDQV D��E��� DQG�UHWXUQV D�E�E�E�

,I�ZH�GHQRWH�WKH�UHODWLRQ�³LV�RI�ORZHU�SULRULW\�WKDQ´�E\ ≺� WKHQ�ZH�KDYH�

, ≺ � ≺ = ≺ PRG ≺ + ≺ ∗ ≺ A ≺ . ≺ IXQFWLRQ�FDOO�

<RX�ILQG�D�FRPSOHWH�OLVW�RI�WKH�RSHUDWRUV�DQG�WKHLU�SULRULWLHV�LQ�6HFWLRQ�³2SHUDWRUV´RI�WKH�0X3$' 4XLFN�5HIHUHQFH� 3DUHQWKHVHV�FDQ�DOZD\V�EH�XVHG�WR�HQIRUFH�DQHYDOXDWLRQ�RUGHU�WKDW�GLIIHUV�IURP�WKH�VWDQGDUG�SULRULW\�RI�WKH�RSHUDWRUV�

� � � PRG �� � � �� PRG ��0, 2

����

Page 77: Mupad Tutorial

Lrf[heb\�>qik^llbhgl

L � �� [�LA�� [��LA��[�2, [�

X� Y � �� �X� Y� � �u, v, v, v

u, v, u, v, u, v

([SUHVVLRQ�7UHHV

$ XVHIXO�PRGHO�IRU�UHSUHVHQWLQJ�D�0X3$'� H[SUHVVLRQ�LV�WKH H[SUHVVLRQ�WUHH� ,WUHIOHFWV�WKH�LQWHUQDO�UHSUHVHQWDWLRQ� 7KH�RSHUDWRUV�RU�WKHLU�FRUUHVSRQGLQJIXQFWLRQV� UHVSHFWLYHO\� DUH�WKH�YHUWLFHV� DQG�WKH�DUJXPHQWV�DUH�VXEWUHHV� 7KHRSHUDWRU�RI�ORZHVW�SULRULW\�LV�DW�WKH�URRW� +HUH�DUH�VRPH�H[DPSOHV�

D � E F � G H VLQ�I�AJ

D

E F

G H A

VLQ

I

J

LQW�H[S�[A��� [ �����

LQW

H[S

A

[ �

[ ��

� �

����

Page 78: Mupad Tutorial

- FnI:=� H[c^\ml

7KH�GLIIHUHQFH D�E LV�LQWHUQDOO\�UHSUHVHQWHG�DV D�E �����

D

E ��

6LPLODUO\� D�TXRWLHQW D�E KDV�WKH�LQWHUQDO�UHSUHVHQWDWLRQ D EA�����

D A

E ��

7KH�OHDYHV�RI�DQ�H[SUHVVLRQ�WUHH�DUH�0X3$' DWRPV�

7KH�IXQFWLRQ SURJ��H[SUWUHH DOORZV�WR�GLVSOD\�H[SUHVVLRQ�WUHHV� 2SHUDWRUV�DUHUHSODFHG�E\�WKH�QDPHV�RI�WKH�FRUUHVSRQGLQJ�V\VWHP�IXQFWLRQV�

SURJ��H[SUWUHH�D�E��BPXOW_��� D_C�� BSRZHU

_��� E_C�� ��

([HUFLVH����� 6NHWFK�WKH�H[SUHVVLRQ�WUHH�RI DAE�VLQ�D�E��

([HUFLVH����� 'HWHUPLQH�WKH�RSHUDQGV�RI ���� [��� ��� ,��DQG [�� ,��([SODLQWKH�GLIIHUHQFHV�WKDW�\RX�REVHUYH�

����

Page 79: Mupad Tutorial

Lrf[heb\�>qik^llbhgl

2SHUDQGV

<RX�FDQ�GHFRPSRVH�H[SUHVVLRQV�V\VWHPDWLFDOO\�E\�PHDQV�RI�WKH�RSHUDQG�IXQFWLRQVRS DQG QRSV� ZKLFK�ZHUH�DOUHDG\�SUHVHQWHG�LQ�6HFWLRQ ���� 7KH�RSHUDQGV�RI�DQH[SUHVVLRQ�FRUUHVSRQG�WR�WKH�VXEWUHHV�EHORZ�WKH�URRW�LQ�WKH�H[SUHVVLRQ�WUHH�

H[SUHVVLRQ � D � E � F � VLQ�[�� QRSV�H[SUHVVLRQ�4

RS�H[SUHVVLRQ�a, b, c, VLQ(x)

$GGLWLRQDOO\� H[SUHVVLRQV�RI�GRPDLQ�W\SH '20B(;35 KDYH�D�³�WK�RSHUDQG�´ ZKLFK�LVDFFHVVLEOH�YLD RS�·���� ,W�FRUUHVSRQGV�WR�WKH�URRW�RI�WKH�H[SUHVVLRQ�WUHH�DQG�WHOOV\RX�ZKLFK�IXQFWLRQ�LV�XVHG�WR�EXLOG�WKH�H[SUHVVLRQ�

RS�D � E F� ��� RS�D EAF� ��� RS�DA�E F�� ���RS�I�VLQ�[�����

BSOXV, BPXOW, BSRZHU, f

VHTXHQFH � D� E� F� RS�VHTXHQFH� ��BH[SUVHT

,I�WKH�H[SUHVVLRQ�LV�D�V\PEROLF�IXQFWLRQ�FDOO� RS�·��� UHWXUQV�WKH�LGHQWLILHU�RI�WKDWIXQFWLRQ�

RS�VLQ���� ��� RS�I�[�� ��� RS�GLII�\�[�� [�� ���RS�LQW�H[S�[A��� [�� ��

VLQ, f, GLII, LQW

<RX�PD\�UHJDUG�WKH��WK�RSHUDQG�RI�DQ�H[SUHVVLRQ�DV�D�³PDWKHPDWLFDO�W\SH�´ )RUH[DPSOH� DQ�DOJRULWKP�IRU�GLIIHUHQWLDWLRQ�RI�DUELWUDU\�H[SUHVVLRQV�PXVW�ILQG�RXWZKHWKHU�WKH�H[SUHVVLRQ�LV�D�VXP� D�SURGXFW� RU�D�IXQFWLRQ�FDOO� 7R�WKLV�HQG� LW�PD\ORRN�DW�WKH��WK�RSHUDQG�WR�GHFLGH�ZKHWKHU�OLQHDULW\� WKH�SURGXFW�UXOH� RU�WKH�FKDLQUXOH�RI�GLIIHUHQWLDWLRQ�DSSOLHV�

����

Page 80: Mupad Tutorial

- FnI:=� H[c^\ml

$V�DQ�H[DPSOH� ZH�GHFRPSRVH�WKH�H[SUHVVLRQ

H[SUHVVLRQ � D � E � VLQ�[� � FA��

ZLWK�WKH�H[SUHVVLRQ�WUHH

D E VLQ

[

A

F �

V\VWHPDWLFDOO\�E\�PHDQV�RI�WKH RS IXQFWLRQ�

RS�H[SUHVVLRQ� ���QRSV�H[SUHVVLRQ��BSOXV, a, b, VLQ(x) , c2

:H�FDQ�SXW�WKHVH�H[SUHVVLRQV�WRJHWKHU�DJDLQ�LQ�WKH�IROORZLQJ�IRUP�

URRW � RS�H[SUHVVLRQ� ��� RSHUDQGV � RS�H[SUHVVLRQ��URRW�RSHUDQGV�c2 + a+ b+ VLQ(x)

1RWH�WKDW�WKH�RXWSXW�RUGHU�YLVLEOH�RQ�WKH�VFUHHQ�LV�GLIIHUHQW�IURP�WKH�LQWHUQDORUGHU�DFFHVVLEOH�E\ RS� <RX�FDQ�DFFHVV�WKH�EXLOGLQJ�EORFNV�RI�DQ�H[SUHVVLRQ�LQRXWSXW�RUGHU�ZLWK�WKH�LQGH[�RSHUDWRU >@� DV�LQ�WKH�IROORZLQJ�

H[SUHVVLRQ>�@� H[SUHVVLRQ>�@c2, VLQ(x)

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�GHFRPSRVH�DQ�H[SUHVVLRQ�FRPSOHWHO\�LQWR�LWV�DWRPV�FRPSDUH�ZLWK�6HFWLRQ �����

H[SUHVVLRQ � VLQ�[ � FRV�D E���

����

Page 81: Mupad Tutorial

Lrf[heb\�>qik^llbhgl

7KH�RSHUDQGV�DQG�VXEH[SUHVVLRQV�DUH�✞

RS�H[SUHVVLRQ� �� � VLQRS�H[SUHVVLRQ� �� � [�FRV�D E�RS�H[SUHVVLRQ� >�� �@� � BSOXVRS�H[SUHVVLRQ� >�� �@� � [RS�H[SUHVVLRQ� >�� �@� � FRV�D E�RS�H[SUHVVLRQ� >�� �� �@� � FRVRS�H[SUHVVLRQ� >�� �� �@� � D ERS�H[SUHVVLRQ� >�� �� �� �@� � BPXOWRS�H[SUHVVLRQ� >�� �� �� �@� � DRS�H[SUHVVLRQ� >�� �� �� �@� � E

([HUFLVH������ 6NHWFK�WKH�H[SUHVVLRQ�WUHH�RI�WKH�IROORZLQJ�%RROHDQ�H[SUHVVLRQ�

FRQGLWLRQ � �QRW D� DQG �E RU F��

+RZ�FDQ�\RX�XVH RS WR�SLFN�WKH�V\PEROLF�LGHQWLILHUV D� E� DQG F RXW�RI�WKH�REMHFWFRQGLWLRQ"

����

Page 82: Mupad Tutorial

- FnI:=� H[c^\ml

K]im]f[]k

6HTXHQFHV�IRUP�DQ�LPSRUWDQW�0X3$'� GDWD�VWUXFWXUH� /LVWV�DQG�VHWV�DUH�EXLOWIURP�VHTXHQFHV� $V�GLVFXVVHG�LQ�6HFWLRQ ���� D�VHTXHQFH�LV�D�VHULHV�RI�0X3$'REMHFWV�VHSDUDWHG�E\�FRPPDV�

VHTXHQFH� � D� E� F� VHTXHQFH� � F� G� Ha, b, c

c, d, e

<RX�PD\�DOVR�XVH�WKH�FRPPD�WR�FRQFDWHQDWH�VHTXHQFHV�

VHTXHQFH� � VHTXHQFH�� VHTXHQFH�a, b, c, c, d, e

6HTXHQFHV�DUH�0X3$' H[SUHVVLRQV�RI�GRPDLQ�W\SH '20B(;35�

,Im DQG n DUH�LQWHJHUV� WKH�FDOO REMHFW�L��L P��Q JHQHUDWHV�WKH�VHTXHQFH

object(m), object(m+ 1), . . . , object(n) :

LA� � L ����� [AL � L ����4, 9, 16, 25, 36, 49, x, x2, x3, x4, x5

7KH�RSHUDWRU � LV�FDOOHG�WKH VHTXHQFH�JHQHUDWRU� 7KH�HTXLYDOHQW�IXQFWLRQDO�IRUP�LVBVHTJHQ�REMHFW�L��L�P��Q�BVHTJHQ�LA�� L� ������ BVHTJHQ�[AL� L� �����

4, 9, 16, 25, 36, 49, x, x2, x3, x4, x5

8VXDOO\� \RX�ZLOO�SUHIHU�WKH�RSHUDWRU�QRWDWLRQ� 7KH�IXQFWLRQDO�IRUP�LV�XVHIXO�LQFRQQHFWLRQ�ZLWK PDS� ]LS RU�VLPLODU�IXQFWLRQV�

<RX�PD\�XVH � LQ�WKH�IROORZLQJ�ZD\�WR�JHQHUDWH�D�VHTXHQFH�RI�VXFFHVVLYH�LQWHJHUV�

� ������23, 24, 25, 26, 27, 28, 29, 30

����

Page 83: Mupad Tutorial

L^jn^g\^l

7KH�FRPPDQG REMHFW�Q UHWXUQV�D�VHTXHQFH�RI n LGHQWLFDO�REMHFWV�

[A� � ��x2, x2, x2, x2, x2, x2, x2, x2, x2, x2

<RX�FDQ�DOVR�XVH�WKH�VHTXHQFH�JHQHUDWRU�LQ�FRQQHFWLRQ�ZLWK�WKH�NH\ZRUG LQ� �7KHIXQFWLRQDO�HTXLYDOHQW�LQ�WKLV�FDVH�LQ BVHTLQ�� 7KH�ORRS�YDULDEOH�WKHQ�UXQV�WKURXJKDOO�RSHUDQGV�RI�WKH�VWDWHG�REMHFW�

I�[� � [ LQ >D� E� F� G@f(a) , f(b) , f(c) , f(d)

I�[� � [ LQ D � E � F � G � VLQ�VTUW����

f(a) , f(b) , f(c) , f(d) , f(VLQ(√

2))

,W�PD\�EH�WHPSWLQJ�WR�DEXVH�WKH � RSHUDWRU�IRU�ORRSV� DV�LQ�WKHVH�FRPPDQGV�

�[�L � VLQ�L�� \�L � [�L� � L �����

)RU�UHDGDELOLW\� PHPRU\�FRQVXPSWLRQ�DQG�WKHUHIRUH�VSHHG� ZH�UHFRPPHQGDJDLQVW�WKLV�XVDJH� 7KH�DVVLJQPHQWV�DERYH�VKRXOG�XVH�D�ORRS�LQVWHDG��FI�&KDSWHU ����

IRU L IURP � WR � GR[�L � VLQ�L��\�L � [�L�

HQGBIRU�

$V�D�VLPSOH�DSSOLFDWLRQ�RI�VHTXHQFHV� ZH�QRZ�FRQVLGHU�WKH�0X3$' GLIIHUHQWLDWRUGLII� 7KH�FDOO GLII�I�[��[� UHWXUQV�WKH�GHULYDWLYH�RI f ZLWK�UHVSHFW�WR x� +LJKHUGHULYDWLYHV�DUH�JLYHQ�E\ GLII�I�[��[�[�� GLII�I�[��[�[�[� HWF� 7KXV� WKH10�WK�GHULYDWLYH�RI f(x) = VLQ(x2) FDQ�EH�FRPSXWHG�FRQYHQLHQWO\�E\�PHDQV�RI�WKHVHTXHQFH�JHQHUDWRU�

GLII�VLQ�[A��� [ � ���� � �

����� FRV�[ � � ������ [ FRV�[ � �

� � � ������ [ FRV�[ � � ������ [ VLQ�[ � �

� � �� ������� [ VLQ�[ � � ���� [ VLQ�[ �

����

Page 84: Mupad Tutorial

- FnI:=� H[c^\ml

7KH�³YRLG´�REMHFW��6HFWLRQ ������PD\�EH�UHJDUGHG�DV�DQ�HPSW\�VHTXHQFH� <RX�PD\JHQHUDWH�LW�E\�FDOOLQJ QXOO�� RU BH[SUVHT��� 7KH V\VWHP�DXWRPDWLFDOO\�HOLPLQDWHVWKLV�REMHFW�IURP�VHTXHQFHV�

6HT � QXOO��� 6HT � 6HT� D� E� QXOO��� Fa, b, c

6RPH�V\VWHP�IXQFWLRQV�VXFK�DV�WKH SULQW FRPPDQG�IRU�VFUHHQ�RXWSXW�6HFWLRQ ������UHWXUQ�WKH QXOO�� REMHFW�

VHTXHQFH � D� E� SULQW�+HOOR�� F+HOOR

a, b, c

<RX�FDQ�DFFHVV�WKH i�WK�HQWU\�RI�D�VHTXHQFH�E\ VHTXHQFH>L@� 5HGHILQLWLRQV�RI�WKHIRUP VHTXHQFH>L@� QHZYDOXH DUH�DOVR�SRVVLEOH�

) � D� E� F� )>�@b

)>�@ � QHZYDOXH� )a, QHZYDOXH, c

$OWHUQDWLYHO\� \RX�PD\�XVH�WKH�RSHUDQG�IXQFWLRQ RS �6HFWLRQ �����WR�DFFHVVVXEVHTXHQFHV��

) � D� E� F� G� H� RS�)� ��� RS�)� �����b

b, c, d

�1RWH�WKDW� LQ�WKLV�H[DPSOH� WKH�LGHQWLILHU ) RI�WKH�VHTXHQFH�LV�SURYLGHG�DV�DUJXPHQW�WR RS� 7KH RSIXQFWLRQ�UHJDUGV�D�GLUHFW�FDOO�RI�WKH�IRUP RS�D�E�F�G�H��� DV�DQ��LQYDOLG��FDOO�ZLWK�VL[�DUJXPHQWV�DQGLVVXHV�DQ�HUURU�PHVVDJH� <RX�PD\�XVH�SDUHQWKHVHV�WR�DYRLG�WKLV�HUURU� RS��D�E�F�G�H�����

����

Page 85: Mupad Tutorial

L^jn^g\^l

<RX�PD\�XVH GHOHWH WR�GHOHWH�HQWULHV�IURP�D�VHTXHQFH� WKXV�VKRUWHQLQJ�WKHVHTXHQFH�

)� GHOHWH )>�@� )� GHOHWH )>�@� )a, b, c, d, e

a, c, d, e

a, c, e

7KH�PDLQ�XVDJH�RI�0X3$' VHTXHQFHV�LV�WKH�JHQHUDWLRQ�RI�OLVWV�DQG�VHWV�DQGVXSSO\LQJ�DUJXPHQWV�WR�IXQFWLRQ�FDOOV� )RU�H[DPSOH� WKH�IXQFWLRQV PD[ DQG PLQFDQ�FRPSXWH�WKH�PD[LPXP�DQG�PLQLPXP� UHVSHFWLYHO\� RI�DUELWUDULO\�PDQ\DUJXPHQWV�

6HT � �� �� ��� �� �� PD[�6HT�� PLQ�6HT�3, −1

([HUFLVH������ $VVLJQ�WKH�YDOXHV x1 = 1� x2 = 2� . . . � x100 = 100 WR�WKH�LGHQWLILHUVx1, x2, . . . , x100�

([HUFLVH������ *HQHUDWH�WKH�VHTXHQFH

x1, x2, x2︸ ︷︷ ︸2

, x3, x3, x3︸ ︷︷ ︸3

, . . . , x10, x10, . . . , x10︸ ︷︷ ︸10

.

([HUFLVH������ 8VH�D�VLPSOH�FRPPDQG�WR�JHQHUDWH�WKH�GRXEOH�VXP

10∑

i=1

i∑

j=1

1i+ j

.

+LQW� WKH�IXQFWLRQ BSOXV DFFHSWV�DUELWUDULO\�PDQ\�DUJXPHQWV� *HQHUDWH�D�VXLWDEOHDUJXPHQW�VHTXHQFH�

����

Page 86: Mupad Tutorial

- FnI:=� H[c^\ml

Daklk

$ OLVW�LV�DQ�RUGHUHG�VHTXHQFH�RI�DUELWUDU\�0X3$'� REMHFWV�HQFORVHG�LQ�VTXDUHEUDFNHWV�

OLVW � >D� �� VLQ�[�A� � �� >D� E� F@� KHOOR����� ������@

[a, 5, VLQ(x)2 + 4, [a, b, c] ,KHOOR,

34, 3.9087

]

$ OLVW�PD\�FRQWDLQ�OLVWV�DV�HOHPHQWV� ,W�PD\�DOVR�EH�HPSW\�

OLVW � >@[]

7KH�SRVVLELOLW\�WR�JHQHUDWH�VHTXHQFHV�YLD � LV�KHOSIXO�IRU�FRQVWUXFWLQJ�OLVWV�

VHTXHQFH � L � L ������ OLVW � >VHTXHQFH@[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

OLVW � >[AL � L �����@[1, x, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12

]

$ OLVW�PD\�RFFXU�RQ�WKH�OHIW�KDQG�VLGH�RI�DQ�DVVLJQPHQW� 7KLV�PD\�EH�XVHG�WRDVVLJQ�YDOXHV�WR�VHYHUDO�LGHQWLILHUV�VLPXOWDQHRXVO\�

>$� %� &@ � >D� E� F@� $ � %A&a+ bc

$ XVHIXO�SURSHUW\�RI�WKLV�FRQVWUXFWLRQ�LV�WKDW�DOO�WKH�DVVLJQPHQWV�DUH�SHUIRUPHG�DWWKH�VDPH�WLPH� VR�\RX�FDQ�VZDS�YDOXHV�

D � �� E� �� D� E1, 2

>D� E@ � >E� D@� D� E2, 1

����

Page 87: Mupad Tutorial

Eblml

7KH�IXQFWLRQ QRSV UHWXUQV�WKH�QXPEHU�RI�HOHPHQWV�RI�D�OLVW� <RX�FDQ�DFFHVV�WKHHOHPHQWV�RI�D�OLVW�E\�PHDQV�RI�LQGH[HG�DFFHVV� OLVW>L@ UHWXUQV�WKH i�WK�OLVWHOHPHQW� DQG OLVW>L��M@ UHWXUQV�WKH�VXE�OLVW�VWDUWLQJ�DW�WKH i�WK�DQG�H[WHQGLQJ�XSWR�DQG�LQFOXGLQJ�WKH j�WK�HOHPHQW� 1HJDWLYH�YDOXHV�IRU i DQG j DUH�FRXQWHG�IURPWKH�HQG�

OLVW � >D� E� F� G� H� I@� OLVW>�@� OLVW>�����@2, [c, d, e]

$V�ZLWK�DOPRVW�DQ\�0X3$' REMHFW� \RX�FDQ�DOVR�DFFHVV�WKH�HOHPHQWV�RI�D�OLVW�ZLWKWKH RS IXQFWLRQ� RS�OLVW� UHWXUQV�WKH�VHTXHQFH�RI�HOHPHQWV� L�H�� WKH�VHTXHQFHWKDW�KDV�EHHQ�XVHG�WR�FRQVWUXFW�WKH�OLVW�E\�HQFORVLQJ�LW�LQ�VTXDUH�EUDFNHWV >@� 7KHFDOO RS�OLVW�L� UHWXUQV�WKH i�WK�OLVW�HOHPHQW� DQG RS�OLVW�L��M� H[WUDFWV�WKHVHTXHQFH�RI�WKH i�WK�XS�WR�WKH j�WK�OLVW�HOHPHQW�

GHOHWH D� E� F� OLVW � >D� E� VLQ�[�� F@� RS�OLVW�a, b, VLQ(x) , c

RS�OLVW� �����b, VLQ(x)

<RX�PD\�FKDQJH�D�OLVW�HOHPHQW�E\�DQ�LQGH[HG�DVVLJQPHQW�

OLVW � >D� E� F@� OLVW>�@ � QHZYDOXH� OLVW[QHZYDOXH, b, c]

:ULWLQJ�WR�D�VXEOLVW�PD\�FKDQJH�WKH�OHQJWK�RI�WKH�OLVW�

OLVW>����@ � >G�H�I�J@� OLVW[QHZYDOXH, d, e, f, g]

$OWHUQDWLYHO\� WKH�FRPPDQG VXEVRS�OLVW�L QHZYDOXH� �&KDSWHU ���UHWXUQV�DFRS\�ZLWK�WKH i�WK�RSHUDQG�UHGHILQHG�

OLVW � >D� E� F@� OLVW� � VXEVRS�OLVW� � QHZYDOXH�[QHZYDOXH, b, c]

����

Page 88: Mupad Tutorial

- FnI:=� H[c^\ml

&DXWLRQ� ,I / LV�DQ�LGHQWLILHU�ZLWKRXW�D�YDOXH� WKHQ�WKH�LQGH[HG�DVVLJQPHQW

/>LQGH[@ � YDOXH�

JHQHUDWHV�D�WDEOH��6HFWLRQ �����DQG�QRW�D�OLVW�

GHOHWH /� />�@ � D� /

1 a

<RX�FDQ�UHPRYH�HOHPHQWV�IURP�D�OLVW�E\�XVLQJ GHOHWH� 7KLV�VKRUWHQV�WKH�OLVW�

OLVW � >D� E� F@� GHOHWH OLVW>�@� OLVW[b, c]

7KH�IXQFWLRQ FRQWDLQV FKHFNV�ZKHWKHU�D�0X3$' REMHFW�EHORQJV�WR�D�OLVW� ,WUHWXUQV�WKH�LQGH[�RI��WKH�ILUVW�RFFXUUHQFH�RI��WKH�HOHPHQW�LQ�WKH�OLVW� ,I�WKH�OLVW�GRHVQRW�FRQWDLQ�WKH�HOHPHQW� WKHQ FRQWDLQV UHWXUQV�WKH�LQWHJHU 0�

FRQWDLQV�>[ � �� D� [ � �@� [ � ��1

FRQWDLQV�>VLQ�D�� E� F@� D�0

7KH�IXQFWLRQ DSSHQG DGGV�HOHPHQWV�WR�WKH�WDLO�RI�D�OLVW�OLVW � >D� E� F@� DSSHQG�OLVW� �� �� ��

[a, b, c, 3, 4, 5]

7KH�GRW�RSHUDWRU � FRQFDWHQDWHV�OLVWV�

OLVW� � >�� �� �@� OLVW� � >�� �� �@�OLVW��OLVW�� OLVW��OLVW�

[1, 2, 3, 4, 5, 6] , [4, 5, 6, 1, 2, 3]

7KH�FRUUHVSRQGLQJ�V\VWHP�IXQFWLRQ�LV BFRQFDW DQG�DFFHSWV�DUELWUDULO\�PDQ\DUJXPHQWV� <RX�FDQ�XVH�LW�WR�FRPELQH�PDQ\�OLVWV�

BFRQFDW�OLVW� � ��[1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3]

����

Page 89: Mupad Tutorial

Eblml

$ OLVW�FDQ�EH�VRUWHG�E\�PHDQV�RI�WKH�IXQFWLRQ VRUW� 7KLV�DUUDQJHV�QXPHULFDO�YDOXHVDFFRUGLQJ�WR�WKHLU�PDJQLWXGH� VWULQJV��6HFWLRQ ������DUH�VRUWHG�OH[LFRJUDSKLFDOO\�

VRUW�>������ �� �� �� ���@�[−1.23,

12, 2, 3, 4

]

VRUW�>�$�� �E�� �D�� �F�� �&�� �F�� �%�� �D��� �DEF�@�[´$´, ´%´, ´&´, ´D´, ´D�´, ´DEF´, ´E´, ´F´, ´F´]

VRUW�>�[������� �[������� �[������@�[´[�����´, ´[�����´, ´[�����´]

1RWH�WKDW�WKH�OH[LFRJUDSKLFDO�RUGHU�RQO\�DSSOLHV�WR�VWULQJV�JHQHUDWHG�ZLWK ��1DPHV�RI�LGHQWLILHUV�DUH�VRUWHG�DFFRUGLQJ�WR�GLIIHUHQW��LQWHUQDO��UXOHV� ZKLFK�RQO\UHGXFH�WR�OH[LFRJUDSKLF�RUGHU�IRU�VKRUW�LGHQWLILHUV��DQG�PD\�FKDQJH�LQ�IXWXUHUHOHDVHV��

GHOHWH $� %� &� D� E� F� D�� DEF�VRUW�>$� E� D� F� &� F� %� D�� DEF@�

[A,B,C, a, D�, DEF, b, c, c]

VRUW�>[������ [������ [�����@�[[�����, [�����, [�����]

0X3$' UHJDUGV�OLVWV�RI�IXQFWLRQ�QDPHV�DV�OLVW�YDOXHG�IXQFWLRQV�

>VLQ� FRV� I@�[�[VLQ(x) , FRV(x) , f(x)]

7KH�IXQFWLRQ PDS DSSOLHV�D�IXQFWLRQ�WR�DOO�HOHPHQWV�RI�D�OLVW�

PDS�>[� �� �� 3,� ���@� VLQ�[VLQ(x) , VLQ(1) , 0, 0, 0.2955202067]

����

Page 90: Mupad Tutorial

- FnI:=� H[c^\ml

,I�WKH�IXQFWLRQ�KDV�PRUH�WKDQ�RQH�DUJXPHQW� WKHQ PDS VXEVWLWXWHV�WKH�OLVW�HOHPHQWVIRU�WKH�ILUVW�DUJXPHQW�DQG�WDNHV�WKH�UHPDLQLQJ�DUJXPHQWV�IURP�LWV�RZQ�DUJXPHQWOLVW�

PDS�>D� E� F@� I� \� ]�[f(a, y, z) , f(b, y, z) , f(c, y, z)]

7KLV PDS FRQVWUXFWLRQ�LV�D�SRZHUIXO�WRRO�IRU�KDQGOLQJ�OLVWV�DV�ZHOO�DV�RWKHU�0X3$'REMHFWV� ,Q�WKH�IROORZLQJ�H[DPSOH� ZH�KDYH�D�QHVWHG�OLVW L� :H�ZDQW�WR�H[WUDFW�WKHILUVW�HOHPHQWV�RI�WKH�VXEOLVWV�XVLQJ RS�·���� 7KLV�LV�HDVLO\�GRQH�XVLQJ PDS�

/ � >>D�� E�@� >D�� E�@� >D�� E�@@� PDS�/� RS� ��[D�, D�, D�]

7KH�0X3$' IXQFWLRQ VHOHFW HQDEOHV�\RX�WR�H[WUDFW�HOHPHQWV�ZLWK�D�FHUWDLQSURSHUW\�IURP�D�OLVW� 7R�WKLV�HQG� \RX�QHHG�D�IXQFWLRQ�WKDW�FKHFNV�ZKHWKHU�DQREMHFW�KDV�WKLV�SURSHUW\�DQG�UHWXUQV 758( RU )$/6(��)RU�H[DPSOH� WKH�FDOOKDV�REMHFW��REMHFW�� UHWXUQV 758( LI REMHFW� LV�DQ�RSHUDQG�RU�VXERSHUDQG�RIREMHFW�� DQG )$/6( RWKHUZLVH�KDV�� � VLQ�� � [�� [�� KDV�� � VLQ�� � [�� \�

758(, )$/6(

1RZ�

VHOHFW�>D � �� [� \� ]� VLQ�D�@� KDV� D�[a+ 2, VLQ(a)]

H[WUDFWV�DOO�WKRVH�OLVW�HOHPHQWV�IRU�ZKLFK KDV�·�D� UHWXUQV 758(��L�H�� WKRVH�ZKLFKFRQWDLQ�WKH�LGHQWLILHU D�

7KH�IXQFWLRQ VSOLW GLYLGHV�D�OLVW�LQWR�WKUHH�OLVWV� DV�IROORZV� 7KH�ILUVW�OLVW�FRQWDLQVDOO�HOHPHQWV�ZLWK�D�FHUWDLQ�SURSHUW\� WKH�VHFRQG�OLVW�FROOHFWV�DOO�WKRVH�HOHPHQWVZLWKRXW�WKH�SURSHUW\� ,I�WKH�WHVW�IRU�WKH�SURSHUW\�UHWXUQV�WKH�YDOXH 81.12:1 IRUVRPH�HOHPHQWV� WKHQ WKHVH�DUH�SXW�LQWR�WKH�WKLUG�OLVW� 2WKHUZLVH� WKH�WKLUG�OLVW�LVHPSW\�

VSOLW�>VLQ�[�� [A�� \� ��@� KDV� [�[[VLQ(x) , x2

], [y, 11] , []

]

����

Page 91: Mupad Tutorial

Eblml

7KH�0X3$' IXQFWLRQ ]LS FRPELQHV�HOHPHQWV�RI�WZR�OLVWV�SDLUZLVH�LQWR�D�QHZ�OLVW�

/� � >D� E� F@� /� � >G� H� I@�]LS�/�� /�� BSOXV�� ]LS�/�� /�� BPXOW��]LS�/�� /�� BSRZHU�

[a+ d, b+ e, c+ f ] , [a d, b e, c f ] ,[ad, be, cf

]

7KH�WKLUG�DUJXPHQW�RI ]LSPXVW�EH�D�IXQFWLRQ�WKDW�WDNHV�WZR�DUJXPHQWV� 7KLVIXQFWLRQ�LV�WKHQ�DSSOLHG�WR�WKH�SDLUV�RI�OLVW�HOHPHQWV� ,Q�WKH�DERYH�H[DPSOH� ZHKDYH�XVHG�WKH�0X3$' IXQFWLRQV BSOXV� BPXOW� DQG BSRZHU IRU�DGGLWLRQ�PXOWLSOLFDWLRQ� DQG�H[SRQHQWLDWLRQ� UHVSHFWLYHO\� ,I�WKH�WZR�LQSXW�OLVWV�KDYHGLIIHUHQW�OHQJWKV� WKHQ�WKH�EHKDYLRU�RI ]LS GHSHQGV�RQ�WKH�RSWLRQDO�IRXUWKDUJXPHQW� ,I�WKLV�LV�QRW�SUHVHQW� WKHQ�WKH�OHQJWK�RI�WKH�UHVXOWLQJ�OLVW�LV�WKHPLQLPXP�RI�WKH�OHQJWKV�RI�WKH�WZR�LQSXW�OLVWV� 2WKHUZLVH� LI�\RX�VXSSO\�DQDGGLWLRQDO�IRXUWK�DUJXPHQW� WKHQ ]LS UHSODFHV�WKH�³PLVVLQJ´�OLVW�HQWULHV�E\�WKLVDUJXPHQW�

/� � >D� E� F� �� �@� /� � >G� H� I@�]LS�/�� /�� BSOXV�

[a+ d, b+ e, c+ f ]

]LS�/�� /�� BSOXV� KHOOR�[a+ d, b+ e, c+ f,KHOOR+ 1,KHOOR+ 2]

7DEOH ��� JLYHV�D�VXPPDU\�RI�DOO�OLVW�RSHUDWLRQV�WKDW�ZH�KDYH�GLVFXVVHG�

([HUFLVH������ *HQHUDWH�WZR�OLVWV�ZLWK�WKH�HOHPHQWV a, b, c, d DQG 1, 2, 3, 4�UHVSHFWLYHO\� &RQFDWHQDWH�WKH�OLVWV� 0XOWLSO\�WKH�OLVWV�SDLUZLVH�

([HUFLVH������ 0XOWLSO\�DOO�HQWULHV�RI�WKH�OLVW >��[��@ E\ 2� 6XSSRVH�\RX�DUHJLYHQ�D�OLVW� ZKRVH�HOHPHQWV�DUH�OLVWV�RI�QXPEHUV�RU�H[SUHVVLRQV� VXFK�DV>>��[��@�>3,@�>�����@@� KRZ�FDQ�\RX�PXOWLSO\�DOO�HQWULHV�E\ 2"

����

Page 92: Mupad Tutorial

- FnI:=� H[c^\ml

� RU BFRQFDW � FRQFDWHQDWLQJ�OLVWVDSSHQG � DSSHQGLQJ�HOHPHQWVFRQWDLQV�OLVW�[� � GRHV OLVW FRQWDLQ�WKH�HOHPHQW ["OLVW>L@ � DFFHVVLQJ�WKH i�WK�HOHPHQWPDS � DSSO\LQJ�D�IXQFWLRQQRSV � OHQJWKRS � DFFHVVLQJ�HOHPHQWVVHOHFW � VHOHFW�DFFRUGLQJ�WR�SURSHUWLHVVRUW � VRUWLQJVSOLW � VSOLW�DFFRUGLQJ�WR�SURSHUWLHVVXEVRS � UHSODFLQJ�HOHPHQWVGHOHWH � GHOHWLQJ�HOHPHQWV]LS � FRPELQLQJ�WZR�OLVWV

7DEOH ���� 0X3$'� IXQFWLRQV�DQG�RSHUDWRUV�IRU�OLVWV

([HUFLVH������ /HWX = [x1, . . . , xn] DQG Y = [y1, . . . , yn] EH�WZR�OLVWV�RI�WKH�VDPHOHQJWK� )LQG�D�VLPSOH�PHWKRG�WR�FRPSXWH

� WKHLU�³LQQHU�SURGXFW´��X DV�URZ�YHFWRU�DQG Y DV�FROXPQ�YHFWRU�

x1 y1 + · · ·+ xn yn,

� WKHLU�³PDWUL[�SURGXFW´��X DV�FROXPQ�YHFWRU�DQG Y DV�URZ�YHFWRU�

[ [x1 y1, x1 y2, . . . , x1 yn], [x2 y1, x2 y2, . . . , x2 yn],. . . , [xn y1, xn y2, . . . , xn yn]] .

<RX�FDQ�DFKLHYH�WKLV�E\�XVLQJ ]LS� BSOXV� PDS DQG�DSSURSULDWH�IXQFWLRQV�6HFWLRQ ������ZLWKLQ�D�VLQJOH�FRPPDQG�OLQH�LQ�HDFK�FDVH� /RRSV��&KDSWHU ����DUHQRW�UHTXLUHG�

([HUFLVH������ ,Q�QXPEHU�WKHRU\� RQH�LV�RIWHQ�LQWHUHVWHG�LQ�WKH�GHQVLW\�RI�SULPHQXPEHUV�LQ�VHTXHQFHV�RI�WKH�IRUP f(1), f(2), . . . � ZKHUH f LV�D�SRO\QRPLDO� )RUHDFK�YDOXH�RIm = 0, 1, . . . , 41� ILQG�RXW�KRZ�PDQ\�RI�WKH�LQWHJHUV n2 + n+m ZLWKn = 1, 2, . . . , 100 DUH�SULPHV�

����

Page 93: Mupad Tutorial

Eblml

([HUFLVH������ ,Q�ZKLFK�RUGHULQJ�ZLOO n FKLOGUHQ�EH�VHOHFWHG�IRU�³UHPRYDO´�E\�DFRXQWLQJ�RXW�UK\PH�FRPSRVHG�RIm ZRUGV" )RU�H[DPSOH� XVLQJ�WKH�UK\PH

³HHQLH±PHHQLH±PLQH\±PRH±FDWFK�D±WLJHU±E\�WKH±WRH´

ZLWK 8 ³ZRUGV�´ 12 FKLOGUHQ�DUH�FRXQWHG�RXW�LQ�WKH�RUGHU�±�±�±��±��±��±�±�±�±�±�±�� +LQW� UHSUHVHQW�WKH�FKLOGUHQ�E\�D�OLVW >����«@DQG�UHPRYH�DQ�HOHPHQW�IURP�WKLV�OLVW�DIWHU�LW�LV�FRXQWHG�RXW�

����

Page 94: Mupad Tutorial

- FnI:=� H[c^\ml

K]lk

$ VHW LV�DQ�XQRUGHUHG�VHTXHQFH�RI�DUELWUDU\�REMHFWV�HQFORVHG�LQ�FXUO\�EUDFHV� 6HWVDUH�RI�GRPDLQ�W\SH '20B6(7�

^��� �� ��� [� ��� �`{−9, 1, 8, 34, 89, x}

7KH�RUGHU�RI�WKH�HOHPHQWV�LQ�D�0X3$'� OLVW�VHHPV�WR�EH�UDQGRP� 7KH�0X3$'NHUQHO�VRUWV�WKH�HOHPHQWV�DFFRUGLQJ�WR�LQWHUQDO�SULQFLSOHV� <RX�VKRXOG�XVH�VHWVRQO\�LI�WKH�RUGHU�RI�WKH�HOHPHQWV�GRHV�QRW�PDWWHU� ,I�\RX�ZDQW�WR�SURFHVV�D�VHTXHQFHRI�H[SUHVVLRQV�LQ�D�FHUWDLQ�RUGHU� XVH�OLVWV�DV�GLVFXVVHG�LQ�WKH�SUHYLRXV�VHFWLRQ�

6HWV�PD\�EH�HPSW\�

HPSW\VHW � ^`∅

$ VHW�FRQWDLQV�HDFK�HOHPHQW�RQO\�RQFH� L�H�� GXSOLFDWH�HOHPHQWV�DUH�UHPRYHGDXWRPDWLFDOO\�

VHW � ^D� �� �� �� �� D� E� �� �� D`{1, 2, 3, 4, a, b}

7KH�IXQFWLRQ QRSV GHWHUPLQHV�WKH�QXPEHU�RI�HOHPHQWV�LQ�D�VHW� $V�IRU�VHTXHQFHVDQG�OLVWV� RS H[WUDFWV�HOHPHQWV�IURP�D�VHW�RS�VHW�a, 1, 2, 3, 4, b

RS�VHW� �����1, 2, 3

:DUQLQJ� 6LQFH�HOHPHQWV�RI�D�VHW�PD\�EH�UHRUGHUHG�LQWHUQDOO\� \RX�VKRXOG�FKHFNFDUHIXOO\�ZKHWKHU�LW�PDNHV�VHQVH�WR�DFFHVV�WKH i�WK�HOHPHQW� )RU�H[DPSOH�VXEVRS�VHW�L QHZYDOXH� �6HFWLRQ ���UHSODFHV�WKH i�WK�HOHPHQW�E\�D�QHZ�YDOXH�+RZHYHU� \RX�VKRXOG�FKHFN�LQ�DGYDQFH��XVLQJ RS��WKDW�WKH�HOHPHQW�WKDW�\RX�ZDQWWR�UHSODFH�UHDOO\�LV�WKH i�WK�HOHPHQW� (VSHFLDOO\�QRWH�WKDW�UHSODFLQJ�DQ�HOHPHQW�RI�DVHW�RIWHQ�UHRUGHUV�RWKHU�HQWULHV�

����

Page 95: Mupad Tutorial

L^ml

7KH�FRPPDQG RS�VHW�L� UHWXUQV�WKH i�WK�HOHPHQW�RI VHW LQ�WKH�LQWHUQDO�RUGHU�ZKLFK�XVXDOO\�LV�GLIIHUHQW�IURP�WKH i�WK�HOHPHQW�RI VHW WKDW�\RX�VHH�RQ�WKH�VFUHHQ�+RZHYHU� \RX�FDQ�DFFHVV�HOHPHQWV�E\�XVLQJ VHW>L@� ZKHUH�WKH�UHWXUQHG�HOHPHQWVLV�WKH i�WK�HOHPHQW�DV�SULQWHG�RQ�WKH�VFUHHQ�

7KH�IXQFWLRQV XQLRQ� LQWHUVHFW� DQG PLQXV IRUP�WKH�XQLRQ� WKH�LQWHUVHFWLRQ� DQGWKH�VHW�WKHRUHWLF�GLIIHUHQFH� UHVSHFWLYHO\� RI�VHWV�

0� � ^�� �� �� D� E`� 0� � ^D� E� F� �� �`�0� XQLRQ 0�� 0� LQWHUVHFW 0�� 0� PLQXV 0�� 0� PLQXV 0�{1, 2, 3, 4, 5, a, b, c} , {a, b} , {1, 2, 3} , {4, 5, c}

,Q�SDUWLFXODU� \RX�FDQ�XVH PLQXV WR�UHPRYH�HOHPHQWV�IURP�D�VHW�

^�� �� �� D� E` PLQXV ^�� D`{1, 2, b}

<RX�FDQ�DOVR�UHSODFH�DQ�HOHPHQW�E\�D�QHZ�YDOXH�ZLWKRXW�FDULQJ�DERXW�WKH�RUGHU�RIWKH�HOHPHQWV�

VHW � ^D� E� ROGYDOXH� F� G`{a, b, c, d, ROGYDOXH}

VHW PLQXV ^ROGYDOXH` XQLRQ ^QHZYDOXH`{a, b, c, d,QHZYDOXH}

7KH�IXQFWLRQ FRQWDLQV FKHFNV�ZKHWKHU�DQ�HOHPHQW�EHORQJV�WR�D�VHW� DQG�UHWXUQVHLWKHU 758( RU )$/6(��

FRQWDLQV�^D� E� F`� D�� FRQWDLQV�^D� E� F � G`� F�758(, )$/6(

0X3$' UHJDUGV�VHWV�RI�IXQFWLRQ�QDPHV�DV�VHW�YDOXHG�IXQFWLRQV�

^VLQ� FRV� I`�[�{FRV(x) , f(x) , VLQ(x)}

�1RWH�WKH�GLIIHUHQFH�WR�WKH�EHKDYLRU�RI FRQWDLQV IRU�OLVWV� WKHUH�WKH�RUGHULQJ�RI�WKH�HOHPHQWV�LVGHWHUPLQHG�ZKHQ�\RX�JHQHUDWH�WKH�OLVW� DQG FRQWDLQV UHWXUQV�WKH�SRVLWLRQ�RI�WKH�HOHPHQW�LQ�WKH�OLVW�

����

Page 96: Mupad Tutorial

- FnI:=� H[c^\ml

FRQWDLQV�0�[� � GRHV 0 FRQWDLQ�WKH�HOHPHQW ["LQWHUVHFW � LQWHUVHFWLRQPDS � DSSO\LQJ�D�IXQFWLRQPLQXV � VHW�WKHRUHWLF�GLIIHUHQFHQRSV � QXPEHU�RI�HOHPHQWVRS � DFFHVVLQJ�HOHPHQWVVHOHFW � VHOHFW�DFFRUGLQJ�WR�SURSHUWLHVVSOLW � VSOLW�DFFRUGLQJ�WR�SURSHUWLHVVXEVRS � UHSODFLQJ�HOHPHQWVXQLRQ � VHW�WKHRUHWLF�XQLRQ

7DEOH ���� 0X3$'� IXQFWLRQV�DQG�RSHUDWRUV�IRU�VHWV

<RX�FDQ�DSSO\�D�IXQFWLRQ�WR�DOO�HOHPHQWV�RI�D�VHW�E\�PHDQV�RI PDS�

PDS�^[� �� �� 3,� ���`� VLQ�{0, 0.2955202067, VLQ(1) , VLQ(x)}

<RX�FDQ�XVH�WKH�IXQFWLRQ VHOHFW WR H[WUDFW�HOHPHQWV�ZLWK�D�FHUWDLQ�SURSHUW\�IURPD�VHW� 7KLV�ZRUNV�DV�IRU�OLVWV� EXW�WKH�UHWXUQHG�REMHFW�LV�D�VHW�

VHOHFW�^^D� [� E`� ^D`� ^[� �``� FRQWDLQV� [�{{1, x} , {a, b, x}}

6LPLODUO\� \RX�FDQ�XVH�WKH�IXQFWLRQ VSOLW WR�GLYLGH�D�VHW�LQWR�WKUHH�VXEVHWV�RIHOHPHQWV�ZLWK�D�FHUWDLQ�SURSHUW\� HOHPHQWV�ZLWKRXW�WKDW�SURSHUW\� DQG�HOHPHQWVIRU�ZKLFK�WKH�V\VWHP�FDQQRW�GHFLGH�WKLV�DQG�UHWXUQV 81.12:1� 7KH�UHVXOW�LV�D�OLVWFRPSULVLQJ�WKHVH�WKUHH�VHWV�

VSOLW�^^D� [� E`� ^D`� ^[� �``� FRQWDLQV� [�[{{1, x} , {a, b, x}} , {{a}} , ∅]

7DEOH ��� FRQWDLQV�D�VXPPDU\�RI�WKH�VHW�RSHUDWLRQV�GLVFXVVHG�VR�IDU�

0X3$' DOVR�SURYLGHV�WKH�GDWD�VWUXFWXUH 'RP��,PDJH6HW IRU�KDQGOLQJ�LQILQLWH�VHWV�VHH�SDJH ��� LQ�&KDSWHU ��

([HUFLVH������ +RZ�FDQ�\RX�FRQYHUW�D�OLVW�WR�D�VHW�DQG�YLFH�YHUVD"

����

Page 97: Mupad Tutorial

L^ml

([HUFLVH������ *HQHUDWH�WKH�VHWV A = {a, b, c}� B = {b, c, d}� DQG C = {b, c, e}�&RPSXWH�WKH�XQLRQ�DQG�WKH�LQWHUVHFWLRQ�RI�WKH�WKUHH�VHWV� DV�ZHOO�DV�WKH�GLIIHUHQFHA \ (B ∪ C)�

([HUFLVH������ ,QVWHDG�RI�WKH�ELQDU\�RSHUDWRUV LQWHUVHFW DQG XQLRQ� \RX�FDQDOVR�XVH�WKH�FRUUHVSRQGLQJ�0X3$' IXQFWLRQV BLQWHUVHFW DQG BXQLRQ WR�FRPSXWHXQLRQV�DQG�LQWHUVHFWLRQV�RI�VHWV� 7KHVH�IXQFWLRQV�DFFHSW�DUELWUDULO\�PDQ\DUJXPHQWV� 8VH�VLPSOH�FRPPDQGV�WR�FRPSXWH�WKH�XQLRQ�DQG�WKH�LQWHUVHFWLRQ�RIDOO�VHWV�EHORQJLQJ�WR 0�

0 � ^^�� �`� ^�� �`� ^�� �`� ^�� �`� ^�� �� �� �``�

����

Page 98: Mupad Tutorial

- FnI:=� H[c^\ml

LYZd]k

$ WDEOH�LV�D�0X3$'� REMHFW�RI�GRPDLQ�W\SH '20B7$%/(� ,W�LV�D�FROOHFWLRQ�RIHTXDWLRQV�RI�WKH�IRUP LQGH[ YDOXH� %RWK�LQGLFHV�DQG�YDOXHV�PD\�EH�DUELWUDU\0X3$' REMHFWV� <RX�FDQ�JHQHUDWH�D�WDEOH�E\�XVLQJ�WKH�V\VWHP�IXQFWLRQ WDEOH�³H[SOLFLW�JHQHUDWLRQ´��

7 � WDEOH�D E� F G�

a b

c d

<RX�FDQ�JHQHUDWH�PRUH�HQWULHV�RU�FKDQJH�H[LVWLQJ�RQHV�E\�³LQGH[HG�DVVLJQPHQWV´RI�WKH�IRUP 7DEOH>LQGH[@� YDOXH�7>I�[�@ � VLQ�[�� 7>�� �@ � ��7>�� �� �@ � ^D� E� F`� 7>D@ � %�7

a B

c d

f(x) VLQ(x)1, 2 5

1, 2, 3 {a, b, c}

,W�LV�QRW�QHFHVVDU\�WR�LQLWLDOL]H�D�WDEOH�YLD WDEOH� ,I 7 LV�DQ�LGHQWLILHU�WKDW�GRHV�QRWKDYH�D�YDOXH� RU�KDV�D�YDOXH�ZKLFK�GRHV�QRW�DOORZ�LQGH[HG�DFFHVV��VXFK�DV�DQLQWHJHU�RU�D�IXQFWLRQ�� WKHQ�DQ�LQGH[HG�DVVLJQPHQW�RI�WKH�IRUP 7>LQGH[@� YDOXHDXWRPDWLFDOO\�WXUQV 7 LQWR�D�WDEOH��³LPSOLFLW�JHQHUDWLRQ´��

GHOHWH 7� 7>D@ � E� 7>E@ � F� 7

a b

b c

����

Page 99: Mupad Tutorial

MZ[e^l

$ WDEOH�PD\�EH�HPSW\�

7 � WDEOH��

<RX�PD\�GHOHWH�WDEOH�HQWULHV�YLD GHOHWH�7DEOH>LQGH[@�

7 � WDEOH�D E� F G� G D F��GHOHWH 7>D@� 7>F@� 7

d a c

<RX�FDQ�DFFHVV�WDEOH�HQWULHV�LQ�WKH�IRUP 7DEOH>LQGH[@� WKLV�UHWXUQV�WKH�HOHPHQWFRUUHVSRQGLQJ�WR�WKH�LQGH[� ,I�WKHUH�LV�QR�HQWU\�IRU�WKH�LQGH[� WKHQ�0X3$' UHWXUQV7DEOH>LQGH[@ V\PEROLFDOO\�

7 � WDEOH�D E� F G� G D F��7>D@� 7>E@� 7>F@� 7>G@b, Tb, d, a c

7KH�FDOO RS�7DEOH� UHWXUQV�DOO�HQWULHV�RI�D�WDEOH� L�H�� WKH�VHTXHQFH�RI�DOO�HTXDWLRQVLQGH[ YDOXH�RS�WDEOH�D $� E %� F &� G '��a = A, b = B, c = C, d = D

1RWH�WKDW�WKH�LQWHUQDO�RUGHU�RI�WKH�WDEOH�HQWULHV�PD\�GLIIHU�ERWK�IURP�WKH�RUGHU�LQZKLFK�\RX�KDYH�JHQHUDWHG�WKH�WDEOH�DQG�IURP�WKH�RUGHU�RI�HQWULHV�SULQWHG�RQ�WKHVFUHHQ� ,W�PD\�ORRN�TXLWH�UDQGRP�

RS�WDEOH�D�L LA� � L �������D�� ���� D� ��� D�� ���� D� ��� D� ��

D�� ���� D�� ���� D� ��� D� ��� D� ��

D�� ���� D�� ���� D�� ���� D� ��� D� ���

D�� ���� D� �

����

Page 100: Mupad Tutorial

- FnI:=� H[c^\ml

7KH�IXQFWLRQ PDS DSSOLHV�D�JLYHQ�IXQFWLRQ�WR�WKH YDOXHV �QRW�WKH LQGLFHV��RI�DOOWDEOH�HQWULHV�

7 � WDEOH�� 3,� � �� � H[S����� PDS�7� IORDW�

1 3.1415926542 4.03 2.718281828

7KH�IXQFWLRQ FRQWDLQV FKHFNV�ZKHWKHU�D�WDEOH�FRQWDLQV�D�SDUWLFXODU LQGH[� ,WLJQRUHV�WKH YDOXHV�

7 � WDEOH�D E�� FRQWDLQV�7� D�� FRQWDLQV�7� E�758(, )$/6(

<RX�PD\�HPSOR\�WKH�IXQFWLRQV VHOHFW DQG VSOLW WR�LQVSHFW ERWK�LQGLFHV�DQGYDOXHV RI�D�WDEOH�DQG�WR�H[WUDFW�WKHP�DFFRUGLQJ�WR�FHUWDLQ�SURSHUWLHV� 7KLV�ZRUNVVLPLODUO\�DV�IRU�OLVWV��6HFWLRQ �����DQG�IRU�VHWV��6HFWLRQ �����

7 � WDEOH�� �QXPEHU�� ��� �QXPEHU�� [ �V\PERO���VHOHFW�7� KDV� �V\PERO��

x ´V\PERO´

VHOHFW�7� KDV� ����

1.0 ´QXPEHU´

VSOLW�7� KDV� �QXPEHU��[

1 ´QXPEHU´

1.0 ´QXPEHU´, x ´V\PERO´ ,

]

7DEOHV�DUH�SDUWLFXODUO\�ZHOO�VXLWHG�IRU�VWRULQJ�ODUJH�DPRXQWV�RI�GDWD� ,QGH[HGDFFHVVHV�WR LQGLYLGXDO HOHPHQWV�DUH�LPSOHPHQWHG�HIILFLHQWO\�DOVR�IRU�ELJ�WDEOHV�D�ZULWH�RU�UHDG�GRHV�QRW�ILOH�WKURXJK�WKH�ZKROH�GDWD�VWUXFWXUH�

����

Page 101: Mupad Tutorial

MZ[e^l

([HUFLVH������ *HQHUDWH�D�WDEOH WHOHSKRQH'LUHFWRU\ ZLWK�WKH�IROORZLQJ�HQWULHV�

)RUG 1815, 5HDJDQ 4711, %XVK 1234, &OLQWRQ 5678.

/RRN�XS�)RUG¶V�QXPEHU� +RZ�FDQ�\RX�ILQG�RXW�ZKRVH�QXPEHU�LV 5678"

([HUFLVH������ *LYHQ�D�WDEOH� KRZ�FDQ�\RX�JHQHUDWH�D�OLVW�RI�DOO�LQGLFHV�DQG�D�OLVWRI�DOO�YDOXHV� UHVSHFWLYHO\"

([HUFLVH������ *HQHUDWH�WKH�WDEOH WDEOH�� ��� ��«�Q Q� DQG�WKH�OLVW>����«�Q@ RI�OHQJWK n = 100000� $GG�D�QHZ�HQWU\�WR�WKH�WDEOH�DQG�WR�WKH�OLVW�+RZ�ORQJ�GRHV�WKLV�WDNH" +LQW� WKH�FDOO WLPH��D� E�� UHWXUQV�WKH�H[HFXWLRQ�WLPHIRU�DQ�DVVLJQPHQW�

����

Page 102: Mupad Tutorial

- FnI:=� H[c^\ml

9jjYqk

$UUD\V� RI�GRPDLQ�W\SH '20B$55$<� PD\�EH�UHJDUGHG�DV�VSHFLDO�WDEOHV� <RX�PD\WKLQN�RI�WKHP�DV�D�FROOHFWLRQ�RI�HTXDWLRQV�RI�WKH�IRUP LQGH[ YDOXH� EXW�LQFRQWUDVW�WR�WDEOHV� WKH�LQGLFHV�PXVW�EH�LQWHJHUV� $ RQH�GLPHQVLRQDO�DUUD\�FRQVLVWVRI�HTXDWLRQV�RI�WKH�IRUP L YDOXH� 0DWKHPDWLFDOO\� LW�UHSUHVHQWV�D�YHFWRU�ZKRVHi�WK�FRPSRQHQW�LV YDOXH� $ WZR�GLPHQVLRQDO�DUUD\�UHSUHVHQWV�D�PDWUL[� ZKRVH(i, j)�WK�FRPSRQHQW�LV�VWRUHG�LQ�WKH�IRUP �L�M� YDOXH� <RX�PD\�JHQHUDWH�DUUD\VRI�DUELWUDU\�GLPHQVLRQ� ZLWK�HQWULHV�RI�WKH�IRUP �L�M�N�«� YDOXH�

7KH�V\VWHP�IXQFWLRQ DUUD\ JHQHUDWHV�DUUD\V� ,Q�LWV�VLPSOHVW�IRUP� \RX�RQO\�VSHFLI\D�VHTXHQFH�RI�UDQJHV�WKDW�GHWHUPLQH�WKH�GLPHQVLRQ�DQG�WKH�VL]H�RI�WKH�DUUD\�

$ � DUUD\������ ������� ��_ ">�� �@� ">�� �@� ">�� �@ __ __ ">�� �@� ">�� �@� ">�� �@ _�� ��

<RX�FDQ�VHH�KHUH�WKDW�WKH�ILUVW�UDQJH ���� DQG�WKH�VHFRQG�UDQJH ���� GHWHUPLQHWKH�DUUD\¶V�QXPEHU�RI�URZV�DQG�FROXPQV� UHVSHFWLYHO\� 7KH�RXWSXW ">���@ VLJQDOVWKDW�WKH�FRUUHVSRQGLQJ�LQGH[�KDV�QRW�EHHQ�DVVLJQHG�D�YDOXH� \HW� 7KXV� WKH�DERYHFRPPDQG�KDV�JHQHUDWHG�DQ�HPSW\�DUUD\� 1RZ� \RX�FDQ�DVVLJQ�YDOXHV�WR�WKHLQGLFHV�

$>�� �@ � �� $>�� �@ � �� $>�� �@ � ��$>�� �@ � +(//2� $(

1 2 3A1,1 A1,2 +(//2

)

<RX�FDQ�DOVR�LQLWLDOL]H�WKH�FRPSOHWH�DUUD\�GLUHFWO\�ZKHQ�JHQHUDWLQJ�LW�E\ DUUD\�-XVW�VXSSO\�WKH�YDOXHV�DV�D��QHVWHG��OLVW�

$ � DUUD\������ ����� >>�� �� �@� >�� �� �@@�(

1 2 34 5 6

)

����

Page 103: Mupad Tutorial

:kkZrl

<RX�FDQ�DFFHVV�DQG�PRGLI\�DUUD\�HOHPHQWV�LQ�WKH�VDPH�ZD\�DV�WDEOH�HOHPHQWV�

$>�� �@ � $>�� �@ � ��� $(

1 2 34 5 16

)

$JDLQ� \RX�PD\�GHOHWH�DQ�DUUD\�HOHPHQW�E\�XVLQJ GHOHWH�

GHOHWH $>�� �@� $>�� �@� $� $>�� �@(A1,1 2 3

4 5 A2,3

), A2,3

$UUD\V�SRVVHVV�D�³0�WK�RSHUDQG´ RS�$UUD\��� SURYLGLQJ�LQIRUPDWLRQ�RQ�WKHGLPHQVLRQ�DQG�WKH�VL]H�RI�WKH�DUUD\� 7KH�FDOO RS�$UUD\��� UHWXUQV�D�VHTXHQFH d�a1.. b1� . . . � ad.. bd� ZKHUH d LV�WKH�GLPHQVLRQ��L�H�� WKH�QXPEHU�RI�LQGLFHV��DQG ai.. biLV�WKH�YDOLG�UDQJH�IRU�WKH i�WK�LQGH[�

9HF � DUUD\������ >[� \� ]@�� RS�9HF� ��1, 1..3

0DW � DUUD\������ ����� >>D� E� F@� >G� H� I@@��RS�0DW� ��

2, 1..2, 1..3

7KXV� WKH�GLPHQVLRQ�RI�DQm× nPDWUL[ DUUD\����P����Q�PD\�EH�REWDLQHG�DV

P RS�0DW�>�����@�� Q RS�0DW�>�����@��

7KH�LQWHUQDO�VWUXFWXUH�RI�DUUD\V�GLIIHUV�IURP�WKH�VWUXFWXUH�RI�WDEOHV� 7KH�HQWULHVDUH�QRW�VWRUHG�LQ�WKH�IRUP�RI�HTXDWLRQV�

RS�0DW�a, b, c, d, e, f

����

Page 104: Mupad Tutorial

- FnI:=� H[c^\ml

7KH�WDEOH�W\SH�LV�PRUH�IOH[LEOH�WKDQ�WKH�DUUD\�W\SH� WDEOHV�DGPLW�DUELWUDU\�LQGLFHV�DQG�WKHLU�VL]H�PD\�JURZ�G\QDPLFDOO\� $UUD\V�DUH�LQWHQGHG�IRU�VWRULQJ�YHFWRUV�DQGPDWULFHV�RI�D�IL[HG�VL]H� :KHQ�\RX�HQWHU�DQ�LQGH[HG�FDOO� WKH�V\VWHP�FKHFNVZKHWKHU�WKH�LQGLFHV�DUH�ZLWKLQ�WKH�VSHFLILHG�UDQJHV� )RU�H[DPSOH�

0DW>�� �@(UURU� ,OOHJDO DUJXPHQW >DUUD\@

<RX�PD\�DSSO\�D�IXQFWLRQ�WR�DOO�DUUD\�FRPSRQHQWV�YLD PDS� )RU�H[DPSOH� KHUH�LVWKH�VLPSOHVW�ZD\�WR�FRQYHUW�DOO�DUUD\�HQWULHV�WR�IORDWLQJ�SRLQW�QXPEHUV�

$ � DUUD\������ >3,� ���@�� PDS�$� IORDW�(

3.141592654 0.1428571429)

:DUQLQJ� ,I 0 LV�DQ�LGHQWLILHU�ZLWKRXW�D�YDOXH� WKHQ�DQ�LQGH[HG�DVVLJQPHQW�RI�WKHIRUP 0>LQGH[�LQGH[����@� YDOXH JHQHUDWHV�D�WDEOH�DQG�QRW�DQ�DUUD\�RI�W\SH'20B$55$< �6HFWLRQ �����0>�� �@ � D� 0

1, 1 a

$GGLWLRQDOO\� 0X3$' SURYLGHV�WKH�PRUH�SRZHUIXO�GDWD�VWUXFWXUHV�RI�GRPDLQ�W\SH'RP��0DWUL[ IRU�KDQGOLQJ�YHFWRUV�DQG�PDWULFHV� 7KHVH�DUH�GLVFXVVHG�LQ6HFWLRQ ����� 6XFK�REMHFWV�DUH�YHU\�FRQYHQLHQW�WR�XVH� \RX�FDQ�PXOWLSO\�WZRPDWULFHV�RU�D�PDWUL[�DQG�D�YHFWRU�E\�PHDQV�RI�WKH�XVXDO�PXOWLSOLFDWLRQ�V\PERO �6LPLODUO\� \RX�FDQ�DGG�PDWULFHV�RI�HTXDO�GLPHQVLRQ�YLD �� 7R�DFKLHYH�WKH�VDPHIXQFWLRQDOLW\�ZLWK�DUUD\V� \RX�KDYH�WR�ZULWH�\RXU�RZQ�SURFHGXUHV� :H�UHIHU�WR�WKHH[DPSOHV 0DWUL[3URGXFW DQG 0DWUL[0XOW LQ�6HFWLRQV ���� DQG ����� UHVSHFWLYHO\�

([HUFLVH������ *HQHUDWH�D�VR�FDOOHG�+LOEHUW�PDWUL[ H RI�GLPHQVLRQ 20× 20 ZLWKHQWULHVHij = 1/(i+ j − 1)�

����

Page 105: Mupad Tutorial

;hhe^Zg�>qik^llbhgl

:ggd]Yf�=phj]kkagfk

0X3$'� LPSOHPHQWV�WKUHH�ORJLFDO��³%RROHDQ´��YDOXHV� 758(� )$/6(��DQG 81.12:1�

GRPW\SH�758(�� GRPW\SH�)$/6(�� GRPW\SH�81.12:1�'20B%22/, '20B%22/, '20B%22/

7KH�RSHUDWRUV DQG� RU� DQG QRW RSHUDWH�RQ�%RROHDQ�YDOXHV�

758( DQG )$/6(� QRW �758( RU )$/6(�� 758( DQG 81.12:1�758( RU 81.12:1

)$/6(, )$/6(, 81.12:1, 758(

7KH�IXQFWLRQV�FRUUHVSRQGLQJ�WR�WKHVH�RSHUDWRUV�DUH BDQG� BRU� DQG BQRW�UHVSHFWLYHO\�

7KH�IXQFWLRQ ERRO HYDOXDWHV�HTXDWLRQV� LQHTXDOLWLHV� RU�FRPSDULVRQV�YLD !� ! � � �� � WR 758( RU )$/6(�ERRO�� ��� ERRO�� �! ���ERRO�� � �� RU QRW ERRO�� ! ��

)$/6(, 758(, 758(

1RWH�WKDW ERRO FDQ�RQO\�FRPSDUH�UHDO�QXPEHUV�VXIILFLHQWO\�GLVWLQFW�IURP�RQHDQRWKHU� (VSHFLDOO\�H[DFW�V\PEROLF�UHSUHVHQWDWLRQV�RI�WKH�VDPH�QXPEHU�FDQQRW�EHFRPSDUHG�ZLWK ERRO�

ERRO�� � 3,�758(

ERRO�VTUW��� VTUW��� � VTUW����(UURU� &DQW HYDOXDWH WR ERROHDQ >BOHHTXDO@

ERRO����������������������� � 3,�(UURU� &DQW HYDOXDWH WR ERROHDQ >BOHHTXDO@

7\SLFDOO\� \RX�ZLOO�XVH�%RROHDQ�FRQVWUXFWV�LQ�EUDQFKLQJ�FRQGLWLRQV�RI LILQVWUXFWLRQV��&KDSWHU ����RU�LQ�WHUPLQDWLRQ�FRQGLWLRQV�RI UHSHDW ORRSV��&KDSWHU���� ,Q�WKH�IROORZLQJ�H[DPSOH� ZH�WHVW�WKH�LQWHJHUV 1, 2, 3 IRU�SULPDOLW\� 7KH�V\VWHPIXQFWLRQ LVSULPH �³LV�WKH�DUJXPHQW�D�SULPH�QXPEHU"´� UHWXUQV 758( RU )$/6(��7KHUHSHDW ORRS�VWRSV�DV�VRRQ�DV�WKH�WHUPLQDWLRQ�FRQGLWLRQ i = 3 HYDOXDWHV�WR 758(�

����

Page 106: Mupad Tutorial

- FnI:=� H[c^\ml

L � ��UHSHDW

L � L � ��LI LVSULPH�L�

WKHQ SULQW�L� �LV D SULPH��HOVH SULQW�L� �LV QR SULPH��

HQGBLIXQWLO L � HQGBUHSHDW

�� �LV QR SULPH�

�� �LV D SULPH�

�� �LV D SULPH�

+HUH�ZH�KDYH�XVHG�VWULQJV�HQFORVHG�LQ � IRU�WKH�VFUHHQ�RXWSXW� 7KH\�DUH�GLVFXVVHGLQ�GHWDLO�LQ�6HFWLRQ ����� 1RWH�WKDW�LW�LV�QRW�QHFHVVDU\�WR�XVH�WKH�IXQFWLRQ ERRO LQEUDQFKLQJ�RU�WHUPLQDWLRQ�FRQGLWLRQV�LQ�RUGHU�WR�HYDOXDWH�WKH�FRQGLWLRQ�WR 758( RU)$/6(�

([HUFLVH������ /HW ∧ GHQRWH�WKH�ORJLFDO�³DQG�´ OHW ∨ GHQRWH�WKH�ORJLFDO�³RU�´ OHW ¬GHQRWH�ORJLFDO�QHJDWLRQ� 7R�ZKLFK�%RROHDQ�YDOXH�GRHV

758( ∧ ()$/6( ∨ ¬ ()$/6( ∨ ¬)$/6())

HYDOXDWH"

([HUFLVH������ /HW /��/� EH�WZR�0X3$' OLVWV�RI�HTXDO�OHQJWK� +RZ�FDQ�\RX�ILQGRXW�ZKHWKHU /�>L@�/�>L@ KROGV�WUXH�IRU�DOO�OLVW�HOHPHQWV"

����

Page 107: Mupad Tutorial

Lmkbg`l

Kljaf_k

6WULQJV DUH�SLHFHV�RI�WH[W� ZKLFK�PD\�EH�XVHG�IRU�IRUPDWWHG�VFUHHQ�RXWSXW� $ VWULQJLV�D�VHTXHQFH�RI�DUELWUDU\�V\PEROV�HQFORVHG�LQ�³VWULQJ�GHOLPLWHUV´ �� ,WV�GRPDLQW\SH�LV '20B675,1*�

VWULQJ� � �8VH IRU PXOWLSOLFDWLRQ��VWULQJ� � �� ��VWULQJ� � �XVH A IRU H[SRQHQWLDWLRQ��

´8VH� �IRU�PXOWLSOLFDWLRQ´

´� ´

´XVH�Ö�IRU�H[SRQHQWLDWLRQ�´

7KH�FRQFDWHQDWLRQ�RSHUDWRU � FRPELQHV�VWULQJV�

VWULQJ� � VWULQJ��VWULQJ��VWULQJ�´8VH� �IRU�PXOWLSOLFDWLRQ� XVH�Ö�IRU�H[SRQHQWLDWLRQ�´

7KH�GRW�RSHUDWRU�LV�D�VKRUW�IRUP�RI�WKH�0X3$'� IXQFWLRQ BFRQFDW� ZKLFKFRQFDWHQDWHV��DUELWUDULO\�PDQ\��VWULQJV�

BFRQFDW��7KLV LV �� �D VWULQJ�� ����´7KLV�LV�D�VWULQJ�´

7KH�LQGH[�RSHUDWRU >@ H[WUDFWV�WKH�FKDUDFWHUV�IURP�D�VWULQJ�

VWULQJ�>�@� VWULQJ�>�@� VWULQJ�>�@�VWULQJ�>�@� VWULQJ�>�@

´8´, ´V´, ´H´, ´�´, ´ ´

:KHQ�XVLQJ�D�UDQJH�DV�LQGH[� VXEVWULQJV�DUH�UHWXUQHG� QHJDWLYH�LQGLFHV�FRXQW�IURPWKH�HQG�RI�WKH�VWULQJ�

VWULQJ�>������@� VWULQJ�>�������@´FDW´, ´H[SRQHQW´

<RX�PD\�XVH�WKH�FRPPDQG SULQW WR�RXWSXW�LQWHUPHGLDWH�UHVXOWV�LQ�ORRSV�RUSURFHGXUHV�RQ�WKH�VFUHHQ��6HFWLRQ ������ %\�GHIDXOW� WKLV�IXQFWLRQ�SULQWV�VWULQJV

����

Page 108: Mupad Tutorial

- FnI:=� H[c^\ml

ZLWK�WKH�HQFORVLQJ�GRXEOH�TXRWHV� <RX�PD\�FKDQJH�WKLV�EHKDYLRU�E\�XVLQJ�WKHRSWLRQ 8QTXRWHG�

SULQW�VWULQJ���8VH IRU PXOWLSOLFDWLRQ� XVH A IRU H[SRQHQWLDWLRQ��

SULQW�8QTXRWHG� VWULQJ��8VH IRU PXOWLSOLFDWLRQ� XVH A IRU H[SRQHQWLDWLRQ�

6WULQJV�DUH�QRW�YDOLG�LGHQWLILHUV�LQ�0X3$'��VR�\RX�FDQQRW�DVVLJQ�YDOXHV�WR�WKHP�

�QDPH� � VLQ�[�(UURU� ,QYDOLG OHIW�KDQG VLGH LQ DVVLJQPHQW >OLQH �� FRO �@

$OVR� DULWKPHWLF�ZLWK�VWULQJV�LV�QRW�DOORZHG�

� � �[�(UURU� ,OOHJDO RSHUDQG >BSOXV@

+RZHYHU� \RX�PD\�XVH�VWULQJV�LQ�HTXDWLRQV�

�GHULYDWLYH RI VLQ�[�� FRV�[�´GHULYDWLYH�RI�VLQ�[�´ = FRV(x)

7KH�IXQFWLRQ H[SU�WH[W FRQYHUWV�D�0X3$' REMHFW�WR�D�VWULQJ� <RX�FDQ�HPSOR\�WKLVIXQFWLRQ�WR�FXVWRPL]H�SULQW�FRPPDQGV�

L � ��SULQW�8QTXRWHG� H[SU�WH[W�L��� LV D SULPH���

� LV D SULPH�D � VLQ�[��SULQW�8QTXRWHG� �7KH GHULYDWLYH RI � � H[SU�WH[W�D� �

� LV � � H[SU�WH[W�GLII�D� [��� ����7KH GHULYDWLYH RI VLQ�[� LV FRV�[��

7KH�GRFXPHQWDWLRQ�RI SULQW FRQWDLQV�PRUH�DGYDQFHG�H[DPSOHV�RI�XVHU�GHILQHGRXWSXW� VHH "SULQW�

<RX�ILQG�QXPHURXV�RWKHU�XVHIXO�IXQFWLRQV�IRU�KDQGOLQJ�VWULQJV�LQ�WKH�VWDQGDUGOLEUDU\��6HFWLRQ�³0DQLSXODWLRQ�RI�6WULQJV´�RI�WKH�0X3$' 4XLFN�5HIHUHQFH��DQG�LQWKH�VWULQJ�OLEUDU\ �"VWULQJOLE��

����

Page 109: Mupad Tutorial

Lmkbg`l

([HUFLVH������ 7KH�FRPPDQG DQDPHV�$OO� UHWXUQV�D�VHW�RI�DOO�LGHQWLILHUV�WKDWKDYH�D�YDOXH�LQ�WKH�FXUUHQW�VHVVLRQ� *HQHUDWH�D OH[LFRJUDSKLFDOO\ RUGHUHG�OLVW�RIWKHVH�LGHQWLILHUV�

([HUFLVH������ +RZ�FDQ�\RX�REWDLQ�WKH�³PLUURU�LPDJH´�RI�D�VWULQJ" +LQW� WKHIXQFWLRQ OHQJWK UHWXUQV�WKH�QXPEHU�RI�V\PEROV�LQ�D�VWULQJ�

����

Page 110: Mupad Tutorial

- FnI:=� H[c^\ml

>mf[lagfk

7KH�DUURZ�RSHUDWRUV �! �D�PLQXV�VLJQ�IROORZHG�E\�D�³JUHDWHU�WKDQ´�VLJQ��DQG ��!�WZR�PLQXV�VLJQV�IROORZHG�E\�D�³JUHDWHU�WKDQ´�VLJQ��JHQHUDWH�VLPSOH�REMHFWV�WKDWUHSUHVHQW�PDWKHPDWLFDO�IXQFWLRQV�

I � �[� \� �! [A� � \A�(x, y)→ x2 + y2

7KH�IXQFWLRQ I FDQ�QRZ�EH�FDOOHG�OLNH�DQ\�V\VWHP�IXQFWLRQ� ,W�WDNHV�WZR�DUELWUDU\LQSXW�SDUDPHWHUV��RU�³DUJXPHQWV´��DQG�UHWXUQV�WKH�VXP�RI�WKHLU�VTXDUHV�

I�D� E � ��

(b+ 1)2 + a2

,Q�WKH�IROORZLQJ�H[DPSOH� WKH�UHWXUQ�YDOXH�RI�WKH�IXQFWLRQ�LV�JHQHUDWHG�E\�DQ LIVWDWHPHQW�

DEV9DOXH � [ �! LI [ ! � WKHQ [ HOVH �[ HQGBLI�DEV9DOXH������

2.3

$V�GLVFXVVHG�LQ�6HFWLRQ ���� WKH�RSHUDWRU # JHQHUDWHV�WKH�FRPSRVLWLRQh : x→ f(g(x)) RI�WZR�IXQFWLRQV f DQG g�

I � [ �! ���� � [�� J � [ �! VLQ�[A���K � I#J� K�D�

1VLQ(a2) + 1

<RX�FDQ�GHILQH�D�UHSHDWHG�FRPSRVLWLRQ f(f(f(·))) RI�D�IXQFWLRQ�ZLWK�LWVHOI�E\�XVLQJWKH�LWHUDWLRQ�RSHUDWRU ##�

III � I##�� III�D�1

11a+1+1

+ 1

����

Page 111: Mupad Tutorial

?ng\mbhgl

2I�FRXUVH� WKHVH�FRQVWUXFWLRQV�DOVR�ZRUN�IRU�V\VWHP�IXQFWLRQV� )RU�H[DPSOH� WKHIXQFWLRQ DEV#5H FRPSXWHV�WKH�DEVROXWH�YDOXH�RI�WKH�UHDO�SDUW�RI�D�FRPSOH[QXPEHU�

I � DEV#5H� I��� � � ,�2

,Q�V\PEROLF�FRPSXWDWLRQV� \RX�RIWHQ�KDYH�WKH�FKRLFH�WR�UHSUHVHQW�D�PDWKHPDWLFDOIXQFWLRQ�HLWKHU�DV�DPDS DUJXPHQWV #→ YDOXH RU�DV�DQ H[SUHVVLRQ�

0DS � [ �! � [ FRV�[A���([SUHVVLRQ � � [ FRV�[A���LQW�0DS�[�� [�� LQW�([SUHVVLRQ� [�

VLQ(x2), VLQ

(x2)

,I�\RX�WU\�WR�FRQYHUW�DQ�H[SUHVVLRQ�LQWR�D�IXQFWLRQ�E\�PHDQV�RI�WKH �! RSHUDWRU�\RX�ZLOO�ILQG�WKDW�LW�GRHV�QRW�TXLWH�ZRUN�

K � [ �! ([SUHVVLRQ�K���

2x FRV(x2)

,QVWHDG� XVH�WKH�ORQJ�DUURZ�RSHUDWRU ��!� ZKLFK�HYDOXDWHV�WKH�ULJKW�KDQG�VLGHH[SUHVVLRQ� DV�H[SODLQHG�LQ�&KDSWHU � DQG�UHWXUQV�D�IXQFWLRQ� ZKLFK�\RX�FDQPDQLSXODWH�IXUWKHU�

K � [ ��! ([SUHVVLRQ�x→ 2x FRV

(x2)

Kx→ 2 FRV

(x2)− 4x2 VLQ

(x2)

,QGHHG� K LV�WKH�IXQFWLRQDO�HTXLYDOHQW�RI GLII�([SUHVVLRQ�[��

K�[� GLII�([SUHVVLRQ� [�2 FRV

(x2)− 4x2 VLQ

(x2)

= 2 FRV(x2)− 4x2 VLQ

(x2)

����

Page 112: Mupad Tutorial

- FnI:=� H[c^\ml

0X3$' FDQ�UHSUHVHQW�PDSV�E\�PHDQV�RI IXQFWLRQDO�H[SUHVVLRQV� IXQFWLRQV�DUHFRQVWUXFWHG�IURP�VLPSOH�IXQFWLRQV��VXFK�DV VLQ� FRV� H[S� OQ� LG��E\�PHDQV�RIRSHUDWRUV��VXFK�DV�WKH�FRPSRVLWLRQ�RSHUDWRU # RU�WKH�DULWKPHWLF�RSHUDWRUV �� �HWF��� 1RWH�WKDW�WKH�DULWKPHWLF�RSHUDWRUV�JHQHUDWH�IXQFWLRQV�WKDW�DUH�GHILQHGSRLQWZLVH� ZKLFK�LV�PDWKHPDWLFDOO\�VRXQG� )RU�H[DPSOH� f + g UHSUHVHQWV�WKH�PDSx→ f(x) + g(x)� f · g UHSUHVHQWV�WKH�PDS x→ f(x) · g(x)� HWF��

GHOHWH I� J�D � I � J� E � I J� F � I�J�D�[�� E�[�� F�[�

f(x) + g(x) , f(x) g(x) ,f(x)g(x)

<RX�DUH�DOORZHG�WR�KDYH�QXPHULFDO�YDOXHV�LQ�IXQFWLRQDO�H[SUHVVLRQV� 0X3$'UHJDUGV�WKHP�DV�FRQVWDQW�IXQFWLRQV�ZKLFK�DOZD\V�UHWXUQ�WKH�SDUWLFXODU�YDOXH�

��[�� ����[� \� ]�� 3,�[�1, 0.1, π

D � I � �� E � I ���� F � I � ���� G � I � VTUW����D�[�� E�[�� F�[�� G�[�

f(x) + 1,3 f(x)

4, f(x) + 0.1, f(x) +

√2

,W�LV�DOVR�SRVVLEOH�WR�XVH�IXQFWLRQV�ZLWKRXW�DVVLJQLQJ�WKHP�WR�DQ�LGHQWLILHU�DQG�WRSDVV�WKHP�DV�DUJXPHQWV�WR�RWKHU�IXQFWLRQV� :H�KDYH�XVHG�WKLV�SRVVLELOLW\�EHIRUH�VR�IRU�UHFDSLWXODWLRQ� ZH�MXVW�JLYH�DQRWKHU�VLPSOH�H[DPSOH�

PDS�>�� �� �� �@� L �! LA��[1, 4, 9, 16]

7KH�RSHUDWRU �! LV�XVHIXO�IRU�GHILQLQJ�IXQFWLRQV�ZKRVH�UHWXUQ�YDOXH�FDQ�EHREWDLQHG�E\�VLPSOH�RSHUDWLRQV� )XQFWLRQV�LPSOHPHQWLQJ�PRUH�FRPSOH[DOJRULWKPV�XVXDOO\�UHTXLUH�PDQ\�FRPPDQGV�DQG�DX[LOLDU\�YDULDEOHV�WR�VWRUHLQWHUPHGLDWH�UHVXOWV� ,Q�SULQFLSOH� \RX�FDQ�GHILQH�VXFK�IXQFWLRQV�YLD �! DV�ZHOO�+RZHYHU� WKLV�KDV�WKH�GUDZEDFN�WKDW�\RX�RIWHQ�XVH JOREDO�YDULDEOHV� ,QVWHDG� ZHUHFRPPHQG�WR�GHILQH�D�SURFHGXUH�YLD SURF���EHJLQ�����HQGBSURF� 7KLV�FRQFHSWRI�WKH�0X3$' SURJUDPPLQJ�ODQJXDJH�LV�PXFK�PRUH�IOH[LEOH�DQG�LV�GLVFXVVHG�LQ

����

Page 113: Mupad Tutorial

?ng\mbhgl

PRUH�GHWDLO�LQ�&KDSWHU ���

([HUFLVH������ 'HILQH�WKH�IXQFWLRQV f(x) = x2 DQG g(x) =√x� &RPSXWH

f(f(g(2)) DQG f(f(. . . f︸ ︷︷ ︸100 WLPHV

(x) . . . )) .

([HUFLVH������ 'HILQH�D�IXQFWLRQ�WKDW�UHYHUVHV�WKH�RUGHU�RI�WKH�HOHPHQWV�LQ�D�OLVW�

([HUFLVH������ 7KH &KHE\VKHY�SRO\QRPLDOV DUH�GHILQHG�UHFXUVLYHO\�E\�WKHIROORZLQJ�IRUPXODH�

T0(x) = 1 , T1(x) = x , Tk(x) = 2xTk−1(x) − Tk−2(x) .

&RPSXWH�WKH�YDOXHV�RI T2(x), . . . , T5(x) IRU x = 1/3� x = 0.33� DQG�IRU�D�V\PEROLFDOYDOXH x�

����

Page 114: Mupad Tutorial

- FnI:=� H[c^\ml

K]ja]k�=phYfkagfk

([SUHVVLRQV�VXFK�DV �����[� DGPLW�VHULHV�H[SDQVLRQV�IRU�V\PEROLF�SDUDPHWHUV�7KLV�SDUWLFXODUO\�VLPSOH�H[DPSOH�LV�WKH�VXP�RI�WKH�JHRPHWULF�VHULHV�

11− x = 1 + x+ x2 + x3 + · · ·

7KH�IXQFWLRQ WD\ORU FRPSXWHV�WKH�OHDGLQJ�WHUPV�RI�VXFK�VHULHV�

W � WD\ORU����� � [�� [ �� ��1 + x+ x2 + x3 + x4 + x5 + x6 + x7 + x8 +O

(x9)

7KLV�LV�WKH�7D\ORU�VHULHV�H[SDQVLRQ�RI�WKH�H[SUHVVLRQ�DURXQG�WKH�SRLQW x = 0� DVUHTXHVWHG�E\�WKH�VHFRQG�DUJXPHQW� 0X3$'� KDV�WUXQFDWHG�WKH�LQILQLWH�VHULHVEHIRUH�WKH�WHUP x9 DQG�KDV�FROOHFWHG�WKH�WDLO�LQ�WKH�³ELJ�2K´�WHUP O(x9) �DOVRFDOOHG�WKH�³/DQGDX�V\PERO´�� 7KH��RSWLRQDO��WKLUG�DUJXPHQW�RI WD\ORU FRQWUROVWKH�WUXQFDWLRQ� ,I�LW�LV�QRW�SUHVHQW� WKHQ�0X3$' XVHV�WKH�YDOXH�RI�WKH�HQYLURQPHQWYDULDEOH 25'(5 LQVWHDG� ZKRVH�GHIDXOW�YDOXH�LV 6�

W � WD\ORU����� � [�� [ ��1 + x+ x2 + x3 + x4 + x5 +O

(x6)

7KH�UHVXOWLQJ�VHULHV�ORRNV�OLNH�DQ�RUGLQDU\�VXP�ZLWK�DQ�DGGLWLRQDO 2�·� WHUP�,QWHUQDOO\� KRZHYHU� LW�LV�UHSUHVHQWHG�E\�D�VSHFLDO�GDWD�VWUXFWXUH�RI�GRPDLQ�W\SH6HULHV��3XLVHX[GRPW\SH�W�

6HULHV::3XLVHX[

7KH�ELJ�2K�WHUP�LWVHOI�LV�D�GDWD�VWUXFWXUH�RQ�LWV�RZQ� RI�GRPDLQ�W\SH 2 DQG�ZLWKVSHFLDO�UXOHV�RI�PDQLSXODWLRQ�

� 2�[A�� � 2�[A��� [A� 2�[A���� 2�[A�� 2�[A����GLII�2�[A��� [�O(x2), O(x12), O(x25), O(x2)

7KH�RUGHULQJ�RI�WKH�WHUPV�LQ�D�7D\ORU�VHULHV�LV�IL[HG� SRZHUV�ZLWK�VPDOOHUH[SRQHQWV�SUHFHGH�WKRVH�ZLWK�KLJKHU�H[SRQHQWV� 7KLV�LV�LQ�FRQWUDVW�WR�WKH�RUGHULQJLQ�RUGLQDU\�VXPV� ZKHUH�KLJK�H[SRQHQWV�SUHFHGH�VPDOO�H[SRQHQWV�

����

Page 115: Mupad Tutorial

L^kb^l�>qiZglbhgl

6 � H[SU�W�x5 + x4 + x3 + x2 + x+ 1

+HUH�ZH�KDYH�XVHG�WKH�V\VWHP�IXQFWLRQ H[SU WR�FRQYHUW�WKH�VHULHV�WR�DQ�H[SUHVVLRQRI�GRPDLQ�W\SH '20B(;35��$V�\RX�FDQ�VHH�LQ�WKH�RXWSXW� WKH 2�·� WHUP�KDV�EHHQ�FXWRII�

7KH RS FRPPDQG�DFWV�RQ�VHULHV�LQ�D�QRQ�REYLRXV�ZD\�DQG�VKRXOG�QRW�EH�XVHG�

RS�W�0, 1, 0, 6, [1, 1, 1, 1, 1, 1] , x = 0, 8QGLUHFWHG

7KHUHIRUH� WKH�IXQFWLRQ FRHII LV�SURYLGHG�WR�H[WUDFW�WKH�FRHIILFLHQWV� 7KLV�LV�PRUHLQWXLWLYH�WKDQ RS� 7KH�FDOO FRHII�W�L� UHWXUQV�WKH�FRHIILFLHQW�RI xi�W � WD\ORU�FRV�[A��� [� ���

1− x4

2+x8

24− x

12

720+x16

40320+O

(x20)

FRHII�W� ��� FRHII�W� ��� FRHII�W� ���� FRHII�W� ���

1, 0, − 1720, )$,/

,Q�WKH�SUHYLRXV�H[DPSOH� ZH�KDYH�VXSSOLHG [ DV�VHFRQG�DUJXPHQW�WR�VSHFLI\�WKHSRLQW�RI�H[SDQVLRQ� 7KLV�LV�HTXLYDOHQW�WR [ ��

7KH�XVXDO�DULWKPHWLF�RSHUDWLRQV�DOVR�ZRUN�IRU�VHULHV�

D � WD\ORU�FRV�[�� [� ��� E � WD\ORU�VLQ�[�� [� ���D� E

1− x2

2+O

(x4), x− x

3

6+O

(x5)

D � E� � D E� DA�

1 + x− x2

2− x

3

6+O

(x4), 2x− 4x3

3+O

(x5), 1− x2 +O

(x4)

����

Page 116: Mupad Tutorial

- FnI:=� H[c^\ml

%RWK�WKH�FRPSRVLWLRQ�RSHUDWRU # DQG�WKH�LWHUDWLRQ�RSHUDWRU ## DSSO\�WR�VHULHV�DVZHOO�

D � WD\ORU�VLQ�[�� [� ����E � WD\ORU�DUFVLQ�[�� [� ���� D#Ex+O

(x21)

,I�\RX�WU\�WR�FRPSXWH�WKH�7D\ORU�VHULHV�RI�D�IXQFWLRQ�WKDW�GRHV�QRW�KDYH�RQH� WKHQWD\ORU DERUWV�ZLWK�DQ�HUURU� 7KH�IXQFWLRQ VHULHV FDQ�FRPSXWH�PRUH�JHQHUDOH[SDQVLRQV �/DXUHQW�VHULHV� 3XLVHX[�VHULHV��

WD\ORU�FRV�[��[� [ �� ���(UURU� ��[ FRV�[� GRHV QRW KDYH D 7D\ORU VHULHV ?H[SDQVLRQ� WU\ VHULHV >WD\ORU@

VHULHV�FRV�[��[� [ �� ���1x− x

2+x3

24− x

5

720+x7

40320+O

(x9)

<RX�FDQ�JHQHUDWH�VHULHV�H[SDQVLRQV�LQ�WHUPV�RI�QHJDWLYH�SRZHUV�E\�H[SDQGLQJDURXQG�WKH�SRLQW LQILQLW\�

VHULHV��[A� � ����[ � ��� [ LQILQLW\�

x− 1 +2x− 2x2

+2x3− 2x4

+O(

1x5

)

7KLV�LV�DQ�H[DPSOH�IRU�DQ�³DV\PSWRWLF´�H[SDQVLRQ� ZKLFK�DSSUR[LPDWHV�WKHEHKDYLRU�RI�D�IXQFWLRQ�IRU�ODUJH�YDOXHV�RI�WKH�DUJXPHQW� ,Q�VLPSOH�FDVHV� VHULHVUHWXUQV�DQ�H[SDQVLRQ�LQ�WHUPV�RI�QHJDWLYH�SRZHUV�RI x� EXW�RWKHU�IXQFWLRQV�PD\WXUQ�XS�DV�ZHOO�

VHULHV��H[S�[� � H[S��[����H[S�[� � H[S��[���[ LQILQLW\�

1− 2H2 x

+2H4 x− 2H6 x

+2H8 x− 2H10 x

+O(H−12 x

)

����

Page 117: Mupad Tutorial

L^kb^l�>qiZglbhgl

([HUFLVH������ 7KH�RUGHU p RI�D�URRW x RI�D�IXQFWLRQ f LV�WKH�PD[LPDO�QXPEHU�RIGHULYDWLYHV�WKDW�YDQLVK�DW�WKH�SRLQW x�

f(x) = f ′(x) = · · · = f (p−1)(x) = 0 , f (p)(x) ̸= 0 .

:KDW�LV�WKH�RUGHU�RI�WKH�URRW x = 0 RI f(x) = WDQ(VLQ(x))− VLQ(WDQ(x))"

([HUFLVH������ %HVLGHV�WKH�DULWKPHWLFDO�RSHUDWRUV� VRPH�RWKHU�V\VWHP�IXQFWLRQVVXFK�DV GLII RU LQW ZRUN�GLUHFWO\�IRU�VHULHV� &RPSDUH�WKH�UHVXOW�RIWD\ORU�GLII������[��[��[� DQG�WKH�GHULYDWLYH�RI WD\ORU������[��[��0DWKHPDWLFDOO\� ERWK�VHULHV�DUH�LGHQWLFDO� &DQ�\RX�H[SODLQ�WKH�GLIIHUHQFH�LQ0X3$'"

([HUFLVH������ 7KH�IXQFWLRQ f(x) =√x+ 1−

√x− 1 WHQGV�WR�]HUR�IRU�ODUJH x�

L�H�� OLPx→∞ f(x) = 0� 6KRZ�WKDW�WKH�DSSUR[LPDWLRQ f(x) ≈ 1/√x LV�YDOLG�IRU�ODUJH

YDOXHV�RI x� )LQG�EHWWHU�DV\PSWRWLF�DSSUR[LPDWLRQV�RI f �

([HUFLVH������ &RPSXWH�WKH�ILUVW�WKUHH�WHUPV�LQ�WKH�VHULHV�H[SDQVLRQ�RI�WKHIXQFWLRQ I� VLQ�[�[A�� DURXQG [ �� 5HDG�WKH�KHOS�SDJH�IRU�WKH�0X3$'IXQFWLRQ UHYHUW� 8VH�WKLV�IXQFWLRQ�WR�FRPSXWH�WKH�OHDGLQJ�WHUPV�RI�WKH�VHULHVH[SDQVLRQ�RI�WKH�LQYHUVH�IXQFWLRQ f−1 �ZKLFK�LV�ZHOO�GHILQHG�LQ�D�FHUWDLQQHLJKERUKRRG�RI x = 0��

����

Page 118: Mupad Tutorial

- FnI:=� H[c^\ml

9d_]ZjYa[�Kljm[lmj]k2 >a]d\k$ Jaf_k$ ]l[&

7KH�0X3$'� NHUQHO�SURYLGHV�GRPDLQ�W\SHV�IRU�WKH�EDVLF�GDWD�VWUXFWXUHV�VXFK�DVQXPEHUV� VHWV� WDEOHV� HWF� ,Q�DGGLWLRQ� \RX�FDQ�GHILQH�\RXU�RZQ�GDWD�VWUXFWXUHV�LQWKH�0X3$' ODQJXDJH�DQG�ZRUN�ZLWK�WKHP�V\PEROLFDOO\� :H�GR�QRW�GLVFXVV�WKHFRQVWUXFWLRQ�RI�VXFK�QHZ�³GRPDLQV´�LQ�WKLV�HOHPHQWDU\�LQWURGXFWLRQ� EXWGHPRQVWUDWH�VRPH�VSHFLDO�³OLEUDU\´�GRPDLQV�SURYLGHG�E\�WKH�V\VWHP�

%HVLGHV�WKH�NHUQHO�GRPDLQV� WKH 'RP OLEUDU\ FRQWDLQV�D�YDULHW\�RI�SUHGHILQHGGRPDLQV�WKDW�ZHUH�LPSOHPHQWHG�E\�WKH�0X3$' GHYHORSHUV� )RU�DQ�RYHUYLHZ� VHHWKH�UHIHUHQFH�GRFXPHQWDWLRQ� ,Q�WKLV�VHFWLRQ� ZH�SUHVHQW�VRPH�SDUWLFXODUO\�XVHIXOGRPDLQV�UHSUHVHQWLQJ�FRPSOH[�PDWKHPDWLFDO�REMHFWV�VXFK�DV�ILHOGV� ULQJV� HWF�6HFWLRQ ���� GLVFXVVHV�D�GDWD�W\SH�IRU�PDWULFHV� ZKLFK�LV�ZHOO�VXLWHG�IRU�SUREOHPVLQ�OLQHDU�DOJHEUD�

7KH�PDLQ�SDUW�RI�D�GRPDLQ�LV�LWV FRQVWUXFWRU� ZKLFK�JHQHUDWHV�REMHFWV�RI�WKHGRPDLQ� (DFK�VXFK�REMHFW�³NQRZV´�LWV�GRPDLQ� ZKLFK�KDV PHWKRGV DWWDFKHG�WR�LW�7KH\�UHSUHVHQW�WKH�PDWKHPDWLFDO�RSHUDWLRQV�IRU�WKHVH�REMHFWV�

+HUH�LV�D�OLVW�RI�VRPH�RI�WKH�ZHOO�NQRZQ�PDWKHPDWLFDO�VWUXFWXUHV�LPSOHPHQWHG�LQWKH 'RP OLEUDU\�

� WKH�ULQJ�RI�LQWHJHUV Z � 'RP��,QWHJHU�

� WKH�ILHOG�RI�UDWLRQDO�QXPEHUV Q � 'RP��5DWLRQDO�

� WKH�ILHOG�RI�UHDO�QXPEHUV R � 'RP��5HDO RU 'RP��)ORDW��

� WKH�ILHOG�RI�FRPSOH[�QXPEHUV C � 'RP��&RPSOH[�

� WKH�ULQJ�RI�LQWHJHUV�PRGXOR n� 'RP��,QWHJHU0RG�Q��

:H�FRQVLGHU�WKH�UHVLGXH�FODVV�ULQJ�RI�LQWHJHUV�PRGXOR n� ,WV�HOHPHQWV�DUH�WKHLQWHJHUV 0, 1, . . . , n− 1� DQG�DGGLWLRQ�DQG�PXOWLSOLFDWLRQ�DUH�GHILQHG�³PRGXOR n�´7KLV�ZRUNV�E\�DGGLQJ�RU�PXOWLSO\LQJ�LQ Z� GLYLGLQJ�WKH�UHVXOW�E\ n� DQG�WDNLQJ�WKHUHPDLQGHU�RI�WKLV�GLYLVLRQ�LQ {0, 1, . . . , n− 1}�

�'RP��5HDO LV�IRU�V\PEROLF�UHSUHVHQWDWLRQV�RI�UHDO�QXPEHUV� ZKLOH 'RP��)ORDW UHSUHVHQWV�WKHP�DVIORDWLQJ�SRLQW�QXPEHUV�

����

Page 119: Mupad Tutorial

:e`^[kZb\�Lmkn\mnk^l3 ?b^e]l% Kbg`l% ^m\'

� � PRG �1

,Q�WKLV�H[DPSOH� ZH�KDYH�XVHG�WKH�GDWD�W\SHV�RI�WKH�0X3$' NHUQHO� WKH�RSHUDWRU PXOWLSOLHV�WKH�LQWHJHUV 3 DQG 5 LQ�WKH�XVXDO�ZD\�WR�JHW 15� DQG�WKH�RSHUDWRU PRGFRPSXWHV�WKH�GHFRPSRVLWLRQ 15 = 2 · 7 + 1 DQG�UHWXUQV 1 DV�UHPDLQGHU�PRGXOR 7�

$OWHUQDWLYHO\� \RX�PD\�WHOO�0X3$' WKDW�\RX�ZDQW�WR�FRPSXWH�LQ Z7 = Z/7 Z E\XVLQJ�WKH�LQSXW�V\QWD[ 'RP��,QWHJHU0RG���� 7KH�ODWWHU�REMHFW�DFWV�DV�DFRQVWUXFWRU�IRU�HOHPHQWV�RI�WKH�UHVLGXH�FODVV�ULQJ� PRGXOR 7�

FRQVWUXFWRU � 'RP��,QWHJHU0RG����[ � FRQVWUXFWRU���� \ � FRQVWUXFWRU���

3PRG 7

5PRG 7

$V�\RX�FDQ�VHH�IURP�WKH�VFUHHQ�RXWSXW� WKH�LGHQWLILHUV [ DQG \ GR�QRW�KDYH�WKHLQWHJHUV 3 DQG 5� UHVSHFWLYHO\� DV�YDOXHV� ,QVWHDG� WKH�QXPEHUV�DUH�HOHPHQWV�RI�WKHUHVLGXH�FODVV�ULQJ�RI�LQWHJHUV�PRGXOR 7�

GRPW\SH�[�� GRPW\SH�\�Z7, Z7

1RZ� \RX�FDQ�XVH�WKH�XVXDO�DULWKPHWLF�RSHUDWLRQV� DQG�0X3$' DXWRPDWLFDOO\�XVHVWKH�FRPSXWDWLRQ�UXOHV�RI�WKH�UHVLGXH�FODVV�ULQJ�

[ \� [A��� \A�� � [ � \1PRG 7, 6PRG 7

7KH�ULQJ 'RP��,QWHJHU0RG��� HYHQ�KDV�D�ILHOG�VWUXFWXUH� VR�WKDW�\RX�FDQ�GLYLGH�E\DOO�ULQJ�HOHPHQWV�H[FHSW ��PRG���

�,I�\RX�ZDQW�WR�H[HFXWH�RQO\�D�VPDOO�QXPEHU�RI�PRGXOR�RSHUDWLRQV� LW�LV�RIWHQ�SUHIHUDEOH�WR�XVHWKH�RSHUDWRU PRG� ZKLFK�LV�LPSOHPHQWHG�LQ�WKH�0X3$' NHUQHO�DQG�TXLWH�IDVW� 7KLV�DSSURDFK�PD\UHTXLUH�VRPH�DGGLWLRQDO�XQGHUVWDQGLQJ�KRZ�WKH�V\VWHP�IXQFWLRQV�ZRUN� )RU�H[DPSOH� WKH�FRPSX�WDWLRQ�RI ��A������PRG�� WDNHV�TXLWH�D�ORQJ�WLPH� VLQFH�0X3$' ILUVW�FRPSXWHV�WKH�YHU\�ELJ�QXP�EHU 1729999 DQG�WKHQ�UHGXFHV�WKH�UHVXOW�PRGXOR 7� ,Q�WKLV�FDVH� WKH�FRPSXWDWLRQ [A������ ZKHUH[� 'RP��,QWHJHU0RG�������� LV�PXFK�IDVWHU�VLQFH�WKH�LQWHUQDO�PRGXODU�DULWKPHWLF�DYRLGV�VXFK�ELJQXPEHUV� $OWHUQDWLYHO\� WKH�FDOO SRZHUPRG������������ XVHV�WKH�V\VWHP�IXQFWLRQ SRZHUPRG WR�FRP�SXWH�WKH�UHVXOW�TXLFNO\�ZLWKRXW�HPSOR\LQJ 'RP��,QWHJHU0RG����

����

Page 120: Mupad Tutorial

- FnI:=� H[c^\ml

[�\2PRG 7

$ PRUH�DEVWUDFW�H[DPSOH�LV�WKH�ILHOG�H[WHQVLRQ

K = Q[√

2 ] = {p+ q√

2 ; p, q ∈ Q } .

<RX�PD\�GHILQH�WKLV�ILHOG�YLD

. � 'RP��$OJHEUDLF([WHQVLRQ�'RP��5DWLRQDO�6TUW�A� �� 6TUW���

+HUH�WKH�LGHQWLILHU 6TUW� �=̂√

2�� GHILQHG�E\�LWV�DOJHEUDLF�SURSHUW\ 6TUW�A� �� LVXVHG�WR�H[WHQG�WKH�UDWLRQDO�QXPEHUV 'RP��5DWLRQDO� 1RZ� \RX�FDQ�FRPSXWH�LQWKLV�ILHOG�

[ � .���� � � 6TUW��� \ � .�� � ��� 6TUW���[A� \ � \A�

6776TUW�54

+5845324

7KH�GRPDLQ 'RP��([SUHVVLRQ)LHOG�QRUPDOL]HU�]HURWHVW� UHSUHVHQWV�WKH³ILHOG´�RI��V\PEROLF��0X3$' H[SUHVVLRQV� 7KH�FRQVWUXFWRU�LV�SDUDPHWUL]HG�E\�WZRIXQFWLRQV QRUPDOL]HU DQG ]HURWHVW� ZKLFK�PD\�EH�FKRVHQ�E\�WKH�XVHU�

7KH�IXQFWLRQ ]HURWHVW LV�FDOOHG�LQWHUQDOO\�E\�DOO�DOJRULWKPV�WKDW�ZDQW�WR�GHFLGHZKHWKHU�D�GRPDLQ�REMHFW�LV�PDWKHPDWLFDOO\ 0� 7\SLFDOO\� \RX�ZLOO�XVH�WKH�V\VWHPIXQFWLRQ LV]HUR� ZKLFK�UHFRJQL]HV�QRW�RQO\�WKH�LQWHJHU � DV�]HUR� EXW�DOVR�RWKHUREMHFWV�VXFK�DV�WKH�IORDWLQJ�SRLQW�QXPEHU ��� RU�WKH�SRO\QRPLDO SRO\���>[@��6HFWLRQ ������

7KH�WDVN�RI�WKH�IXQFWLRQ QRUPDOL]HU LV�WR�JHQHUDWH�D�QRUPDO�IRUP�IRU�0X3$'REMHFWV�RI�W\SH 'RP��([SUHVVLRQ)LHOG�·� ·�� 2SHUDWLRQV�RQ�VXFK�REMHFWV�ZLOO�XVHWKLV�IXQFWLRQ�WR�VLPSOLI\�WKH�UHVXOW�EHIRUH�UHWXUQLQJ�LW� )RU�H[DPSOH� LI�\RX�VXSSO\WKH�LGHQWLW\�IXQFWLRQ LG IRU�WKH QRUPDOL]HU DUJXPHQW� WKHQ�RSHUDWLRQV�RQ�REMHFWVRI�WKLV�GRPDLQ�ZRUN�OLNH�IRU�WKH�XVXDO�0X3$' H[SUHVVLRQV�ZLWKRXW�DGGLWLRQDOQRUPDOL]DWLRQ�

FRQVWUXFWRU � 'RP��([SUHVVLRQ)LHOG�LG� LV]HUR��[ � FRQVWUXFWRU�D��D � E�A���\ � FRQVWUXFWRU�E��D � E�A���

����

Page 121: Mupad Tutorial

:e`^[kZb\�Lmkn\mnk^l3 ?b^e]l% Kbg`l% ^m\'

[ � \a

(a+ b)2+

b

(a+ b)2

,I�\RX�VXSSO\�WKH�V\VWHP�IXQFWLRQ QRUPDO LQVWHDG� WKH�UHVXOW�LV�VLPSOLILHGDXWRPDWLFDOO\��6HFWLRQ �����

FRQVWUXFWRU � 'RP��([SUHVVLRQ)LHOG�QRUPDO� LV]HUR��[ � FRQVWUXFWRU�D��D � E�A���\ � FRQVWUXFWRU�E��D � E�A���[ � \

1a+ b

:H�QRWH�WKDW�WKH�SXUSRVH�RI�VXFK�0X3$' GRPDLQV�LV�QRW�QHFHVVDULO\�WKH�GLUHFWJHQHUDWLRQ�RI�GDWD�VWUXFWXUHV�RU�WKH�FRPSXWDWLRQ�ZLWK�WKH�FRUUHVSRQGLQJ�REMHFWV�,QGHHG� VRPH�FRQVWUXFWRUV�VLPSO\�UHWXUQ�REMHFWV�RI�WKH�XQGHUO\LQJ�NHUQHOGRPDLQV� LI�VXFK�GRPDLQV�H[LVW�

GRPW\SH�'RP��,QWHJHU�����GRPW\SH�'RP��5DWLRQDO�������GRPW\SH�'RP��)ORDW�3,���GRPW\SH�'RP��([SUHVVLRQ)LHOG�LG� LV]HUR��D � E��

'20B,17, '20B5$7, '20B)/2$7, '20B(;35

,Q�WKHVH�FDVHV� WKHUH�LV�QR�LPPHGLDWH�EHQHILW�LQ�XVLQJ�VXFK�D�FRQVWUXFWRU� \RX�PD\DV�ZHOO�FRPSXWH�GLUHFWO\�ZLWK�WKH�FRUUHVSRQGLQJ�NHUQHO�REMHFWV� 7KH�PDLQDSSOLFDWLRQ�RI�VXFK�VSHFLDO�GDWD�VWUXFWXUHV�LV�WKH�FRQVWUXFWLRQ�RI�PRUH�FRPSOH[PDWKHPDWLFDO�VWUXFWXUHV� $ VLPSOH�H[DPSOH�LV�WKH�JHQHUDWLRQ�RI�PDWULFHV�6HFWLRQ ������RU�SRO\QRPLDOV��6HFWLRQ ������ZLWK�HQWULHV�LQ�D�SDUWLFXODU�ULQJ� VXFKWKDW�PDWUL[�RU�SRO\QRPLDO�DULWKPHWLF� UHVSHFWLYHO\� LV�SHUIRUPHG�DFFRUGLQJ�WR�WKHFRPSXWDWLRQ�UXOHV�RI�WKH�FRHIILFLHQW�ULQJ�

����

Page 122: Mupad Tutorial

- FnI:=� H[c^\ml

N][lgjk�Yf\�EYlja[]k

,Q�6HFWLRQ ����� ZH�KDYH�JLYHQ�H[DPSOHV�RI�VSHFLDO�GDWD�W\SHV��³GRPDLQV´��IRUGHILQLQJ�DOJHEUDLF�VWUXFWXUHV�VXFK�DV�ULQJV� ILHOGV� HWF��LQ�0X3$'�� ,Q�WKLV�VHFWLRQ�ZH�GLVFXVV�WZR�IXUWKHU�GRPDLQV�IRU�JHQHUDWLRQ�DQG�FRQYHQLHQW�FRPSXWDWLRQ�ZLWKYHFWRUV�DQG�PDWULFHV� 'RP��0DWUL[ DQG 'RP��6TXDUH0DWUL[� ,Q�SULQFLSOH� \RXPD\�XVH�DUUD\V�IRU�ZRUNLQJ�ZLWK�YHFWRUV�RU�PDWULFHV��6HFWLRQ ����� +RZHYHU� WKHQ\RX�KDYH�WR�GHILQH�\RX�RZQ�URXWLQHV�IRU�DGGLWLRQ� PXOWLSOLFDWLRQ� LQYHUVLRQ� RUGHWHUPLQDQW�FRPSXWDWLRQ� XVLQJ�WKH�0X3$' SURJUDPPLQJ�ODQJXDJH��&KDSWHU ����)RU�WKH�VSHFLDO�PDWUL[�W\SH�WKDW�ZH�SUHVHQW�LQ�ZKDW�IROORZV� VXFK�URXWLQHV�H[LVWDQG�DUH�³DWWDFKHG´�WR�WKH�PDWULFHV�DV�PHWKRGV� 0RUHRYHU� \RX�PD\�XVH�WKHIXQFWLRQV�RI�WKH OLQDOJ OLEUDU\��OLQHDU�DOJHEUD� 6HFWLRQ ������ ZKLFK�FDQ�KDQGOHPDWULFHV�RI�WKLV�W\SH�

'HILQLWLRQ�RI�0DWULFHV�DQG�9HFWRUV

9HFWRUV�LQ�0X3$'� DUH�UHJDUGHG�DV�VSHFLDO�PDWULFHV�RI�GLPHQVLRQ 1× n RU n× 1�UHVSHFWLYHO\� 7KH�FRPPDQG PDWUL[PD\�EH�XVHG�IRU�FUHDWLQJ�PDWULFHV�DQG�YHFWRUVRI�DUELWUDU\�GLPHQVLRQ�

PDWUL[�>> �� �� �� � @�> D� E� F� G @�>VLQ�[�� FRV�[�� H[S�[�� OQ�[�@@��

PDWUL[�>[�� [�� [�� [�@�

⎜⎝1 2 3 4a b c d

VLQ(x) FRV(x) Hx OQ(x)

⎟⎠ ,

⎜⎜⎜⎜⎝

[�

[�

[�

[�

⎟⎟⎟⎟⎠

$UELWUDU\�DULWKPHWLFDO�H[SUHVVLRQV�PD\�EH�XVHG�DV�HOHPHQWV� $OWKRXJK�PDWULFHVFUHDWHG�E\ PDWUL[ DUH�VXLWDEOH�IRU�PRVW�DSSOLFDWLRQV�DQG�ZLOO�EH�WKH�PRVW�ZLGHO\XVHG�PDWUL[�REMHFWV� 0X3$'¶V�FRQFHSW�IRU�PDWULFHV�LV�PRUH�JHQHUDO� 7\SLFDOO\� DVSHFLILF�FRHIILFLHQW�ULQJ�IRU�WKH�HOHPHQWV�LV�DWWDFKHG�WR�D�0X3$'�PDWUL[� ,Q�IDFW�WKH�IXQFWLRQ PDWUL[ LV�MXVW�D�FRQVWUXFWRU�ZLWK�D�VKRUW�QDPH�IRU�VSHFLDO�PDWULFHV�RIGRPDLQ�W\SH 'RP��0DWUL[�5���ZKHUH 5 'RP��([SUHVVLRQ)LHOG�� UHSUHVHQWVDUELWUDU\�0X3$'�H[SUHVVLRQV�

����

Page 123: Mupad Tutorial

O^\mhkl�Zg]�FZmkb\^l

:H�H[SODLQ�WKH�JHQHUDO�FRQFHSW� 0X3$' SURYLGHV�WKH�GDWD�W\SH 'RP��0DWUL[ IRUPDWULFHV�RI�DUELWUDU\�GLPHQVLRQm× n� 7KH�W\SH 'RP��6TXDUH0DWUL[ UHSUHVHQWVVTXDUH�PDWULFHV�RI�GLPHQVLRQ n× n� 7KH\�EHORQJ�WR�WKH�OLEUDU\ 'RP� ZKLFK�DOVRFRPSULVHV�GDWD�W\SHV�IRU�PDWKHPDWLFDO�VWUXFWXUHV�VXFK�DV�ULQJV�DQG�ILHOGV�6HFWLRQ ������ 0DWULFHV�PD\�KDYH�HQWULHV�IURP�D�VHW�WKDW�PXVW�EH�HTXLSSHG�ZLWK�DULQJ�VWUXFWXUH�LQ�WKH�PDWKHPDWLFDO�VHQVH� )RU�H[DPSOH� \RX�PD\�XVH�WKHSUHGHILQHG�ULQJV�DQG�ILHOGV�VXFK�DV 'RP��,QWHJHU� 'RP��,QWHJHU0RG�Q�� HWF��IURPWKH 'RP OLEUDU\�

7KH�FDOO 'RP��0DWUL[�5� FUHDWHV�WKH�FRQVWUXFWRU�IRU�PDWULFHV�RI�DUELWUDU\GLPHQVLRQm× n ZLWK�FRHIILFLHQWV�LQ�WKH�ULQJ 5��:KHQ�\RX�FRQVWUXFW�VXFK�DPDWUL[� \RX�DUH�UHTXLUHG�WR�HQVXUH�WKDW�LWV�HQWULHV�EHORQJ�WR��RU�PD\�EH�FRQYHUWHGWR��WKLV�ULQJ� <RX�VKRXOG�NHHS�WKLV�LQ�PLQG�ZKHQ�WU\LQJ�WR�JHQHUDWH�PDWULFHV�ZLWKHQWULHV�RXWVLGH�WKH�FRHIILFLHQW�ULQJ�LQ�D�FRPSXWDWLRQ��IRU�H[DPSOH� WKH�LQYHUVH�RIDQ�LQWHJHU�PDWUL[�LQ�JHQHUDO�KDV�QRQ�LQWHJUDO�UDWLRQDO�HQWULHV��

7KH�IROORZLQJ�H[DPSOH�\LHOGV�WKH�FRQVWUXFWRU�IRU�PDWULFHV�ZLWK�UDWLRQDO�QXPEHUHQWULHV�

FRQVWUXFWRU � 'RP��0DWUL[�'RP��5DWLRQDO�'RP::0DWUL[(Q)

1RZ� \RX�PD\�JHQHUDWH�PDWULFHV�RI�DUELWUDU\�GLPHQVLRQV� ,Q�WKH�IROORZLQJH[DPSOH� ZH�JHQHUDWH�D 2× 3PDWUL[�ZLWK�DOO�HQWULHV�LQLWLDOL]HG�WR 0�

$ � FRQVWUXFWRU��� ��(

0 0 00 0 0

)

:KHQ�JHQHUDWLQJ�D�PDWUL[� \RX�PD\�VXSSO\�D�IXQFWLRQ f WKDW�WDNHV�WZR�DUJXPHQWV�7KHQ�WKH�HQWU\�LQ�URZ i DQG�FROXPQ j LV�LQLWLDOL]HG�ZLWK f(i, j)�

I � �L� M� �! �L M�� $ � FRQVWUXFWRU��� �� I�(

1 2 32 4 6

)

����

Page 124: Mupad Tutorial

- FnI:=� H[c^\ml

$OWHUQDWLYHO\� \RX�FDQ�LQLWLDOL]H�D�PDWUL[�E\�VSHFLI\LQJ�D��QHVWHG��OLVW� (DFK�OLVWHOHPHQW�LV�LWVHOI�D�OLVW�DQG�FRUUHVSRQGV�WR�RQH�URZ�RI�WKH�PDWUL[� 7KH�IROORZLQJFRPPDQG�JHQHUDWHV�WKH�VDPH�PDWUL[�DV�LQ�WKH�SUHYLRXV�H[DPSOH�

FRQVWUXFWRU��� �� >>�� �� �@� >�� �� �@@��

7KH�SDUDPHWHUV�IRU�WKH�GLPHQVLRQ�DUH�RSWLRQDO�KHUH� VLQFH�WKH\�DUH�DOVR�JLYHQ�E\WKH�VWUXFWXUH�RI�WKH�OLVW� 7KXV�

FRQVWUXFWRU�>>�� �� �@� >�� �� �@@��

DOVR�UHWXUQV�WKH�VDPH�PDWUL[� $Q�DUUD\�RI�GRPDLQ�W\SH '20B$55$< �6HFWLRQ �����LVDOVR�YDOLG�IRU�LQLWLDOL]LQJ�D�PDWUL[�

$UUD\ � DUUD\������ ����� >>�� �� �@� >�� �� �@@��0DW � FRQVWUXFWRU�$UUD\��

<RX�PD\�GHILQH�FROXPQ�DQG�URZ�YHFWRUV�DVm× 1 DQG 1× nPDWULFHV� UHVSHFWLYHO\�3ODLQ�OLVWV�FDQ�EH�XVHG�IRU�GHILQLQJ�FROXPQ�RU�URZ�YHFWRUV�

FROXPQ � FRQVWUXFWRU��� �� >�� �� �@�⎛

⎜⎝123

⎟⎠

URZ � FRQVWUXFWRU��� �� >�� �� �@�(

1 2 3)

,I�QR�GLPHQVLRQV�DUH�VSHFLILHG� D�SODLQ�OLVW�JHQHUDWHV�D�FROXPQ�YHFWRU�

FROXPQ � FRQVWUXFWRU�>�� �� �@�⎛

⎜⎝123

⎟⎠

����

Page 125: Mupad Tutorial

O^\mhkl�Zg]�FZmkb\^l

7KH�HQWULHV�RI�D�PDWUL[�RU�D�YHFWRU�PD\�EH�DFFHVVHG�LQ�WKH�IRUPV PDWUL[>L�M@�URZ>L@� RU FROXPQ>M@� 6LQFH�YHFWRUV�DUH�VSHFLDO�PDWULFHV� \RX�PD\�DFFHVV�WKHFRPSRQHQWV�RI�D�YHFWRU�DOVR�LQ�WKH�IRUP URZ>��L@ RU FROXPQ>M��@� UHVSHFWLYHO\�

$>�� �@� URZ>�@� URZ>�� �@�FROXPQ>�@� FROXPQ>�� �@

6, 3, 3, 2, 2

6XEPDWULFHV�DUH�JHQHUDWHG�DV�IROORZV�

0DW>����� ����@� URZ>����� ����@�FROXPQ>����� ����@(

1 22 4

),(

1 2),

(12

)

<RX�PD\�FKDQJH�D�PDWUL[�HQWU\�E\�DQ�LQGH[HG�DVVLJQPHQW�

0DW>�� �@ � ��� URZ>�@ � �� FROXPQ>�� �@ � ��0DW� URZ� FROXPQ(

1 2 32 4 23

),(

1 5 3),

⎜⎝153

⎟⎠

<RX�FDQ�XVH�ORRSV��&KDSWHU ����WR�FKDQJH�DOO�FRPSRQHQWV�RI�D�PDWUL[�

P � �� Q � �� 0DW � FRQVWUXFWRU�P� Q��IRU L IURP � WR P GR

IRU M IURP � WR Q GR0DW>L� M@ � L M

HQGBIRUHQGBIRU�

<RX�FDQ�JHQHUDWH�GLDJRQDO�PDWULFHV E\�VXSSO\LQJ�WKH�RSWLRQ 'LDJRQDO� ,Q�WKLVFDVH� WKH�WKLUG�DUJXPHQW�WR�WKH�FRQVWUXFWRU�PD\�HLWKHU�EH�D�OLVW�RI�WKH�GLDJRQDOHOHPHQWV�RU�D�IXQFWLRQ f VXFK�WKDW�WKH i�WK�GLDJRQDO�HOHPHQW�LV f(i)�

FRQVWUXFWRU��� �� >��� ��@� 'LDJRQDO�(

11 00 12

)

����

Page 126: Mupad Tutorial

- FnI:=� H[c^\ml

,Q�WKH�QH[W�H[DPSOH� ZH�JHQHUDWH�DQ�LGHQWLW\�PDWUL[�E\�VXSSO\LQJ � DV�D�IXQFWLRQ�GHILQLQJ�WKH�GLDJRQDO�HOHPHQWV�

FRQVWUXFWRU��� �� �� 'LDJRQDO�(

1 00 1

)

$OWHUQDWLYHO\� LGHQWLW\�PDWULFHV�FDQ�EH�JHQHUDWHG�E\�WKH �LGHQWLW\�PHWKRG�RI�WKHFRQVWUXFWRU�

FRQVWUXFWRU��LGHQWLW\���(

1 00 1

)

7KH�FRQVWUXFWRU�FRQVLGHUHG�VR�IDU�UHWXUQV�PDWULFHV�ZLWK�UDWLRQDO��L�H�� UHDO�QXPEHU�HQWULHV� 7KXV� WKH�IROORZLQJ�DWWHPSW�WR�JHQHUDWH�D�PDWUL[�ZLWK�FRPSOH[FRHIILFLHQWV�GRHV�QRW�ZRUN�

FRQVWUXFWRU�>>�� �� �@� >�� �� � � ,@@�(UURU� XQDEOH WR GHILQH PDWUL[ RYHU 'RP��5DWLRQDO ?>�'RP��0DWUL[�'RP��5DWLRQDO����QHZ@

<RX�KDYH�WR�FKRRVH�D�VXLWDEOH�FRHIILFLHQW�ULQJ�WR�JHQHUDWH�D�PDWUL[�ZLWK�WKH�DERYHHQWULHV� ,Q�WKH�IROORZLQJ�H[DPSOH� ZH�GHILQH�D�QHZ�FRQVWUXFWRU�IRU�PDWULFHV�ZLWKFRPSOH[�QXPEHU�HQWULHV�

FRQVWUXFWRU � 'RP��0DWUL[�'RP��&RPSOH[��FRQVWUXFWRU�>>�� �� �@� >�� �� � � ,@@�(

1 2 32 4 1 + L

)

<RX�PD\�JHQHUDWH�PDWULFHV�ZKRVH�HQWULHV�DUH�DUELWUDU\�0X3$' H[SUHVVLRQV�E\PHDQV�RI�WKH�ILHOG 'RP��([SUHVVLRQ)LHOG�LG�LV]HUR� �VHH�6HFWLRQ ������ 7KLV�LVWKH�VWDQGDUG�FRHIILFLHQW�ULQJ�IRU�PDWULFHV� <RX�PD\�DOZD\V�XVH�WKLV�ULQJ�ZKHQ�WKHFRHIILFLHQWV�DQG�WKHLU�SURSHUWLHV�DUH�LUUHOHYDQW�

�&I��SDJH �����

����

Page 127: Mupad Tutorial

O^\mhkl�Zg]�FZmkb\^l

'RP��0DWUL[�� ZLWK�QR�DUJXPHQW�LV�D�FRQVWUXFWRU�IRU�VXFK�PDWULFHV� $V�DVKRUWKDQG�QRWDWLRQ� WKLV�PDWUL[�ULQJ�LV�DWWDFKHG�WR�WKH�LGHQWLILHU PDWUL[�

PDWUL['RP::0DWUL[()

PDWUL[�>>�� [ � \� ��[A�@� >VLQ�[�� �� FRV�[�@�>[ 3,� � � ,� �[ 3,@@�

⎜⎝1 x+ y 1

x2

VLQ(x) 0 FRV(x)π x 1 + L −π x

⎟⎠

,I�\RX�XVH 'RP��([SUHVVLRQ)LHOG�QRUPDO�LV]HUR� DV�WKH�FRHIILFLHQW�ULQJ� WKHQDOO�PDWUL[�HQWULHV�DUH�VLPSOLILHG�YLD�WKH�IXQFWLRQ QRUPDO� DV�GHVFULEHG�LQ6HFWLRQ ����� $ULWKPHWLFDO�RSHUDWLRQV�ZLWK�VXFK�PDWULFHV�DUH�FRPSDUDWLYHO\�VORZ�VLQFH�D�FDOO�WR QRUPDOPD\�EH�TXLWH�WLPH�FRQVXPLQJ� +RZHYHU� WKH�UHVXOWV�DUH�LQJHQHUDO�VLPSOHU�WKDQ�WKH��HTXLYDOHQW��UHVXOWV�RI�FRPSXWDWLRQV�ZLWK�WKH�VWDQGDUGFRHIILFLHQW�ULQJ 'RP��([SUHVVLRQ)LHOG�LG�LV]HUR� XVHG�E\ 'RP��0DWUL[���

7KH�FRQVWUXFWRU 'RP��6TXDUH0DWUL[�Q�5� FRUUHVSRQGV�WR�WKH�ULQJ�RIn�GLPHQVLRQDO�VTXDUH�PDWULFHV�ZLWK�FRHIILFLHQW�ULQJ 5� ,I�WKH�DUJXPHQW 5 LVPLVVLQJ� WKHQ�0X3$' DXWRPDWLFDOO\�XVHV�WKH�FRHIILFLHQW�ULQJ�RI�DOO�0X3$'H[SUHVVLRQV� 7KH�IROORZLQJ�VWDWHPHQW�\LHOGV�WKH�FRQVWUXFWRU�IRU 2× 2PDWULFHV�ZKRVH�HQWULHV�PD\�EH�DUELWUDU\�0X3$' H[SUHVVLRQV�

FRQVWUXFWRU � 'RP��6TXDUH0DWUL[���'RP::6TXDUH0DWUL[(2)

FRQVWUXFWRU�>>�� \@� >[A�� �@@�(

0 y

x2 1

)

����

Page 128: Mupad Tutorial

- FnI:=� H[c^\ml

&RPSXWLQJ�ZLWK�0DWULFHV

<RX�FDQ�XVH�WKH�VWDQGDUG�DULWKPHWLFDO�RSHUDWRUV�IRU�GRLQJ�EDVLF�DULWKPHWLF�ZLWKPDWULFHV�

$ � PDWUL[�>>�� �@� >�� �@@��% � PDWUL[�>>D� E@� >F� G@@��$ � %� $ %�$ % � % $� $A� � %(a+ 1 b+ 2c+ 3 d+ 4

),

(a+ 2 c b+ 2 d

3 a+ 4 c 3 b+ 4 d

)

(2 c− 3 b 2 d− 3 b− 2 a

3 a+ 3 c− 3 d 3 b− 2 c

),

(a+ 7 b+ 10c+ 15 d+ 22

)

0XOWLSOLFDWLRQ�RI�D�PDWUL[�DQG�D�QXPEHU�ZRUNV�FRPSRQHQWZLVH��VFDODUPXOWLSOLFDWLRQ��

� %(

2 a 2 b2 c 2 d

)

7KH�LQYHUVH�RI�D�PDWUL[�LV�UHSUHVHQWHG�E\ ��$ RU $A�����

& � $A����(−2 132 −1

2

)

$ VLPSOH�WHVW�VKRZV�WKDW�WKH�FRPSXWHG�LQYHUVH�LV�FRUUHFW�

$ &� & $(

1 00 1

),

(1 00 1

)

����

Page 129: Mupad Tutorial

O^\mhkl�Zg]�FZmkb\^l

$Q�LQYHUVLRQ�UHWXUQV )$,/ ZKHQ�0X3$' LV�XQDEOH�WR�FRPSXWH�WKH�UHVXOW� 7KHIROORZLQJ�PDWUL[�LV�QRW�LQYHUWLEOH�

& � PDWUL[�>>�� �@� >�� �@@��&A����

)$,/

7KH�FRQFDWHQDWLRQ�RSHUDWRU �� ZKLFK�FRPELQHV�OLVWV��6HFWLRQ �����RU�VWULQJV�6HFWLRQ ������ LV�³RYHUORDGHG´�IRU�PDWULFHV� <RX�FDQ�XVH�LW�WR�FRPELQH�PDWULFHVZLWK�WKH�VDPH�QXPEHU�RI�URZV�

$� %� $�%(

1 23 4

),

(a b

c d

),

(1 2 a b3 4 c d

)

%HVLGHV�WKH�DULWKPHWLF�RSHUDWRUV� RWKHU�V\VWHP�IXQFWLRQV�DUH�DSSOLFDEOH�WRPDWULFHV� +HUH�LV�D�OLVW�RI�H[DPSOHV�

� FRQMXJDWH�$� UHSODFHV�DOO�FRPSRQHQWV�E\�WKHLU�FRPSOH[�FRQMXJDWHV�

� GLII�$�[� GLIIHUHQWLDWHV�FRPSRQHQWZLVH�ZLWK�UHVSHFW�WR [�

� H[S�$� FRPSXWHV eA =∞∑

i=0

1i!Ai�

� H[SDQG�$� DSSOLHV H[SDQG WR�DOO�FRPSRQHQWV�RI $�

� H[SU�$� FRQYHUWV $ WR�DQ�DUUD\�RI�GRPDLQ�W\SH '20B$55$<�

� IORDW�$� DSSOLHV IORDW WR�DOO�FRPSRQHQWV�RI $�

� KDV�$�H[SUHVVLRQ� FKHFNV�ZKHWKHU�DQ�H[SUHVVLRQ�LV�FRQWDLQHG�LQ�DW�OHDVWRQH�HQWU\�RI $�

� LQW�$�[� LQWHJUDWHV�FRPSRQHQWZLVH�ZLWK�UHVSHFW�WR [�

� LV]HUR�$� FKHFNV�ZKHWKHU�DOO�FRPSRQHQWV�RI $ YDQLVK�

� PDS�$�IXQFWLRQ� DSSOLHV�WKH�IXQFWLRQ�WR�DOO�FRPSRQHQWV�

����

Page 130: Mupad Tutorial

- FnI:=� H[c^\ml

� QRUP�$� �LGHQWLFDO�ZLWK QRUP�$�,QILQLW\���FRPSXWHV�WKH�LQILQLW\�QRUP���

� $_HTXDWLRQ DSSOLHV ·_HTXDWLRQ WR�DOO�HQWULHV�RI $�

� VXEV�$�HTXDWLRQ� DSSOLHV VXEV�·�HTXDWLRQ� WR�DOO�HQWULHV�RI $�

� &� ]LS�$�%�I� UHWXUQV�WKH�PDWUL[�GHILQHG�E\ Cij = f(Aij , Bij)�

7KH�OLQHDU�DOJHEUD�OLEUDU\ OLQDOJ DQG�WKH�QXPHULFV�OLEUDU\ QXPHULF �6HFWLRQ �����FRPSULVH�PDQ\�RWKHU�IXQFWLRQV�IRU�KDQGOLQJ�PDWULFHV�

([HUFLVH������ *HQHUDWH�WKH 15× 15+LOEHUW�PDWUL[H = (Hij) ZLWKHij = 1/(i+ j − 1)� *HQHUDWH�WKH�YHFWRU b⃗ = H e⃗� ZKHUH e⃗ = (1, . . . , 1)t� &RPSXWHWKH�H[DFW�VROXWLRQ�YHFWRU x⃗ RI�WKH�V\VWHP�RI�HTXDWLRQV H x⃗ = b⃗ �RI�FRXUVH� WKLVVKRXOG�\LHOG x⃗ = e⃗�� &RQYHUW�DOO�HQWULHV�RI H WR�IORDWLQJ�SRLQW�YDOXHV�DQG�VROYH�WKHV\VWHP�RI�HTXDWLRQV�DJDLQ� &RPSDUH�WKH�UHVXOW�WR�WKH�H[DFW�VROXWLRQ� <RX�ZLOO�QRWHD�GUDPDWLF�GLIIHUHQFH� ZKLFK�LV�FDXVHG�E\�QXPHULFDO�URXQGLQJ�HUURUV� /DUJHU+LOEHUW�PDWULFHV�FDQQRW�EH�LQYHUWHG�ZLWK�WKH�VWDQGDUG�SUHFLVLRQ�RI�FRPPRQQXPHULFDO�VRIWZDUH�

6SHFLDO�0HWKRGV�IRU�0DWULFHV

$ FRQVWUXFWRU�WKDW�KDV�EHHQ�JHQHUDWHG�E\�PHDQV�RI�HLWKHU 'RP��0DWUL[�·� RU'RP��6TXDUH0DWUL[�·� FRQWDLQV�PDQ\�VSHFLDO�IXQFWLRQV�IRU�WKH�FRUUHVSRQGLQJGDWD�W\SH� ,I 0� 'RP��0DWUL[�ULQJ� LV�D�FRQVWUXFWRU�DQG $� 0�·� LV�D�PDWUL[JHQHUDWHG�ZLWK�WKLV�FRQVWUXFWRU� DV�GHVFULEHG�LQ�6HFWLRQ ����� WKHQ� DPRQJ�PD\RWKHUV� WKH�IROORZLQJ�PHWKRGV�DUH�DYDLODEOH�

� 0��FRO�$�L� UHWXUQV�WKH i�WK�FROXPQ�RI $�

� 0��GHO&RO�$�L� UHPRYHV�WKH i�WK�FROXPQ�IURP $�

� 0��GHO5RZ�$�L� UHPRYHV�WKH i�WK�URZ�IURP $�

� 0��PDWGLP�$� UHWXUQV�WKH�GLPHQVLRQ >P�Q@ RI�WKHm× nPDWUL[ $�

��QRUP�$��� UHWXUQV � WKH � RQH�QRUP� QRUP�$�)UREHQLXV� \LHOGV � WKH � )UREHQLXV � QRUP(∑

i,j

|Aij |2)1/2

����

Page 131: Mupad Tutorial

O^\mhkl�Zg]�FZmkb\^l

� 0��UDQGRP�� UHWXUQV�D�PDWUL[�ZLWK�UDQGRP�HQWULHV�

� 0��URZ�$�L� UHWXUQV�WKH i�WK�URZ�RI $�

� 0��VZDS&RO�$�L�M� H[FKDQJHV�FROXPQV i DQG j�

� 0��VZDS5RZ�$�L�M� H[FKDQJHV�URZV i DQG j�

� 0��WU�$� UHWXUQV�WKH�WUDFH∑iAii RI A�

� 0��WUDQVSRVH�$� UHWXUQV�WKH�WUDQVSRVH (Aji) RI A = (Aij)�

0 � 'RP��0DWUL[��� $ � 0�>>[� �@� >�� \@@�(x 12 y

)

0��FRO�$� ��� 0��GHO&RO�$� ��� 0��PDWGLP�$�(x

2

),

(1y

), [2, 2]

0��VZDS&RO�$� �� ��� 0��WU�$�� 0��WUDQVSRVH�$�(

1 xy 2

), x+ y,

(x 21 y

)

6LQFH�WKH�GRPDLQ�LV�DWWDFKHG�WR�WKH�REMHFW $ DV $��GRP� RQH�FDQ�DOVR�FDOO�WKHGRPDLQ�PHWKRGV�YLD $��GRP��PHWKRG� )RU�H[DPSOH�

$��GRP��WU�$�x+ y

7KH�GRFXPHQWDWLRQ�RI 'RP��0DWUL[ FRQWDLQV�D�VXUYH\�RI�WKHVH�PHWKRGV�

����

Page 132: Mupad Tutorial

- FnI:=� H[c^\ml

7KH�/LEUDULHV OLQDOJ DQG QXPHULF

%HVLGHV�V\VWHP�IXQFWLRQV�RSHUDWLQJ�RQ�PDWULFHV� WKH�OLEUDU\�� OLQDOJ FRQWDLQV�DYDULHW\�RI�RWKHU�OLQHDU�DOJHEUD�IXQFWLRQV�

LQIR�OLQDOJ�/LEUDU\ OLQDOJ� WKH OLQHDU DOJHEUD SDFNDJH

�� ,QWHUIDFH�OLQDOJ��DGG&RO� OLQDOJ��DGG5RZ�OLQDOJ��DGMRLQW� OLQDOJ��DQJOH�OLQDOJ��EDVLV� OLQDOJ��FKDUPDW�OLQDOJ��FKDUSRO\� OLQDOJ��FRO�OLQDOJ��FRPSDQLRQ� OLQDOJ��FRQFDW0DWUL[�OLQDOJ��FURVV3URGXFW� OLQDOJ��FXUO�OLQDOJ��GHO&RO� OLQDOJ��GHO5RZ�OLQDOJ��GHW� OLQDOJ��GLYHUJHQFH�OLQDOJ��HLJHQYDOXHV� OLQDOJ��HLJHQYHFWRUV�OLQDOJ��H[SU�0DWUL[� OLQDOJ��IDFWRU&KROHVN\�OLQDOJ��IDFWRU/8� OLQDOJ��IDFWRU45�OLQDOJ��IUREHQLXV)RUP� OLQDOJ��JDXVV(OLP�OLQDOJ��JDXVV-RUGDQ� OLQDOJ��JUDG�OLQDOJ��KHUPLWH)RUP� OLQDOJ��KHVVHQEHUJ�OLQDOJ��KHVVLDQ� OLQDOJ��KLOEHUW�OLQDOJ��LQW%DVLV� OLQDOJ��LQYHUVH/8�OLQDOJ��LQYKLOEHUW� OLQDOJ��LQYSDVFDO�OLQDOJ��LQYYDQGHUPRQGH� OLQDOJ��LV+HUPLWHDQ�OLQDOJ��LV3RV'HI� OLQDOJ��LV8QLWDU\�OLQDOJ��MDFRELDQ� OLQDOJ��MRUGDQ)RUP�OLQDOJ��NURQHFNHU3URGXFW� OLQDOJ��ODSODFLDQ�OLQDOJ��PDWGLP� OLQDOJ��PDWOLQVROYH�OLQDOJ��PDWOLQVROYH/8� OLQDOJ��PLQSRO\�OLQDOJ��PXOW&RO� OLQDOJ��PXOW5RZ�OLQDOJ��QFROV� OLQDOJ��QRQ=HURV�OLQDOJ��QRUPDOL]H� OLQDOJ��QURZV�OLQDOJ��QXOOVSDFH� OLQDOJ��RJ&RRUG7DE�OLQDOJ��RUWKRJ� OLQDOJ��SDVFDO�OLQDOJ��SHUPDQHQW� OLQDOJ��SRWHQWLDO�OLQDOJ��SVHXGR,QYHUVH� OLQDOJ��UDQGRP0DWUL[�OLQDOJ��UDQN� OLQDOJ��URZ�

��:H�UHIHU�WR�&KDSWHU � IRU�D�GHVFULSWLRQ�RI�JHQHUDO�OLEUDU\�RUJDQL]DWLRQ� H[SRUWLQJ� HWF�

����

Page 133: Mupad Tutorial

O^\mhkl�Zg]�FZmkb\^l

OLQDOJ��VFDODU3URGXFW� OLQDOJ��VHW&RO�OLQDOJ��VHW5RZ� OLQDOJ��VPLWK)RUP�OLQDOJ��VWDFN0DWUL[� OLQDOJ��VXEPDWUL[�OLQDOJ��VXEVWLWXWH� OLQDOJ��VXP%DVLV�OLQDOJ��VZDS&RO� OLQDOJ��VZDS5RZ�OLQDOJ��V\OYHVWHU� OLQDOJ��WRHSOLW]�OLQDOJ��WRHSOLW]6ROYH� OLQDOJ��WU�OLQDOJ��WUDQVSRVH� OLQDOJ��YDQGHUPRQGH�OLQDOJ��YDQGHUPRQGH6ROYH� OLQDOJ��YHFGLP�OLQDOJ��YHFWRU2I� OLQDOJ��YHFWRU3RWHQWLDO�OLQDOJ��ZLHGHPDQQ

6RPH�RI�WKHVH�IXQFWLRQV� VXFK�DV OLQDOJ��FRO RU OLQDOJ��GHO&RO� VLPSO\�FDOO�WKHLQWHUQDO�PHWKRGV�IRU�PDWULFHV�WKDW�ZH�KDYH�GHVFULEHG�LQ�6HFWLRQ ����� DQG�KHQFHGR�QRW�DGG�QHZ�IXQFWLRQDOLW\� +RZHYHU� OLQDOJ DOVR�FRQWDLQV�PDQ\�DGGLWLRQDODOJRULWKPV� 7KH�FRPPDQG "OLQDOJ \LHOGV�D�VKRUW�GHVFULSWLRQ�RI�DOO�IXQFWLRQV� <RXFDQ�ILQG�D�GHWDLOHG�GHVFULSWLRQ�RI�D�IXQFWLRQ�RQ�WKH�FRUUHVSRQGLQJ�KHOS�SDJH�

<RX�PD\�XVH�WKH�IXOO�QDPH OLEUDU\��IXQFWLRQ WR�FDOO�D�IXQFWLRQ�

$ � PDWUL[�>>D� E@� >F� G@@�� OLQDOJ��GHW�$�a d− b c

7KH�FKDUDFWHULVWLF�SRO\QRPLDO GHW(x In −A) RI�WKLV�PDWUL[�LV

OLQDOJ��FKDUSRO\�$� [�x2 + (−a− d)x+ a d− b c

7KH�HLJHQYDOXHV�DUH

OLQDOJ��HLJHQYDOXHV�$�{a

2+d

2−√a2 − 2 a d+ d2 + 4 b c

2,a

2+d

2+√a2 − 2 a d+ d2 + 4 b c

2

}

����

Page 134: Mupad Tutorial

- FnI:=� H[c^\ml

7KH�QXPHULFV�OLEUDU\ QXPHULF �VHH "QXPHULF��FRQWDLQV�PDQ\�IXQFWLRQV�IRUQXPHULFDO�FRPSXWDWLRQV�ZLWK�PDWULFHV�

QXPHULF��GHW � GHWHUPLQDQWQXPHULF��H[S0DWUL[ � H[S�0DWUL[�QXPHULF��IDFWRU&KROHVN\ � &KROHVN\�IDFWRUL]DWLRQQXPHULF��IDFWRU/8 � LU IDFWRUL]DWLRQQXPHULF��IDFWRU45 � QR IDFWRUL]DWLRQQXPHULF��I0DWUL[ � IXQFWLRQDO�FDOFXOXVQXPHULF��LQYHUVH � LQYHUVLRQQXPHULF��HLJHQYDOXHV � HLJHQYDOXHVQXPHULF��HLJHQYHFWRUV � HLJHQYDOXHV�DQG�±YHFWRUVQXPHULF��VLQJXODUYDOXHV � VLQJXODU�YDOXHVQXPHULF��VLQJXODUYHFWRUV � VLQJXODU�YDOXHV�DQG�YHFWRUV

3DUWLDOO\� WKHVH�URXWLQHV�ZRUN�IRU�PDWULFHV�ZLWK�V\PEROLF�HQWULHV�RI�W\SH'RP��([SUHVVLRQ)LHOG DQG�WKHQ�DUH�PRUH�HIILFLHQW�IRU�ODUJH�PDWULFHV�WKDQ�WKHOLQDOJ IXQFWLRQV� +RZHYHU� WKH�ODWWHU�FDQ�KDQGOH�DUELWUDU\�FRHIILFLHQW�ULQJV�

([HUFLVH������ )LQG�WKH�YDOXHV�RI a, b, c IRU�ZKLFK�WKH�PDWUL[

⎝1 a b1 1 c1 1 1

⎠ LV�QRW

LQYHUWLEOH�

([HUFLVH������ &RQVLGHU�WKH�IROORZLQJ�PDWULFHV�

A =

⎝1 3 0−1 2 70 8 1

⎠, B =

⎝7 −12 30 1

⎠ .

/HW BT EH�WKH�WUDQVSRVH�RI B� &RPSXWH�WKH�LQYHUVH�RI 2A+BBT � ERWK�RYHU�WKHUDWLRQDO�QXPEHUV�DQG�RYHU�WKH�UHVLGXH�FODVV�ULQJ�PRGXOR 7�

����

Page 135: Mupad Tutorial

O^\mhkl�Zg]�FZmkb\^l

([HUFLVH������ *HQHUDWH�WKH 3× 3PDWUL[

Aij ={

0 IRU i = j ,1 IRU i ̸= j .

&RPSXWH�LWV�GHWHUPLQDQW� LWV�FKDUDFWHULVWLF�SRO\QRPLDO� DQG�LWV�HLJHQYDOXHV� )RUHDFK�HLJHQYDOXH� FRPSXWH�D�EDVLV�RI�WKH�FRUUHVSRQGLQJ�HLJHQVSDFH�

6SDUVH�0DWULFHV

7KH�LQWHUQDO�VWRUDJH�RI�PDWULFHV�LV�RSWLPL]HG�IRU�VSDUVH�GDWD� L�H�� IRU�PDWULFHVFRQVLVWLQJ�ODUJHO\�RI�]HURHV� VLQFH�VXFK�PDWULFHV�DSSHDU�RIWHQ�LQ�SUDFWLFH� +HUH�DUHVRPH�UHPDUNV�FRQFHUQLQJ�HIILFLHQF\�ZKHQ�\RXU�PDWULFHV�DUH�ODUJH�DQG�VSDUVH�

� 8VH�WKH�VWDQGDUG�FRHIILFLHQW�ULQJ 'RP��([SUHVVLRQ)LHOG�� RI�DUELWUDU\0X3$'� H[SUHVVLRQV�ZKHQHYHU�SRVVLEOH� $V�GLVFXVVHG�LQ�WKH�SUHYLRXVVHFWLRQV� WKH�FRQVWUXFWRU 'RP��0DWUL[�� FUHDWHV�PDWULFHV�RI�WKLV�W\SH� )RUFRQYHQLHQFH� WKLV�FRQVWUXFWRU�LV�DOVR�DYDLODEOH�DV�WKH�IXQFWLRQ PDWUL[�

� ,QGH[HG�UHDGLQJ�DQG�ZULWLQJ�WR�ODUJH�PDWULFHV�LV�VRPHZKDW�H[SHQVLYH� ,ISRVVLEOH� RQH�VKRXOG�DYRLG�FUHDWLQJ�ODUJH�HPSW\�PDWULFHV�RI�WKH�GHVLUHGGLPHQVLRQ�E\ PDWUL[�P� Q� DQG�ILOOLQJ�LQ�WKH�QRQ�]HUR�HQWULHV�E\�LQGH[HGDVVLJQPHQWV� 2QH�VKRXOG�VHW�WKH�HQWULHV�GLUHFWO\�ZKHQ�FUHDWLQJ�WKH�PDWUL[�

/LVWV�RI�HTXDWLRQV�FDQ�EH�XVHG�WR�VSHFLI\�WKH�HQWULHV�ZKHQ�FUHDWLQJ�D�PDWUL[� 7KHIROORZLQJ�PDWUL[ A RI�GLPHQVLRQ 1000× 1000 FRQVLVWV�RI�D�GLDJRQDO�EDQG�DQG�WZRDGGLWLRQDO�HQWULHV�LQ�WKH�XSSHU�ULJKW�DQG�WKH�ORZHU�OHIW�FRUQHU� :H�GLVSOD\�VRPHHQWULHV�RI�LWV 10�WK�SRZHU�

Q � �����$ � PDWUL[�Q� Q� >�L� L� L � L ���Q�

�Q� �� �� ��� Q� �@��% � $A���%>�� �@� %>�� Q@

1002010022050086122130089, 1001009015040066101119105055

����

Page 136: Mupad Tutorial

- FnI:=� H[c^\ml

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�JHQHUDWH�D 100× 100 WUL�GLDJRQDO�7RHSOLW]�PDWUL[ $DQG�VROYH�WKH�HTXDWLRQ Ax = b� ZKHUH b LV�WKH�FROXPQ�YHFWRU (1, . . . , 1)�

$ � PDWUL[����� ���� >��� �� ��@� %DQGHG��E � PDWUL[����� �� >� � ���@��[ � ���$� E⎛

⎜⎜⎜⎜⎝

5099. . .

50

⎟⎟⎟⎟⎠

+HUH�ZH�KDYH�DSSOLHG�WKH�LQYHUVH ��$ RI $ WR�WKH�ULJKW�KDQG�VLGH�RI�WKH�HTXDWLRQ�1RWH� KRZHYHU� WKDW�LQYHUWLQJ�D�VSDUVH�PDWUL[�LV�QRW�D�JRRG�LGHD�EHFDXVH�WKHLQYHUVH�RI�D�VSDUVH�PDWUL[�LV� LQ�JHQHUDO� QRW�VSDUVH� ,W�LV�PXFK�PRUH�HIILFLHQW�WRXVH�VSDUVH�PDWUL[�IDFWRUL]DWLRQ�WR�FRPSXWH�WKH�VROXWLRQ�RI�D�VSDUVH�V\VWHP�RIOLQHDU�HTXDWLRQV� :H�XVH�WKH�OLQHDU�VROYHU QXPHULF��PDWOLQVROYH ZLWK�WKH�RSWLRQ6\PEROLF WR�FRPSXWH�WKH�VROXWLRQ�RI�D�VSDUVH�V\VWHP�UHSUHVHQWLQJ������HTXDWLRQVIRU������XQNQRZQV� 7KH QXPHULF URXWLQH�HPSOR\V�VSDUVLW\�LQ�DQ�RSWLPDO�ZD\�

$ � PDWUL[������ ����� >��� �� ��@� %DQGHG��E � PDWUL[������ �� >� � ����@��>[� NHUQHO@ � QXPHULF��PDWOLQVROYH�$� E� 6\PEROLF��

:H�GLVSOD\�RQO\�D�IHZ�FRPSRQHQWV�RI�WKH�VROXWLRQ�YHFWRU�

[>�@� [>�@� [>�@� [>�@� [>�@� [>���@� [>����@500, 999, 1497, 1994, 2490, 999, 500

$Q�$SSOLFDWLRQ

:H�ZDQW�WR�FRPSXWH�WKH�V\PEROLF�VROXWLRQ a(t), b(t) RI�WKH�V\VWHP�RI�VHFRQG�RUGHUGLIIHUHQWLDO�HTXDWLRQV

G2

Gt2a(t) = 2 c

GGtb(t) ,

G2

Gt2b(t) = −2 c

GGta(t) + 3 c2 b(t)

ZLWK�DQ�DUELWUDU\�FRQVWDQW c� :ULWLQJ a′(t) = GGta(t)� b

′(t) = GGtb(t)� WKHVH�HTXDWLRQV

PD\�EH�HTXLYDOHQWO\�ZULWWHQ�DV�D�V\VWHP�RI�ILUVW�RUGHU�GLIIHUHQWLDO�HTXDWLRQV�LQ�WKH

����

Page 137: Mupad Tutorial

O^\mhkl�Zg]�FZmkb\^l

YDULDEOHV x(t) = (a(t), a′(t), b(t), b′(t))�

GGtx(t) = Ax(t), A =

⎜⎜⎝

0 1 0 00 0 0 2 c0 0 0 10 −2 c 3 c2 0

⎟⎟⎠ .

7KH�VROXWLRQ�RI�WKLV�V\VWHP�LV�JLYHQ�E\�DSSO\LQJ�WKH�H[SRQHQWLDO�PDWUL[ etA WR�WKHLQLWLDO�FRQGLWLRQ x(0)�

x(t) = etAx(0) .

GHOHWH F� W�$ � PDWUL[�>>�� �� �� � @�

>�� �� �� � F@�>�� �� �� � @�>�� �� F� � FA�� � @@��

:H�XVH�WKH�IXQFWLRQ H[S WR�FRPSXWH B = etA�

% � H[S�W $�⎛

⎜⎜⎜⎜⎝

1 −3 t+ H−c t L 2 Lc − Hc t L 2 L

c Hc t L 3 L+ 6 c t− H−c t L 3 L 2c −

Hc t L

c −H−c t L

c

0 2 H−c t L + 2 Hc t L − 3 6 c− 3 c H−c t L − 3 c Hc t L H−c t L L− Hc t L L0 H−c t L

c + Hc t L

c −2c 4− 3 Hc t L

2 − 3 H−c t L

2H−c t L Lc 2 −

Hc t L Lc 2

0 −H−c t L L+ Hc t L L c H−c t L 3 L2 − c H

c t L 3 L2

H−c t L

2 + Hc t L

2

⎟⎟⎟⎟⎠

7R�VLPSOLI\�WKLV�PDWUL[� ZH�XVH�WKH�IXQFWLRQ 6LPSOLI\� ZKLFK�IRU�PDWULFHV�VLPSO\PDSV�LWVHOI�WR�WKH�HQWULHV�

% � 6LPSOLI\�%�⎛

⎜⎜⎜⎜⎝

1 4 VLQ(c t)c − 3 t 6 c t− 6 VLQ(c t) −2 (FRV(c t)−1)

c

0 4 FRV(c t)− 3 −6 c (FRV(c t)− 1) 2 VLQ(c t)0 2 (FRV(c t)−1)

c 4− 3 FRV(c t) VLQ(c t)c

0 −2 VLQ(c t) 3 c VLQ(c t) FRV(c t)

⎟⎟⎟⎟⎠

����

Page 138: Mupad Tutorial

- FnI:=� H[c^\ml

:H�DVVLJQ�D�V\PEROLF�LQLWLDO�FRQGLWLRQ�WR�WKH�YHFWRU x0� 7KH�IROORZLQJ�FDOO�FUHDWHV�D4× 1PDWUL[�WR�EH�LQWHUSUHWHG�DV�D�FROXPQ�YHFWRU�

[B� � PDWUL[�>DB�� DB�� EB�� EB�@�⎛

⎜⎜⎜⎜⎝

a0

a′0b0

b′0

⎟⎟⎟⎟⎠

7KXV� WKH�GHVLUHG�V\PEROLF�VROXWLRQ�RI�WKH�V\VWHP�RI�GLIIHUHQWLDO�HTXDWLRQV�LV

[ � W ��! % [B��

7KH�VROXWLRQ�IXQFWLRQV a(t) DQG b(t) ZLWK�WKH�V\PEROLF�LQLWLDO�FRQGLWLRQV a(0) = a0�a′(0) = a′0� b(0) = b0� b′(0) = b′0 DUH�

D � W ��! H[SDQG�[�W�>�@�� D�W�

a0 − 6 b0 VLQ(c t)− 3 t a′0 +2 b′0c− 2 FRV(c t) b′0

c+

4 a′0 VLQ(c t)c

+ 6 b0 c t

E � W ��! H[SDQG�[�W�>�@�� E�W�

4 b0 − 3 b0 FRV(c t)−2 a′0c

+2 FRV(c t) a′0

c+b′0 VLQ(c t)c

)LQDOO\� ZH�FKHFN�WKDW�WKH�DERYH�H[SUHVVLRQV�UHDOO\�VROYH�WKH�V\VWHP�RI�GLIIHUHQWLDOHTXDWLRQV�

H[SDQG�GLII�D�W��W�W� � � F GLII�E�W��W���H[SDQG�GLII�E�W��W�W� � � F GLII�D�W��W� � � FA� E�W��

0, 0

����

Page 139: Mupad Tutorial

IherghfbZel

HgdqfgeaYdk

&RPSXWDWLRQ�ZLWK�SRO\QRPLDOV�LV�DQ�LPSRUWDQW�WDVN�IRU�D�FRPSXWHU�DOJHEUDV\VWHP� 2I�FRXUVH� \RX�PD\�UHDOL]H�D�SRO\QRPLDO�LQ�0X3$'� DV�DQ�H[SUHVVLRQ�LQWKH�VHQVH�RI�6HFWLRQ ��� DQG�XVH�WKH�VWDQGDUG�DULWKPHWLF�

SRO\QRPLDO([SUHVVLRQ � � � [ � [A��H[SDQG�SRO\QRPLDO([SUHVVLRQA��x4 + 2x3 + 3x2 + 2x+ 1

+RZHYHU� WKHUH�H[LVWV�D�VSHFLDO�GDWD�W\SH '20B32/< WRJHWKHU�ZLWK�VRPH�NHUQHO�DQGOLEUDU\�IXQFWLRQV� ZKLFK�VLPSOLILHV�VXFK�FRPSXWDWLRQV�DQG�PDNHV�WKHP�PRUHHIILFLHQW�

'HILQLWLRQ�RI�3RO\QRPLDOV

7KH�V\VWHP�IXQFWLRQ SRO\ JHQHUDWHV�SRO\QRPLDOV�

SRO\�� � � [ � � [A��SRO\

(3x2 + 2x+ 1, [x]

)

+HUH�ZH�KDYH�VXSSOLHG�WKH�H[SUHVVLRQ 1 + 2x+ 3x2 �RI�GRPDLQ�W\SH '20B(;35� WRSRO\� ZKLFK�FRQYHUWV�WKLV�H[SUHVVLRQ�WR�D�QHZ�REMHFW�RI�GRPDLQ�W\SH '20B32/<�7KH�LQGHWHUPLQDWH >[@ LV�D�IL[HG�SDUW�RI�WKLV�GDWD�W\SH� 7KLV�LV�UHOHYDQW�IRUGLVWLQJXLVKLQJ�EHWZHHQ�LQGHWHUPLQDWHV�DQG��V\PEROLF��FRHIILFLHQWV�RU�SDUDPHWHUV�)RU�H[DPSOH� LI�\RX�ZDQW�WR�UHJDUG�WKH�H[SUHVVLRQ a0 + a1 x+ a2 x2 DV�DSRO\QRPLDO�LQ x ZLWK�FRHIILFLHQWV a0, a1, a2� WKHQ�WKH�DERYH�IRUP�RI�WKH�FDOO�WR SRO\GRHV�QRW�\LHOG�WKH�GHVLUHG�UHVXOW�

SRO\�DB� � DB� [ � DB� [A��SRO\

(a0 + a1 x+ a2 x2, [a0, a1, a2, x]

)

7KLV�GRHV�QRW�UHSUHVHQW�D�SRO\QRPLDO�LQ x EXW�D�³PXOWLYDULDWH´�SRO\QRPLDO�LQ�IRXULQGHWHUPLQDWHV x, a0, a1, a2� <RX�FDQ�VSHFLI\�WKH�LQGHWHUPLQDWHV�RI�D�SRO\QRPLDO�LQIRUP�RI�D�OLVW�DV�DUJXPHQW�WR SRO\� 7KH�V\VWHP�WKHQ�UHJDUGV�DOO�RWKHU�V\PEROLFLGHQWLILHUV�DV�V\PEROLF�FRHIILFLHQWV�

����

Page 140: Mupad Tutorial

- FnI:=� H[c^\ml

SRO\�DB� � DB� [ � DB� [A�� >[@�SRO\

(a2 x

2 + a1 x+ a0, [x])

,I�\RX�GR�QRW�VSHFLI\�D�OLVW�RI�LQGHWHUPLQDWHV� WKHQ SRO\ FDOOV�WKH�IXQFWLRQ LQGHWVWR�GHWHUPLQH�DOO�V\PEROLF�LGHQWLILHUV�LQ�WKH�H[SUHVVLRQ�DQG�LQWHUSUHWV�WKHP�DVLQGHWHUPLQDWHV�RI�WKH�SRO\QRPLDO�

LQGHWV�DB� � DB� [ � DB� [A�� 3RO\([SU�{a0, a1, a2, x}

7KH�GLVWLQFWLRQ�EHWZHHQ�LQGHWHUPLQDWHV�DQG�FRHIILFLHQWV�LV�UHOHYDQW�IRU�WKHUHSUHVHQWDWLRQ�RI�WKH�SRO\QRPLDO�

H[SUHVVLRQ � � � [ � [A� � D [ � 3, [A� � Ex− b+ a x+ π x2 + x2 + 1

SRO\�H[SUHVVLRQ� >D� [@�SRO\

(a x+ (π + 1) x2 + x+ (1− b) , [a, x]

)

SRO\�H[SUHVVLRQ� >[@�SRO\

((π + 1) x2 + (a+ 1) x+ (1− b) , [x]

)

<RX�FDQ�VHH�WKDW�0X3$' FROOHFWV�WKH�FRHIILFLHQWV�RI�HTXDO�SRZHUV�RI�WKHLQGHWHUPLQDWH� 7KH�WHUPV�DUH�VRUWHG�DFFRUGLQJ�WR�IDOOLQJ�H[SRQHQWV�

,QVWHDG�RI�XVLQJ�DQ�H[SUHVVLRQ� \RX�PD\�DOVR�JHQHUDWH�D�SRO\QRPLDO�E\�VSHFLI\LQJD�OLVW�RI�WKH�QRQ�]HUR�FRHIILFLHQWV�WRJHWKHU�ZLWK�WKH�UHVSHFWLYH�H[SRQHQWV� 7KHFRPPDQG SRO\�OLVW�>[@� JHQHUDWHV�WKH�SRO\QRPLDO

∑ki=0 ai x

ni IURP�WKH�OLVW[[a0, n0], [a1, n1], . . . , [ak, nk]]�

OLVW � >>�� �@� >D� �@� >E� �@@� SRO\�OLVW� >[@�SRO\

(b x5 + a x3 + 1, [x]

)

,I�\RX�ZDQW�WR�FRQVWUXFW�D�PXOWLYDULDWH�SRO\QRPLDO�LQ�WKLV�ZD\� VSHFLI\�OLVWV�RIH[SRQHQWV�IRU�DOO�YDULDEOHV�

SRO\�>>�� >�� �@@� >�� >�� �@@@� >[� \@�SRO\

(2x3 y4 + 3x2 y, [x, y]

)

����

Page 141: Mupad Tutorial

IherghfbZel

&RQYHUVHO\� WKH�IXQFWLRQ SRO\�OLVW FRQYHUWV�D�SRO\QRPLDO�WR�D�OLVW�RI�FRHIILFLHQWVDQG�H[SRQHQWV�

SRO\�OLVW�SRO\�E [A� � D [A� � �� >[@� �[[b, 5] , [a, 3] , [1, 0]]

)RU�PRUH�DEVWUDFW�FRPSXWDWLRQV� \RX�PD\�ZDQW�WR�UHVWULFW�WKH�FRHIILFLHQWV�RI�DSRO\QRPLDO�WR�D�FHUWDLQ�VHW��PDWKHPDWLFDOO\� D�ULQJ��ZKLFK�LV�UHSUHVHQWHG�E\�DVSHFLDO�GDWD�VWUXFWXUH� :H�KDYH�DOUHDG\�VHHQ�W\SLFDO�H[DPSOHV�RI�ULQJV�DQG�WKHLUFRUUHVSRQGLQJ�0X3$' GRPDLQV�LQ�6HFWLRQ ����� WKH�LQWHJHUV 'RP��,QWHJHU� WKHUDWLRQDO�QXPEHUV 'RP��5DWLRQDO� RU�WKH�UHVLGXH�FODVV�ULQJ 'RP��,QWHJHU0RG�Q�RI�LQWHJHUV�PRGXOR n� <RX�PD\�VSHFLI\�WKH�FRHIILFLHQW�ULQJ�DV�DUJXPHQW�WR SRO\�

SRO\�[ � �� >[@� 'RP��,QWHJHU�SRO\ (x+ 1, [x] ,Z)

SRO\�� [ � ���� >[@� 'RP��5DWLRQDO�

SRO\

(2x−

(12

), [x] ,Q

)

SRO\�� [ � ��� >[@� 'RP��,QWHJHU0RG����SRO\ (x+ 2, [x] ,Z3)

1RWH�WKDW�LQ�WKH�ODVW�H[DPSOH� WKH�V\VWHP�KDV�DXWRPDWLFDOO\�VLPSOLILHG�WKHFRHIILFLHQWV�DFFRUGLQJ�WR�WKH�UXOHV�IRU�FRPSXWLQJ�ZLWK�LQWHJHUV�PRGXOR 3���

� PRG �� �� PRG �1, 2

,Q�WKH�IROORZLQJ�H[DPSOH� SRO\ FRQYHUWV�WKH�FRHIILFLHQWV�WR�IORDWLQJ�SRLQWQXPEHUV� DV�VSHFLILHG�E\�WKH�WKLUG�DUJXPHQW�

SRO\�3, [ � ���� >[@� 'RP��)ORDW�SRO\ (3.141592654x− 0.5, [x] ,'RP::)ORDW)

��)RU �SRO\QRPLDOV� \RX �PD\ �DOVR �XVH ,QW0RG��� LQVWHDG �RI 'RP��,QWHJHU0RG���� LQ � WKH � IRUPSRO\�� [����>[@�,QW0RG����� 7KHQ�WKH�LQWHJHUV�PRGXOR 3 DUH�UHSUHVHQWHG�E\ −1, 0, 1 DQG�QRW� DVIRU 'RP��,QWHJHU0RG���� E\ 0, 1, 2� 3RO\QRPLDO�DULWKPHWLF�LV�PXFK�IDVWHU�ZKHQ�\RX�XVH ,QW0RG����

����

Page 142: Mupad Tutorial

- FnI:=� H[c^\ml

,I�QR�FRHIILFLHQW�ULQJ�LV�VSHFLILHG� WKHQ�0X3$' E\�GHIDXOW�XVHV�WKH�ULQJ ([SU ZKLFKV\PEROL]HV�DUELWUDU\�0X3$' H[SUHVVLRQV� ,Q�WKLV�FDVH� \RX�PD\�XVH�V\PEROLFLGHQWLILHUV�DV�FRHIILFLHQWV�

SRO\QRPLDO � SRO\�D � [ � E \� >[� \@��RS�SRO\QRPLDO�

SRO\ (x+ b y + a, [x, y])

a+ x+ b y, [x, y] , ([SU

:H�VXPPDUL]H�WKDW�D�0X3$' SRO\QRPLDO�FRPSULVHV�WKUHH�SDUWV�

�� D�SRO\QRPLDO�H[SUHVVLRQ�RI�WKH�IRUP∑ai1i2... x

i11 xi22 . . .�

�� D�OLVW�RI�LQGHWHUPLQDWHV [x1, x2, . . .]�

�� WKH�FRHIILFLHQW�ULQJ�

7KHVH�DUH�WKH�WKUHH�RSHUDQGV�RI�D�0X3$' SRO\QRPLDO S� ZKLFK�FDQ�EH�DFFHVVHG�YLDRS�S���� RS�S���� DQG RS�S���� UHVSHFWLYHO\� 7KXV� \RX�PD\�FRQYHUW�DSRO\QRPLDO�WR�D�PDWKHPDWLFDOO\�HTXLYDOHQW�H[SUHVVLRQ�� RI�GRPDLQ�W\SH '20B(;35E\

H[SUHVVLRQ � RS�SRO\QRPLDO� ���

+RZHYHU� \RX�VKRXOG�SUHIHUDEO\�XVH�WKH�V\VWHP�IXQFWLRQ H[SU� ZKLFK�FDQ�FRQYHUWYDULRXV�GRPDLQ�W\SHV�VXFK�DV�SRO\QRPLDOV�WR�H[SUHVVLRQV�

SRO\QRPLDO � SRO\�[A� � � [ � ��SRO\

(x3 + 5x+ 3, [x]

)

RS�SRO\QRPLDO� �� H[SU�SRO\QRPLDO�x3 + 5x+ 3 = x3 + 5x+ 3

��,I�WKH�SRO\QRPLDO�LV�GHILQHG�RYHU�D�ULQJ�RWKHU�WKDQ 'RP��([SUHVVLRQ)LHOG RU ([SU� WKH�UHVXOW�PD\QRW�EH�HTXLYDOHQW�

����

Page 143: Mupad Tutorial

IherghfbZel

&RPSXWLQJ�ZLWK�3RO\QRPLDOV

7KH�IXQFWLRQ GHJUHH GHWHUPLQHV�WKH�GHJUHH�RI�D�SRO\QRPLDO�

S � SRO\�� � [ � D [A� \� >[� \@��GHJUHH�S� [�� GHJUHH�S� \�

2, 1

,I�\RX�GR�QRW�VSHFLI\�WKH�QDPH�RI�DQ�LQGHWHUPLQDWH�DV�VHFRQG�DUJXPHQW� WKHQGHJUHH UHWXUQV�WKH�³WRWDO�GHJUHH´�

GHJUHH�S�� GHJUHH�SRO\�[A�� � [ � ���3, 27

7KH�IXQFWLRQ FRHII H[WUDFWV�FRHIILFLHQWV�IURP�D�SRO\QRPLDO�

S � SRO\�� � D [ � � [A�� >[@��FRHII�S� ��� FRHII�S� ��� FRHII�S� ��a, 0, 0

)RU�PXOWLYDULDWH�SRO\QRPLDOV� WKH�FRHIILFLHQW�RI�D�SRZHU�RI�RQH�SDUWLFXODULQGHWHUPLQDWH�LV�DJDLQ�D�SRO\QRPLDO�LQ�WKH�UHPDLQLQJ�LQGHWHUPLQDWHV�

S � SRO\�� � [ � D [A� \� >[� \@��FRHII�S� \� ��� FRHII�S� \� ��

SRO\ (x+ 1, [x]) , SRO\(a x2, [x]

)

7KH�VWDQGDUG�RSHUDWRUV �� �� DQG A ZRUN�IRU�SRO\QRPLDO�DULWKPHWLF�DV�ZHOO�

S � SRO\�� � D [A�� >[@�� T � SRO\�E � F [� >[@��S � T� S � T�S T� SA�

SRO\(a x2 + c x+ (b+ 1) , [x]

), SRO\

(a x2 + (−c) x+ (1− b) , [x]

)

SRO\((a c) x3 + (a b) x2 + c x+ b, [x]

), SRO\

(a2 x4 + (2 a) x2 + 1, [x]

)

����

Page 144: Mupad Tutorial

- FnI:=� H[c^\ml

7KH�IXQFWLRQ GLYLGH SHUIRUPV�D�³GLYLVLRQ�ZLWK�UHPDLQGHU´�

S � SRO\�[A� � ��� T � SRO\�[A� � ��� GLYLGH�S� T�SRO\ (x, [x]) , SRO\ (x+ 1, [x])

7KH�UHVXOW�LV�D�VHTXHQFH�ZLWK�WZR�RSHUDQGV� WKH�TXRWLHQW�DQG�WKH�UHPDLQGHU�RI�WKHGLYLVLRQ�

>TXRWLHQW� UHPDLQGHU@ � >GLYLGH�S� T�@�

:H�FKHFN�

S TXRWLHQW T � UHPDLQGHUSRO\

(x3 + 1, [x]

)= SRO\

(x3 + 1, [x]

)

7KH�SRO\QRPLDO�GHQRWHG�E\ UHPDLQGHU LV�RI�ORZHU�GHJUHH�WKDQ T� ZKLFK�PDNHV�WKHGHFRPSRVLWLRQ S TXRWLHQW T�UHPDLQGHU XQLTXH� 'LYLGLQJ�WZR�SRO\QRPLDOV�E\PHDQV�RI�WKH�XVXDO�GLYLVLRQ�RSHUDWRU � LV�RQO\�DOORZHG�LQ�WKH�VSHFLDO�FDVH�ZKHQ�WKHUHPDLQGHU�WKDW GLYLGH ZRXOG�UHWXUQ�YDQLVKHV�S � SRO\�[A� � ��� T � SRO\�[ � ��� S�T

SRO\ (x+ 1, [x])

S � SRO\�[A� � ��� T � SRO\�[ � ��� S�T)$,/

1RWH�WKDW�WKH�DULWKPHWLF�RSHUDWRUV�SURFHVV�RQO\�SRO\QRPLDOV�RI�H[DFWO\�LGHQWLFDOW\SHV�

SRO\�[ � \� >[� \@� � SRO\�[A�� >[� \@��SRO\�[� � SRO\�[� >[@� ([SU�

SRO\(x2 + x+ y, [x, y]

), SRO\ (2x, [x])

%RWK�WKH�OLVW�RI�LQGHWHUPLQDWHV�DQG�WKH�FRHIILFLHQW�ULQJ�PXVW�FRLQFLGH� RWKHUZLVHWKH�V\VWHP�UHWXUQV�WKH�LQSXW�DV�D�V\PEROLF�H[SUHVVLRQ�

SRO\�[ � \� >[� \@� � SRO\�[A�� >[@�SRO\ (x+ y, [x, y]) + SRO\

(x2, [x]

)

����

Page 145: Mupad Tutorial

IherghfbZel

SRO\�[� 'RP��,QWHJHU� � SRO\�[�SRO\ (x, [x] ,Z) + SRO\ (x, [x])

7KH�SRO\QRPLDO�DULWKPHWLF�SHUIRUPV�FRHIILFLHQW�DGGLWLRQV�DQG�PXOWLSOLFDWLRQVDFFRUGLQJ�WR�WKH�UXOHV�RI�WKH�FRHIILFLHQW�ULQJ�

S � SRO\�� [ � ��� >[@� 'RP��,QWHJHU0RG�����S� S � S� S S

SRO\ (x+ 2, [x] ,Z3)

SRO\ (2x+ 1, [x] ,Z3)

SRO\(x2 + x+ 1, [x] ,Z3

)

7KH�VWDQGDUG�RSHUDWRU GRHV�QRW�ZRUN�LPPHGLDWHO\�IRU�PXOWLSO\LQJ�D�SRO\QRPLDOE\�D�VFDODU� \RX�QHHG�WR�FRQYHUW�WKH�VFDODU�IDFWRU�WR�D�SRO\QRPLDO�ILUVW�

S � SRO\�[A� � \��VFDODU S� SRO\�VFDODU� RS�S� ������ S

VFDODU SRO\(x2 + y, [x, y]

)

SRO\(VFDODUx2 + VFDODU y, [x, y]

)

+HUH�ZH�KDYH�HQVXUHG�WKDW�WKH�SRO\QRPLDO�JHQHUDWHG�IURP�WKH�VFDODU�IDFWRU�LV�RIWKH�VDPH�W\SH�DV S E\�SDVVLQJ RS�S������ � >[�\@�([SU��DV�IXUWKHU�DUJXPHQWV�WRSRO\� $OWHUQDWLYHO\� WKH�IXQFWLRQ PXOWFRHIIVPXOWLSOLHV�DOO�FRHIILFLHQWV�RI�DSRO\QRPLDO�E\�D�VFDODU�IDFWRU�

PXOWFRHIIV�S� VFDODU�SRO\

(VFDODUx2 + VFDODU y, [x, y]

)

7KH�IXQFWLRQ PDSFRHIIV DSSOLHV�DQ�DUELWUDU\�IXQFWLRQ�WR�DOO�FRHIILFLHQWV�RI�DSRO\QRPLDO�

S � SRO\�� [A� � � \�� PDSFRHIIV�S� I�SRO\

(f(2) x2 + f(3) y, [x, y]

)

����

Page 146: Mupad Tutorial

- FnI:=� H[c^\ml

7KLV�\LHOGV�DQRWKHU�ZD\�RI�PXOWLSOLFDWLRQ�E\�D�VFDODU�

PDSFRHIIV�S� BPXOW� VFDODU�SRO\

((2 VFDODU) x2 + (3 VFDODU) y, [x, y]

)

$QRWKHU�LPSRUWDQW�RSHUDWLRQ�LV�WKH�HYDOXDWLRQ�RI�D�SRO\QRPLDO�DW�D�SRLQW�FRPSXWLQJ�WKH�LPDJH�YDOXH�� 7KH�IXQFWLRQ HYDOS DFKLHYHV�WKLV�

S � SRO\�[A� � �� >[@��HYDOS�S� [ ��� HYDOS�S� [ [ � \�

5, (x+ y)2 + 1

7KLV�FRPSXWDWLRQ�LV�DOVR�YDOLG�IRU�PXOWLYDULDWH�SRO\QRPLDOV�DQG�\LHOGV�DSRO\QRPLDO�LQ�WKH�UHPDLQLQJ�LQGHWHUPLQDWHV�RU� IRU�D�XQLYDULDWH�SRO\QRPLDO� DQHOHPHQW�RI�WKH�FRHIILFLHQW�ULQJ�

S � SRO\�[A� � \��T � HYDOS�S� [ ��� HYDOS�T� \ ��

SRO\ (y, [y])

2

<RX�FDQ�DOVR�XVH�WKH�PRUH�JHQHUDO�HYDOXDWLRQ�RSHUDWRU _ IRU�WKH�VDPH�HIIHFW�

S _ [ �� S _ >[ �� \ �@SRO\ (y, [y])

2

<RX�PD\�DOVR�UHJDUG�D�SRO\QRPLDO�DV�D�IXQFWLRQ�RI�WKH�LQGHWHUPLQDWHV�DQG�FDOO�WKLVIXQFWLRQ�ZLWK�DUJXPHQWV�

S��� ]�z + 4

$ YDULHW\�RI�0X3$' IXQFWLRQV�DFFHSW�SRO\QRPLDOV�DV�LQSXW� $Q�LPSRUWDQWRSHUDWLRQ�LV�IDFWRUL]DWLRQ� ZKLFK�LV�SHUIRUPHG�DFFRUGLQJ�WR�WKH�UXOHV�RI�WKHFRHIILFLHQW�ULQJ� <RX�FDQ�GR�IDFWRUL]DWLRQ�ZLWK�WKH�0X3$' IXQFWLRQ IDFWRU�

����

Page 147: Mupad Tutorial

IherghfbZel

IDFWRU�SRO\�[A� � ���SRO\ (x− 1, [x]) · SRO\

(x2 + x+ 1, [x]

)

IDFWRU�SRO\�[A� � �� 'RP��,QWHJHU0RG�����

SRO\ (x+ 1, [x] ,Z2)2

7KH�IXQFWLRQ ' GLIIHUHQWLDWHV�SRO\QRPLDOV�

'�SRO\�[A� � [ � ���SRO\

(7x6 + 1, [x]

)

(TXLYDOHQWO\� \RX�PD\�DOVR�XVH GLII�SRO\QRPLDO�[��

,QWHJUDWLRQ�DOVR�ZRUNV�IRU�SRO\QRPLDOV�

S � SRO\�[A� � [ � ��� LQW�S� [�

SRO\

((18

)x8 +

(12

)x2 + x, [x]

)

7KH�IXQFWLRQ JFG FRPSXWHV�WKH�JUHDWHVW�FRPPRQ�GLYLVRU�RI�SRO\QRPLDOV�

S � SRO\��[ � ��A� �[ � ����T � SRO\��[ � �� �[ � ��A���IDFWRU�JFG�S� T��

SRO\ (x+ 2, [x]) · SRO\ (x+ 1, [x])

7KH�LQWHUQDO�UHSUHVHQWDWLRQ�RI�D�SRO\QRPLDO�VWRUHV�RQO\�WKRVH�SRZHUV�RI�WKHLQGHWHUPLQDWHV�ZLWK�QRQ�]HUR�FRHIILFLHQWV� 7KLV�LV�SDUWLFXODUO\�DGYDQWDJHRXV�IRU³VSDUVH´�SRO\QRPLDOV�RI�KLJK�GHJUHH�DQG�PDNHV�DULWKPHWLF�ZLWK�VXFK�SRO\QRPLDOVHIILFLHQW� 7KH�IXQFWLRQ QWHUPV UHWXUQV�WKH�QXPEHU�RI�QRQ�]HUR�WHUPV�RI�DSRO\QRPLDO� 7KH�IXQFWLRQ QWKPRQRPLDO H[WUDFWV�LQGLYLGXDO�PRQRPLDOV��FRHIILFLHQWWLPHV�SRZHUV�RI�WKH�LQGHWHUPLQDWHV�� QWKFRHII DQG QWKWHUP UHWXUQ�WKHDSSURSULDWH�FRHIILFLHQW�DQG�SURGXFW�RI�SRZHUV�RI�WKH�LQGHWHUPLQDWHV� UHVSHFWLYHO\�

S � SRO\�D [A��� � E [A�� � F� >[@��

����

Page 148: Mupad Tutorial

- FnI:=� H[c^\ml

QWHUPV�S�� QWKPRQRPLDO�S� ���QWKFRHII�S� ��� QWKWHUP�S� ��

3, SRO\(b x10, [x]

), b, SRO\

(x10, [x]

)

7DEOH ��� LV�D�VXPPDU\�RI�WKH�RSHUDWLRQV�IRU�SRO\QRPLDOV�GLVFXVVHG�DERYH� 6HFWLRQ³)XQFWLRQV�IRU�3RO\QRPLDOV´�RI�WKH�0X3$' 4XLFN�5HIHUHQFH�OLVWV�IXUWKHU�IXQFWLRQVIRU�SRO\QRPLDOV�LQ�WKH�VWDQGDUG�OLEUDU\� 7KH JURHEQHU OLEUDU\�FRPSULVHV�IXQFWLRQVIRU�KDQGOLQJ�PXOWLYDULDWH�SRO\QRPLDO�LGHDOV��VHH "JURHEQHU��

([HUFLVH������ &RQVLGHU�WKH�SRO\QRPLDOV p = x7 − x4 + x3 − 1 DQG q = x3 − 1�&RPSXWH p− q2� 'RHV q GLYLGH p" )DFWRU p DQG q�

([HUFLVH������ $ SRO\QRPLDO�LV�FDOOHG�LUUHGXFLEOH��RYHU�D�FRHIILFLHQW�ILHOG��LI�LWFDQQRW�EH�IDFWRUHG�LQWR�D�SURGXFW�RI�PRUH�WKDQ�RQH�QRQFRQVWDQW�SRO\QRPLDOV� 7KHIXQFWLRQ LUUHGXFLEOH WHVWV�D�SRO\QRPLDO�IRU�LUUHGXFLELOLW\� )LQG�DOO�LUUHGXFLEOHTXDGUDWLF�SRO\QRPLDOV a x2 + b x+ c , a ̸= 0 RYHU�WKH�ILHOG�RI�LQWHJHUV�PRGXOR 3�

����

Page 149: Mupad Tutorial

IherghfbZel

�� �� � A � DULWKPHWLFFRHII � H[WUDFW�FRHIILFLHQWVGHJUHH � SRO\QRPLDO�GHJUHHGLII� ' � GLIIHUHQWLDWLRQGLYLGH � GLYLVLRQ�ZLWK�UHPDLQGHUHYDOS � HYDOXDWLRQH[SU � FRQYHUVLRQ�WR�H[SUHVVLRQIDFWRU � IDFWRUL]DWLRQJFG � JUHDWHVW�FRPPRQ�GLYLVRUPDSFRHIIV � DSSO\�D�IXQFWLRQPXOWFRHIIV � PXOWLSOLFDWLRQ�E\�D�VFDODUQWHUPV � QXPEHU�RI�QRQ�]HUR�FRHIILFLHQWVQWKFRHII � n�WK�FRHIILFLHQWQWKPRQRPLDO � n�WK�PRQRPLDOQWKWHUP � n�WK�WHUPSRO\ � FRQVWUXFW�D�SRO\QRPLDOSRO\�OLVW � FRQYHUVLRQ�WR�OLVW

7DEOH ���� 0X3$'� IXQFWLRQV�RSHUDWLQJ�RQ�SRO\QRPLDOV

����

Page 150: Mupad Tutorial

- FnI:=� H[c^\ml

@Yj\oYj]�>dgYl�9jjYqk

8VXDOO\� IORDWLQJ�SRLQW�FRPSXWDWLRQV�LQ�FRPSXWHU�DOJHEUD�V\VWHPV�DUH�SHUIRUPHGLQ�³VRIWZDUH�IORDWV�´ SURYLGLQJ�XVHU�FRQWUROODEOH�SUHFLVLRQ��LQ�0X3$'�� YLD�WKHHQYLURQPHQW�YDULDEOH ',*,76� DW�WKH�H[SHQVH�RI�PHPRU\�DQG�VSHHG� 7KHUH�DUHFDVHV� KRZHYHU� ZKHUH�HVSHFLDOO\�WKH�PHPRU\�UHTXLUHPHQWV�RI�VRIWZDUH�IORDWV�DUHSURKLELWLYH� HVSHFLDOO\�ZKHQ�LPSRUWLQJ�GDWD�IURP�DQ�H[WHUQDO�VRXUFH� VXFK�DVLPDJH�ILOHV� )RU�VXFK�RFFDVLRQV� 0X3$' YHUVLRQ���LQWURGXFHG�D�QHZ�GDWD�W\SH�DUUD\V�RI�KDUGZDUH�IORDWLQJ�SRLQW�QXPEHUV� '20B+)$55$<�

$ KDUGZDUH�IORDWLQJ�SRLQW�DUUD\�LV�FUHDWHG�E\�WKH�IXQFWLRQ KIDUUD\� ZKLFK�PLPLFVDUUD\ �FI� &KDSWHU �����FORVHO\� ,W�FDQ�FRQWDLQ�RQO\�IORDWLQJ�SRLQW�QXPEHUV��UHDODQG�RU�FRPSOH[�� DQG�FRPSOHWHO\�LJQRUHV�WKH�YDOXHV�RI ',*,76 IRU�LWVFRPSXWDWLRQV� ZKLFK�DUH�SHUIRUPHG�RQ�KDUGZDUH�IORDWV�ZLWK�ZKDWHYHU�SUHFLVLRQWKH�XQGHUO\LQJ�KDUGZDUH�DQG�RU�RSHUDWLQJ�V\VWHP�SURYLGH� �8VXDOO\� WKLV�PHDQVVRPHWKLQJ�DURXQG 15 GHFLPDO�GLJLWV�� $FFHVVLQJ�HOHPHQWV�FRQYHUWV�WKHP�WR�DQGIURP�VRIWZDUH�IORDWV�

$ � KIDUUD\������� ������ >IUDQGRP�� � L ������@��GRPW\SH�$�� GRPW\SH�$>�� �@�

'20B+)$55$<, '20B)/2$7

0RVW�RI�WKH QXPHULF OLEUDU\�DFFHSWV�KDUGZDUH�IORDWLQJ�SRLQW�DUUD\V�DV�LQSXWV�DQGUHWXUQV�REMHFWV�RI�WKH�VDPH�W\SH�

% � QXPHULF��IIW�$�� GRPW\SH�%�'20B+)$55$<

QXPHULF��HLJHQYDOXHV�$�[5.173121022, 0.8239615051, 0.3962245484 + 0.3259514491 i, . . . ]

����

Page 151: Mupad Tutorial

AZk]pZk^�?ehZm�:kkZrl

,Q�DGGLWLRQ�WR�VWDQGDUG�DUUD\�RSHUDWLRQV� KI�DUUD\V�DOORZ�DGGLWLRQ�RI�DUUD\V�RI�WKHVDPH�VL]H�DV�ZHOO�DV�PDWUL[�PXOWLSOLFDWLRQ�IRU�WZR�GLPHQVLRQDO�KI�DUUD\V�RIFRPSDWLEOH�VL]HV�

$ � KIDUUD\������ ����� >� �����@��% � KIDUUD\������ ����� >���L�M� � M ���� � L ����@�⎛

⎜⎝1.0 2.0 3.0 4.05.0 6.0 7.0 8.09.0 10.0 11.0 12.0

⎟⎠

⎜⎜⎜⎜⎝

0.5 0.33333333330.3333333333 0.25

0.25 0.20.2 0.1666666667

⎟⎟⎟⎟⎠

$ � $� $ %⎛

⎜⎝2.0 4.0 6.0 8.010.0 12.0 14.0 16.018.0 20.0 22.0 24.0

⎟⎠

⎜⎝2.716666667 2.1

7.85 5.912.98333333 9.7

⎟⎠

����

Page 152: Mupad Tutorial

- FnI:=� H[c^\ml

Afl]jnYd�9jal`e]la[

7KHUH�LV�DQRWKHU�NHUQHO�GDWD�W\SH� FDOOHG '20B,17(59$/��2EMHFWV�RI�WKLV�W\SHUHSUHVHQW�UHDO�RU�FRPSOH[�LQWHUYDOV�RI�IORDWLQJ�SRLQW�QXPEHUV� 7KH\�SURYLGH�DPRQJ�RWKHU�SRVVLELOLWLHV� D�PHDQV�RI�FRQWUROOLQJ�RQH�RI�WKH�PRVW�IXQGDPHQWDOSUREOHPV�RI�IORDWLQJ�SRLQW�DULWKPHWLF� URXQG�RII�HUURUV�

7KH�EDVLF�LGHD�LV�DV�IROORZV� ,QVWHDG�RI�IORDWLQJ�SRLQW�QXPEHUV x1� x2 HWF�� ZKHUHDOPRVW�HDFK�RSHUDWLRQ�OHDGV�WR��XVXDOO\�VPDOO��HUURUV� FRQVLGHU�LQWHUYDOV X1�X2

HWF��ZKLFK�DUH�NQRZQ�WR�FRQWDLQ�WKH�SUHFLVH�QXPEHUV�FRPLQJ�IURP�WKH�DSSOLFDWLRQ�2QH�ZRXOG�OLNH�WR�KDYH�D�YHULILHG�VWDWHPHQW�WKDW�WKH�YDOXH y = f(x1, x2, . . . ) RI�DIXQFWLRQ f OLHV�LQ�VRPH�LQWHUYDO Y � 0DWKHPDWLFDOO\� WKH�LPDJH�VHW

Y = f(X1, X2, . . . ) = {f(ξ1, ξ2, . . . ); ξ1 ∈ X1; ξ2 ∈ X2; . . .}

LV�ZDQWHG� &RPSXWLQJ�WKLV�LPDJH�VHW H[DFWO\ LV�D�IRUPLGDEOH�WDVN� ,Q�IDFW� LW�LV�WRRDPELWLRXV�WR�DVN�IRU�DQ�H[DFW�UHSUHVHQWDWLRQ�ZKHQ�XVLQJ�IDVW�DQG�PHPRU\�HIILFLHQWIORDWLQJ�SRLQW�DULWKPHWLF� ,QVWHDG� WKH�LQWHUYDO�YHUVLRQ�RI�D�IXQFWLRQ f LV�DQDOJRULWKP f̂ WKDW�SURGXFHV�D�ODUJHU�VHW f̂(X1, X2, . . . ) ZKLFK�LV JXDUDQWHHG WRFRQWDLQ�WKH�H[DFW�LPDJH�VHW�RI f �

f(X1, X2, . . . ) ⊂ f̂(X1, X2, . . . ).

,I�ZH�WKLQN�DERXW�WKH�LQWHUYDOV X1�X2 HWF��DV�LQFRUSRUDWLQJ�WKH�LQDFFXUDFLHV�LQ x1�x2 HWF� FDXVHG�E\�URXQG�RII� WKH�LQWHUYDO�YHUVLRQ f̂ RI�WKH�IXQFWLRQ f SURYLGHVYHULILHG�XSSHU�DQG�ORZHU�ERXQGV�IRU�WKH�UHVXOW y = f(x1, x2, . . . ) LQ�ZKLFK�WKHURXQG�RII�HUURUV�RI x1� x2 HWF� KDYH�SURSDJDWHG� $OWHUQDWLYHO\� \RX�FDQ�DOVR�UHJDUGX1, . . . DV�LPSUHFLVH�PHDVXUHPHQWV�RU�LQ�VRPH�RWKHU�ZD\�XQGHUGHWHUPLQHGTXDQWLW\��³WKLV�SDUDPHWHU�LV�VRPHZKHUH�EHWZHHQ 0 DQG 1�´ ³WKLV�UHVLVWRU�KDV�DWROHUDQFH�RI 10�´�� ,Q�VXFK�D�VHWWLQJ� WKH�UHVXOW f̂(X1, X2, . . . ) LV�D�VHW JXDUDQWHHGWR�FRQWDLQ DOO SRVVLEOH�WUXH�UHVXOWV�

)ORDWLQJ�SRLQW�LQWHUYDOV�DUH�FUHDWHG�IURP�H[DFW�QXPEHUV�RU�IORDWLQJ�SRLQWQXPEHUV�XVLQJ�WKH�IXQFWLRQ KXOO RU�LWV�RSHUDWRU�HTXLYDOHQW ����

;� � KXOO�3,�3.141592653 . . . 3.141592654

;� � FRV�� 3,�� ��� �� 3,���−1.0 . . . − 0.1736481776

����

Page 153: Mupad Tutorial

Bgm^koZe�:kbmaf^mb\

&RPSOH[�LQWHUYDOV�DUH�UHFWDQJXODU�DUHDV�LQ�WKH�FRPSOH[�SODLQ�FRQVLVWLQJ�RI�DQLQWHUYDO�IRU�WKH�UHDO�SDUW�DQG�DQ�LQWHUYDO�IRU�WKH�LPDJLQDU\�SDUW� 8VLQJ KXOO RU ����\RX�PD\�VSHFLI\�WKH�UHFWDQJOH�E\�LWV�³ORZHU�OHIW´�DQG�³XSSHU�ULJKW´�FRUQHUV�

;� � �� � � ,� ��� �� � � ,�(2.0 . . . 4.0) + L (−3.0 . . . 5.0)

7KH�0X3$'� IXQFWLRQV�WKDW�VXSSRUW�LQWHUYDO�DULWKPHWLF�LQFOXGH�WKH�EDVLFDULWKPHWLFDO�RSHUDWLRQV �� �� � �� A DV�ZHOO�DV�PRVW�RI�WKH�VSHFLDO�IXQFWLRQV�VXFK�DVVLQ� FRV� H[S� OQ� DEV HWF��;�A� � ;�

8.869604401 . . . 9.695956224

;� � , ;� � ;�(5.141592653 . . . 7.141592654) + L (−2.826351823 . . . 6.0)

VLQ�;�� � H[S�DEV�;���7.389056098 . . . 603.728285

:KHQ�GLYLGLQJ�E\�DQ�LQWHUYDO�FRQWDLQLQJ 0� LQILQLWH�LQWHUYDOV�DUH�SURGXFHG� 7KHREMHFWV 5'B,1) DQG 5'B1,1) �³URXQGHG�LQILQLW\´�DQG�³URXQGHG�QHJDWLYH�LQILQLW\�´UHVSHFWLYHO\��UHSUHVHQW�WKH�YDOXHV ±∞ LQ�DQ�LQWHUYDO�FRQWH[W�

VLQ�;�A� � ����−0.4527492553 . . . 0.4794255387

���5'B1,1) . . . − 2.208728094 ∪ 2.085829642 . . . 5'B,1)

7KH�ODVW�H[DPSOH�VKRZV�WKDW�WKH�DULWKPHWLF�PD\�SURGXFH�³V\PEROLF´�XQLRQV�RIIORDWLQJ�SRLQW�LQWHUYDOV� $FWXDOO\� WKH�XQLRQ�LV�VWLOO�RI�W\SH '20B,17(59$/�

GRPW\SH���'20B,17(59$/

����

Page 154: Mupad Tutorial

- FnI:=� H[c^\ml

,Q�IDFW� WKH�IXQFWLRQV XQLRQ DQG LQWHUVHFW IRU�FUHDWLQJ�XQLRQV�DQG�LQWHUVHFWLRQV�RIVHWV�FDQ�EH�XVHG�IRU�FUHDWLQJ�LQWHUYDOV��ZKLFK�DUH� DIWHU�DOO� D�VSHFLDO�NLQG�RI�VHWV��

;� XQLRQ ;�A�0.0301536896 . . . 1.0 ∪ 3.141592653 . . . 3.141592654

FRV�;� ;�� LQWHUVHFW ;�−1.0 . . . − 0.1736481776

6\PEROLF�REMHFWV�VXFK�DV�LGHQWLILHUV��&KDSWHU �����DQG�IORDWLQJ�SRLQW�LQWHUYDOV�FDQEH�PL[HG� 7KH�IXQFWLRQ LQWHUYDO UHSODFHV�DOO�QXPHULFDO�VXEH[SUHVVLRQV�RI�DQH[SUHVVLRQ��&KDSWHU �����E\�IORDWLQJ�SRLQW�LQWHUYDOV�

LQWHUYDO�� [A� � 3,�(2.0 . . . 2.0) x2 + (3.141592653 . . . 3.141592654)

,Q�0X3$'��LGHQWLILHUV�DUH�LPSOLFLWO\�DVVXPHG�WR�UHSUHVHQW�DUELWUDU\�FRPSOH[YDOXHV� &RQVHTXHQWO\� WKH�IXQFWLRQ KXOO UHSODFHV [ E\�WKH�LQWHUYDO�UHSUHVHQWLQJWKH�ZKROH�FRPSOH[�SODQH�

KXOO���(5'B1,1) . . . 5'B,1)) + L (5'B1,1) . . . 5'B,1))

7KHUH�LV�D�QXPEHU�RI�VSHFLDOL]HG�IXQFWLRQV�IRU�IORDWLQJ�SRLQW�LQWHUYDOV�DWWDFKHG�DVPHWKRGV�WR�WKH�NHUQHO�GRPDLQ '20B,17(59$/��+HUH� ZH�RQO\�PHQWLRQ'20B,17(59$/��FHQWHU �WKH�FHQWHU�RI�DQ�LQWHUYDO�RU�D�XQLRQ�RI�LQWHUYDOV��DQG'20B,17(59$/��ZLGWK �WKH�ZLGWK�RI�DQ�LQWHUYDO�RU�D�XQLRQ�RI�LQWHUYDOV��'20B,17(59$/��FHQWHU�� ��� � XQLRQ � ��� ��

5.5

'20B,17(59$/��ZLGWK�� ��� � XQLRQ � ��� ��7.0

6HH "'20B,17(59$/ IRU�D�VXUYH\�RI�WKH�DYDLODEOH�PHWKRGV�

)XUWKHU� D�OLEUDU\�GRPDLQ 'RP��)ORDW,9 H[LVWV�ZKLFK�LV�MXVW�D�IDoDGH�IRU�WKHIORDWLQJ�SRLQW�LQWHUYDOV�RI�WKH�NHUQHO�W\SH '20B,17(59$/��,W�LV�XVHIXO�IRUHPEHGGLQJ�IORDWLQJ�SRLQW�LQWHUYDOV�LQ�³FRQWDLQHUV´�VXFK�DV�PDWULFHV��&KDSWHU������RU�SRO\QRPLDOV��&KDSWHU ������ 7KHVH�FRQWDLQHUV�UHTXLUH�WKH�VSHFLILFLDWLRQ�RID�FRHIILFLHQW�GRPDLQ�WKDW�KDV�WKH�PDWKHPDWLFDO�SURSHUWLHV�RI�D�ULQJ� ,Q�RUGHU�WR

����

Page 155: Mupad Tutorial

Bgm^koZe�:kbmaf^mb\

PDNH�IORDWLQJ�SRLQW�LQWHUYDOV�HPEHGGDEOH�LQ�VXFK�FRQWDLQHUV� WKH�GRPDLQ'RP��)ORDW,9 ZDV�JLYHQ�D�YDULHW\�RI�PDWKHPDWLFDO�DWWULEXWHV��³FDWHJRULHV´�UHTXLUHG�IRU�VXFK�SXUSRVHV�

'RP��)ORDW,9��DOO&DWHJRULHV��>&DW��)LHOG� ���� &DW��5LQJ� ���@

,Q�SDUWLFXODU� WKH�VHW�RI�DOO�IORDWLQJ�SRLQW�LQWHUYDOV�LV�QRW�RQO\�UHJDUGHG�DV�D�ULQJ�EXW�HYHQ�DV�D�ILHOG� 6WULFWO\�VSHDNLQJ� KRZHYHU� WKHVH�PDWKHPDWLFDO�FDWHJRULHV�DUHQRW�UHDOO\�DGHTXDWH�IRU�LQWHUYDO�REMHFWV� )RU�H[DPSOH� VXEWUDFWLQJ�DQ�LQWHUYDO�IURPLWVHOI�GRHV�QRW�\LHOG�WKH�]HUR�HOHPHQW��WKH�QHXWUDO�HOHPHQW�ZLWK�UHVSHFW�WRDGGLWLRQ��

�� ��� �� � �� ��� ��−1.0 . . . 1.0

3UDJPDWLFDOO\� KRZHYHU� WKH�PDWKHPDWLFDO�FDWHJRULHV�ZHUH�VHW� DQ\ZD\� WR�HQDEOHHPEHGGLQJ� $IWHU�DOO� WKH�RSHUDWLRQV�LPSOLHG�E\�WKHVH�FDWHJRULHV�GR�ZRUN� )RUH[DPSOH� ZH�FRQVLGHU�WKH�LQYHUVH�RI�D�+LOEHUW�PDWUL[ �DOVR�VHH�([HUFLVH ������7KHVH�PDWULFHV�DUH�QRWRULRXVO\�LOO�FRQGLWLRQHG� )RU�WKH 8× 8+LOEHUW�PDWUL[� WKHFRQGLWLRQ�QXPEHU��WKH�UDWLR�RI�WKH�ODUJHVW�DQG�WKH�VPDOOHVW�HLJHQYDOXH��LV

$ � OLQDOJ��KLOEHUW����HY � QXPHULF��HLJHQYDOXHV�$��PD[�RS�HY���PLQ�RS�HY��

15257575299.0

5RXJKO\�VSHDNLQJ� ZKHQ�LQYHUWLQJ�WKLV�PDWUL[�E\�D�IORDWLQJ�SRLQW�DOJRULWKP� RQHVKRXOG�H[SHFW�WR�ORRVH�DERXW����GHFLPDO�GLJLWV�RI�DFFXUDF\�

ORJ���� ��10.18348552

&RQVHTXHQWO\� ZLWK�ORZ�YDOXHV�RI ',*,76� WKH�LQYHUVH�VKRXOG�EH�VXEVWDQWLDOO\PDUUHG�E\�URXQG�RII� ,QGHHG� DIWHU�FRQYHUVLRQ�RI $ WR�D�PDWUL[�RI�IORDWLQJ�SRLQWLQWHUYDOV� ZH�FDQ�XVH�WKH�JHQHULF�DOJRULWKP�IRU�PDWUL[�LQYHUVLRQ�

����

Page 156: Mupad Tutorial

- FnI:=� H[c^\ml

% � ��'RP��0DWUL[�'RP��)ORDW,9��$�DUUD\������ �����

��� �� ������������ ��� ���������������

��� �� ������������ ��� ����������������

��� �� ����������� ��� ������������

7KH�HQWULHV�RI�WKH�LQYHUVH�DUH JXDUDQWHHG WR�OLH�LQ�WKH�LQGLFDWHG�LQWHUYDOV� 7KHFRPSRQHQW (8, 8) LV�GHWHUPLQHG�WR�D�SUHFLVLRQ�RI 6 OHDGLQJ�GHFLPDO�GLJLWV� ZKHUHDVWKH�FRPSRQHQW (1, 1) LV�RQO\�NQRZQ�WR�OLH�VRPHZKHUH�EHWZHHQ −73.29 DQG 201.3�1RWH� KRZHYHU� WKDW�WKH�JHQHULF�LQYHUVLRQ�DOJRULWKP�WHQGV�WR�RYHUHVWLPDWH�WKHLQWHUYDOV�GUDVWLFDOO\� 7KH�UHVXOWV�UHWXUQHG�E\�D�QXPHULFDO�LQYHUVLRQ�ZLWK³VWDQGDUG´�IORDWLQJ�SRLQW�QXPEHUVPLJKW DFWXDOO\�EH�PRUH�DFFXUDWH�WKDQSUHGLFWHG�E\�WKLV�LQWHUYDO�FRPSXWDWLRQ�

7KH�H[DFW�FRPSRQHQWV�RI�WKH�LQYHUVH�DUH�DYDLODEOH� WRR� $OO�HQWULHV�RI�LQYHUVH+LOEHUW�PDWULFHV�DUH�LQWHJHUV�

& � OLQDOJ��LQYKLOEHUW���DUUD\������ �����

��� �� ������

��� �� �����������

��� �� ����������

8VLQJ�WKH�RSHUDWRU LQ� LW�LV�SRVVLEOH�WR�GHWHUPLQH�ZKHWKHU�D�QXPEHU�RU�H[SUHVVLRQLV�LQVLGH�DQ�LQWHUYDO� 7R�FRPELQH�WKH�PDWULFHV� ZH�XVH ]LS �&KDSWHU ������DQGFKHFN�IRU�HDFK�HQWU\�RI�WKH�H[DFW�LQYHUVH & LI�LW�LV�LQVLGH�WKH�LQWHUYDO�LQ�WKH�PDWUL[ %�7R�WKLV�HQG� WKH�IROORZLQJ�LQSXW�FRQYHUWV�ERWK�PDWULFHV�LQWR�OLVWV�RI�WKHLU�HQWULHVDQG�WKHQ�SHUIRUPV�WKH�FKHFN�

]LS�>RS�&�@� >RS�%�@� �F� E� �! ERRO�F LQ E��>758(� 758(� 758(� 758(� ���� 758(@

����

Page 157: Mupad Tutorial

Gnee�H[c^\ml3 QXOO��% 1,/% )$,/% XQGHILQHG

Fmdd�GZb][lk2 QXOO��$ 1,/$ )$,/$ XQGHILQHG

7KHUH�DUH�VHYHUDO�REMHFWV�UHSUHVHQWLQJ�WKH�³YRLG´�LQ�0X3$'�� )LUVW� ZH�KDYH�WKH³HPSW\�VHTXHQFH´�JHQHUDWHG�E\ QXOO��� ,W�LV�RI�GRPDLQ�W\SH '20B18// DQGJHQHUDWHV�QR�RXWSXW�RQ�WKH�VFUHHQ� 6\VWHP�IXQFWLRQV�VXFK�DV UHVHW �6HFWLRQ ������SULQW �6HFWLRQ ������ RU LI �&KDSWHU ����ZLWKRXW�DQ HOVH EUDQFK� ZKLFK�FDQQRWUHWXUQ�PDWKHPDWLFDOO\�XVHIXO�YDOXHV� UHWXUQ�WKLV�0X3$' REMHFW�LQVWHDG�

D � UHVHW��� E � SULQW��KHOOR����KHOOR�

GRPW\SH�D�� GRPW\SH�E�'20B18//, '20B18//

7KH�REMHFW QXOO�� LV�SDUWLFXODUO\�XVHIXO�LQ�FRQQHFWLRQ�ZLWK�VHTXHQFHV�6HFWLRQ ����� 7KH�V\VWHP�DXWRPDWLFDOO\�UHPRYHV�WKLV�REMHFW�IURP�VHTXHQFHV� DQG\RX�FDQ�XVH�LW� IRU�H[DPSOH� WR�UHPRYH�VHTXHQFH�HQWULHV�VHOHFWLYHO\�

GHOHWH D� E� F�6HT � D� E� F� 6HT � VXEV�6HT� E QXOO���a, c

+HUH�ZH�KDYH�XVHG�WKH�VXEVWLWXWLRQ�FRPPDQG VXEV �&KDSWHU �� WR�UHSODFH E E\QXOO���

7KH�0X3$' REMHFW 1,/��ZKLFK�LV�GLVWLQFW�IURP QXOO��� LQWXLWLYHO\�PHDQV�³QRYDOXH�´ 6RPH�V\VWHP�IXQFWLRQV�UHWXUQ�WKH 1,/ REMHFW�ZKHQ�\RX�FDOO�WKHP�ZLWKDUJXPHQWV�IRU�ZKLFK�WKH\�QHHG�QRW�FRPSXWH�DQ\WKLQJ� $ W\SLFDO�H[DPSOH�DUHXQLQLWLDOL]HG�RSHUDQGV�RI�DUUD\V�

$ � DUUD\������� RS�$� ��1,/

8QLQLWLDOL]HG�ORFDO�YDULDEOHV�DQG�SDUDPHWHUV�RI�0X3$' SURFHGXUHV�DOVR�KDYH�WKHYDOXH 1,/ �6HFWLRQ ������

7KH�0X3$' REMHFW )$,/ LQWXLWLYHO\�PHDQV�³, FRXOG�QRW�ILQG�D�YDOXH�´ 6\VWHPIXQFWLRQV�UHWXUQ�WKLV�REMHFW�ZKHQ�WKHUH�LV�QR�PHDQLQJIXO�UHVXOW�IRU�WKH�JLYHQ�LQSXWSDUDPHWHUV�

����

Page 158: Mupad Tutorial

- FnI:=� H[c^\ml

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�WU\�WR�FRPSXWH�WKH�LQYHUVH�RI�D�VLQJXODU�PDWUL[�

$ � PDWUL[�>>�� �@� >�� �@@�(

1 22 4

)

$A����)$,/

$QRWKHU�REMHFW�ZLWK�D�VLPLODU�PHDQLQJ�LV XQGHILQHG� )RU�H[DPSOH� WKH�0X3$'IXQFWLRQ OLPLW UHWXUQV�WKLV�REMHFW�ZKHQ�WKH�UHTXHVWHG�OLPLW�GRHV�QRW�H[LVW�

OLPLW���[� [ ��XQGHILQHG

$ULWKPHWLFDO�RSHUDWLRQV�ZLWK XQGHILQHG UHWXUQ�DJDLQ XQGHILQHG�

XQGHILQHG � �� �AXQGHILQHGXQGHILQHG, XQGHILQHG

�����

Page 159: Mupad Tutorial

-

(YDOXDWLRQ�DQG�6LPSOLILFDWLRQ

A\]flaÚ]jk�Yf\�L`]aj�NYdm]k

&RQVLGHU�

GHOHWH [� D� \ � D � [a+ x

6LQFH�WKH�LGHQWLILHUV D DQG [ RQO\�UHSUHVHQW�WKHPVHOYHV� WKH�³YDOXH´�RI \ LV�WKHV\PEROLF�H[SUHVVLRQ a+ x� :H�KDYH�WR�GLVWLQJXLVK�FDUHIXOO\�EHWZHHQ�WKH�LGHQWLILHU\ DQG�LWV�YDOXH� 0RUH�SUHFLVHO\� WKH YDOXH RI�DQ�LGHQWLILHU�GHQRWHV�WKH�0X3$'�

REMHFW�WKDW�WKH�V\VWHP�FRPSXWHV�E\�HYDOXDWLRQ�DQG�VLPSOLILFDWLRQ�RI�WKHULJKW�KDQG�VLGH�RI�WKH�DVVLJQPHQW LGHQWLILHU� YDOXH DW�WKH�WLPH�RI�DVVLJQPHQW�

1RWH�WKDW�LQ�WKH�H[DPSOH�DERYH� WKH�YDOXH�RI \ LV�FRPSRVHG�RI�WKH�V\PEROLFLGHQWLILHUV D DQG [� ZKLFK�PD\�EH�DVVLJQHG�YDOXHV�DW�D�ODWHU�WLPH� )RU�H[DPSOH� LIZH�DVVLJQ�WKH�YDOXH 1 WR�WKH�LGHQWLILHU D� WKHQ D LV�UHSODFHG�E\�LWV�YDOXH 1 LQ�WKHH[SUHVVLRQ D�[� DQG�WKH�FDOO \ UHWXUQV x+ 1�

D � �� \x+ 1

:H�VD\�WKDW�WKH HYDOXDWLRQ RI�WKH�LGHQWLILHU \ UHWXUQV�WKH�UHVXOW x+ 1� EXW�LWV YDOXHLV�VWLOO D�[�

:H�GLVWLQJXLVK�EHWZHHQ�DQ�LGHQWLILHU� LWV�YDOXH� DQG�LWV�HYDOXDWLRQ� WKHYDOXH GHQRWHV�WKH�HYDOXDWLRQ DW�WKH�WLPH�RI�DVVLJQPHQW� D�ODWHUHYDOXDWLRQPD\�UHWXUQ�D�GLIIHUHQW ³FXUUHQW�YDOXH´�

Page 160: Mupad Tutorial

. >oZenZmbhg�Zg]�LbfiebÛ\Zmbhg

,I�ZH�QRZ�DVVLJQ�WKH�YDOXH 2 WR [� WKHQ�ERWK D DQG [ DUH�UHSODFHG�E\�WKHLU�YDOXHV�DWWKH�QH[W�HYDOXDWLRQ�RI \� +HQFH�ZH�REWDLQ�WKH�VXP 2 + 1 DV�D�UHVXOW� ZKLFK�0X3$'DXWRPDWLFDOO\�VLPSOLILHV�WR 3�

[ � �� \3

7KH HYDOXDWLRQ RI \ QRZ�UHWXUQV�WKH�QXPEHU 3� LWV YDOXH LV�VWLOO a+ x�

,W�LV�UHDVRQDEOH�WR�VD\�WKDW�WKH�YDOXH�RI \ LV�WKH�UHVXOW�DW�WKH�WLPH�RI�DVVLJQPHQW�1DPHO\� LI�ZH�GHOHWH�WKH�YDOXHV�RI�WKH�LGHQWLILHUV D DQG [ LQ�WKH�DERYH�H[DPSOH�WKHQ�WKH�HYDOXDWLRQ�RI \ \LHOGV�LWV�RULJLQDO�YDOXH�LPPHGLDWHO\�DIWHU�WKH�DVVLJQPHQW�

GHOHWH D� [� \a+ x

,I D RU [ DOUHDG\�KDYH�D�YDOXH EHIRUH ZH�DVVLJQ�WKH�H[SUHVVLRQ D�[ WR \� WKHQ�WKHIROORZLQJ�KDSSHQV�

[ � �� \ � D � [� \a+ 1

$W�WKH�WLPH�RI�DVVLJQPHQW� \ LV�DVVLJQHG�WKH�HYDOXDWLRQ�RI D�[� L�H�� a+ 1� ,QGHHG�WKLV�LV�QRZ�WKH YDOXH RI \� ZKLFK�FRQWDLQV�QR�UHIHUHQFH�WR [�GHOHWH [� \a+ 1

+HUH�DUH�VRPH�IXUWKHU�H[DPSOHV�IRU�WKLV�PHFKDQLVP� :H�ILUVW�DVVLJQ�WKH�UDWLRQDOQXPEHU ��� WR [� WKHQ�ZH�DVVLJQ�WKH�REMHFW >[�[A��[A�@ WR�WKH�LGHQWLILHU OLVW� ,QWKH�DVVLJQPHQW� WKH�V\VWHP�HYDOXDWHV�WKH�ULJKW�KDQG�VLGH�DQG�DXWRPDWLFDOO\UHSODFHV�WKH�LGHQWLILHU [ E\�LWV�YDOXH� 7KXV� DW�WKH�WLPH�RI�DVVLJQPHQW� WKH�LGHQWLILHUOLVW JHWV�WKH�YDOXH [1/3, 1/9, 1/27] DQG�QRW [x, x2, x3]�

[ � ���� OLVW � >[� [A�� [A�@[13,19,

127

]

���

Page 161: Mupad Tutorial

B]^gmbÛ^kl�Zg]�Ma^bk�OZen^l

GHOHWH [� OLVW[13,19,

127

]

0X3$' DSSOLHV�WKH�VDPH�HYDOXDWLRQ�VFKHPH�WR�V\PEROLF�IXQFWLRQ�FDOOV�

GHOHWH I� \ � I�3,�f(π)

$IWHU�WKH�DVVLJQPHQW

I � VLQ�

ZH�REWDLQ�WKH�HYDOXDWLRQ

\0

:KHQ�HYDOXDWLQJ \� WKH�V\VWHP�UHSODFHG�WKH�LGHQWLILHU I E\�LWV�YDOXH� ZKLFK�LV�WKHYDOXH�RI�WKH�LGHQWLILHU VLQ� 7KLV�LV�D�SURFHGXUH�ZKLFK�LV�H[HFXWHG�ZKHQ \ LVHYDOXDWHG�DQG�UHWXUQV VLQ(π) DV 0�

���

Page 162: Mupad Tutorial

. >oZenZmbhg�Zg]�LbfiebÛ\Zmbhg

;gehd]l]$ Af[gehd]l]$ Yf\�=f^gj[]\�=nYdmYlagf

:H�FRQVLGHU�RQFH�DJDLQ�WKH�ILUVW�H[DPSOH�IURP�WKH�SUHYLRXV�VHFWLRQ� 7KHUH�ZH�KDYHDVVLJQHG�WKH�H[SUHVVLRQ D�[ WR�WKH�LGHQWLILHU \� DQG D DQG [ GLG�QRW�KDYH�D�YDOXH�

GHOHWH D� [� \ � D � [� D � �� \x+ 1

:H�QRZ�H[SODLQ�LQ�JUHDWHU�GHWDLO�KRZ�0X3$'� SHUIRUPV�WKH�ILQDO�HYDOXDWLRQ�

)LUVW��³OHYHO 1´��WKH�HYDOXDWRU�FRQVLGHUV�WKH�YDOXH a+ x RI \� 6LQFH�WKLV�YDOXHFRQWDLQV�LGHQWLILHUV [ DQG D� D�VHFRQG�HYDOXDWLRQ�VWHS��³OHYHO 2´��LV�QHFHVVDU\�WRGHWHUPLQH�WKH�YDOXH�RI�WKHVH�LGHQWLILHUV� 7KH�V\VWHP�UHFRJQL]HV�WKDW D KDV�WKHYDOXH 1� ZKLOH [ KDV�QR�YDOXH��DQG�WKXV�PDWKHPDWLFDOO\�UHSUHVHQWV�DQ�XQNQRZQ��1RZ�WKH�V\VWHP¶V�DULWKPHWLF�FRPELQHV�WKHVH�UHVXOWV�WR x+ 1� DQG�WKLV�LV�WKHHYDOXDWLRQ�RI \� )LJXUHV ���±��� LOOXVWUDWH�WKLV�SURFHVV� $ ER[�UHSUHVHQWV�DQLGHQWLILHU�DQG�LWV�YDOXH��RU ·� UHVSHFWLYHO\� LI�LW�KDV�QR�YDOXH�� $Q�DUURZ�UHSUHVHQWVRQH�HYDOXDWLRQ�VWHS�

\ ·

)LJXUH ���� 7KH�LGHQWLILHU\ ZLWKRXW�D�YDOXH�

\

� [ · D ·

)LJXUH ���� $IWHU � WKH � DV�VLJQPHQW \� D�[�

\

� [ · D

)LJXUH ���� $IWHU � WKH � DV�VLJQPHQW D� �� ZH� ILQDOO\REWDLQ [�� DV�WKH�HYDOXD�WLRQ�RI \�

,Q�DQDORJ\�WR�WKH�H[SUHVVLRQ�WUHHV�IRU�UHSUHVHQWLQJ�V\PEROLF�H[SUHVVLRQV�6HFWLRQ ����� \RX�PD\�LPDJLQH�WKH�SURFHVV�RI�HYDOXDWLRQ�DV�DQ HYDOXDWLRQ�WUHH�ZKRVH�YHUWLFHV�DUH�H[SUHVVLRQV�ZLWK�V\PEROLF�LGHQWLILHUV� ZLWK�EUDQFKHV�SRLQWLQJWR�WKH�FRUUHVSRQGLQJ�YDOXHV�RI�WKHVH�LGHQWLILHUV� 7KH�V\VWHP�WUDYHUVHV�WKLV�WUHHXQWLO�WKHUH�DUH�HLWKHU�QR�LGHQWLILHUV�OHIW�RU�DOO�UHPDLQLQJ�LGHQWLILHUV�KDYH�QR�YDOXH�

���

Page 163: Mupad Tutorial

<hfie^m^% Bg\hfie^m^% Zg]�>g_hk\^]�>oZenZmbhg

7KH�XVHU�PD\�FRQWURO�WKH�OHYHOV�RI�WKLV�WUHH�YLD�WKH�V\VWHP�IXQFWLRQ OHYHO� :H�ORRNDW�DQ�H[DPSOH�

GHOHWH D� E� F� [ � D � E� D � E � �� E � F�

7KH�HYDOXDWLRQ�WUHH�IRU [ LV�

[

� D E

� E 1 F ·

F ·

OHYHO 0 �

OHYHO 1 �

OHYHO 2 �

OHYHO 3 �

7KH�LGHQWLILHU [ IRUPV�WKH�WRS�OHYHO��WKH�URRW� OHYHO 0��RI�LWV�RZQ�HYDOXDWLRQ�WUHH�

OHYHO�[� ��x

7KH�QH[W�OHYHO 1 GHWHUPLQHV�WKH�YDOXH�RI [�OHYHO�[� ��a+ b

$W�WKH�IROORZLQJ�OHYHO 2� D DQG E DUH�UHSODFHG�E\�WKHLU�YDOXHV b+ 1 DQG c�UHVSHFWLYHO\�

OHYHO�[� ��b+ c+ 1

���

Page 164: Mupad Tutorial

. >oZenZmbhg�Zg]�LbfiebÛ\Zmbhg

7KH�UHPDLQLQJ E LV�UHSODFHG�E\�LWV�YDOXH c RQO\�LQ�WKH�QH[W�OHYHO 3�

OHYHO�[� ��2 c+ 1

:H�FDOO�WKH�W\SH�RI�HYDOXDWLRQ�GHVFULEHG�KHUH�D FRPSOHWH�HYDOXDWLRQ� 7KLV�PHDQVWKDW�LGHQWLILHUV�DUH�UHSODFHG�E\�WKHLU�YDOXHV�UHFXUVLYHO\�XQWLO�QR�IXUWKHU�HYDOXDWLRQVDUH�SRVVLEOH� 7KH�HQYLURQPHQW�YDULDEOH /(9(/� ZKLFK�KDV�WKH�GHIDXOW�YDOXH 100�GHWHUPLQHV�KRZ�IDU�0X3$' GHVFHQGV�DW�PRVW�LQ�DQ�HYDOXDWLRQ�WUHH�

,Q�LQWHUDFWLYH�PRGH� 0X3$' DOZD\V�HYDOXDWHV�FRPSOHWHO\�

0RUH�SUHFLVHO\� WKLV�PHDQV�WKDW�0X3$' HYDOXDWHV�XS�WR�GHSWK /(9(/ LQ�LQWHUDFWLYHPRGH��

GHOHWH D�� D�� D�� /(9(/ � ��D� � D�� D�

D�

D� � D�� D�D�

8S�WR�QRZ� WKH�HYDOXDWLRQ�WUHH�IRU D� KDV�GHSWK 2� DQG�WKH /(9(/ YDOXH�RI 2DFKLHYHV�D�FRPSOHWH�HYDOXDWLRQ� +RZHYHU� LQ�WKH�QH[W�VWHS� WKH�YDOXH�RI D� LV�QRWWDNHQ�LQWR�DFFRXQW�

D� � D�� D�D�

7R�UHVWRUH�WKH�GHIDXOW�YDOXH�RI /(9(/��ZH�XVH GHOHWH�GHOHWH /(9(/�

$V�VRRQ�DV�0X3$' UHDOL]HV�WKDW�WKH�FXUUHQW�HYDOXDWLRQ�OHYHO�H[FHHGV�WKH�YDOXH�RIWKH�HQYLURQPHQW�YDULDEOH 0$;/(9(/ �ZKRVH�GHIDXOW�YDOXH�LV 100�� WKHQ�LW�DVVXPHVWR�EH�LQ�DQ�LQILQLWH�ORRS�DQG�DERUWV�WKH�HYDOXDWLRQ�ZLWK�DQ�HUURU�PHVVDJH�

�2QH�VKRXOG�QRW�FRQIXVH�WKLV�ZLWK�WKH�HIIHFW�RI�D�V\VWHP�IXQFWLRQ�FDOO� ZKLFK�PD\�UHWXUQ�D QRW�FRP�SOHWHO\�HYDOXDWHG�REMHFW� VXFK�DV VXEV �&KDSWHU ��� 7KH�FDOO VXEV�VLQ�[��[ ��� IRU�H[DPSOH� UHWXUQVVLQ(0) DQG�QRW 0� 7KH�IXQFWLRQDOLW\�RI VXEV LV�WR�SHUIRUP�D�VXEVWLWXWLRQ�DQG�WR�UHWXUQ�WKH�UHVXOWLQJ�REMHFWZLWKRXW�IXUWKHU�HYDOXDWLRQ�

���

Page 165: Mupad Tutorial

<hfie^m^% Bg\hfie^m^% Zg]�>g_hk\^]�>oZenZmbhg

0$;/(9(/ � �� D�(UURU� 5HFXUVLYH GHILQLWLRQ >6HH "0$;/(9(/@

GHOHWH 0$;/(9(/�

:H�QRZ�SUHVHQW�VRPH�LPSRUWDQW�H[FHSWLRQV�WR�WKH�UXOH�RI�FRPSOHWH�HYDOXDWLRQ�7KH�FDOOV ODVW�L�� �L� RU � �&KDSWHU ������GR�QRW�OHDG�WR�DQ�HYDOXDWLRQ� :HFRQVLGHU�WKH�H[DPSOH

GHOHWH [� >VLQ�[�� FRV�[�@� [ � ��

1RZ� �� DFFHVVHV�WKH�OLVW�ZLWKRXW�HYDOXDWLQJ�LW�

��[VLQ(x) , FRV(x)]

+RZHYHU� \RX�FDQ�HQIRUFH�HYDOXDWLRQ�E\�PHDQV�RI HYDO�

HYDO���[0, 1]

&RPSDUH�WKLV�WR�WKH�IROORZLQJ�VWDWHPHQWV� ZKHUH�UHTXHVWLQJ�WKH�LGHQWLILHU OLVWFDXVHV�WKH�XVXDO�FRPSOHWH�HYDOXDWLRQ�

GHOHWH [� OLVW � >VLQ�[�� FRV�[�@� [ � �� OLVW[0, 1]

$UUD\V�RI�GRPDLQ�W\SH '20B$55$< DUH�DOZD\V�HYDOXDWHG�ZLWK�OHYHO 1�

GHOHWH D� E� $ � DUUD\������ >D� E@��E � D� D � �� $(a b

)

$V�\RX�FDQ�VHH� WKH�FDOO�RI $ UHWXUQV�WKH�YDOXH��WKH�DUUD\�� EXW�GRHV�QRW�UHSODFH D�EE\�WKHLU�YDOXHV� <RX�FDQ�HYDOXDWH�WKH�HQWULHV�YLD PDS�$�HYDO��

PDS�$� HYDO�(

1 1)

���

Page 166: Mupad Tutorial

. >oZenZmbhg�Zg]�LbfiebÛ\Zmbhg

1RWH�WKDW�LQ�FRQWUDVW�WR�WKH�DERYH�EHKDYLRU� WKH�LQGH[HG�DFFHVV�RI�DQ�LQGLYLGXDOHQWU\�LV�HYDOXDWHG�FRPSOHWHO\�

$>�@� $>�@1, 1

0DWULFHV��RI�W\SH 'RP��0DWUL[�·��� WDEOHV��RI�GRPDLQ�W\SH '20B7$%/(�� DQGSRO\QRPLDOV��'20B32/<� DUH�WUHDWHG�LQ�WKH�VDPH�ZD\�DV�DUUD\V� 0RUHRYHU� ZLWKLQSURFHGXUHV� 0X3$' DOZD\V�HYDOXDWHV�RQO\�XS�WR�OHYHO 1 �6HFWLRQ ������� ,I�WKLV�LVQRW�VXIILFLHQW� \RX�PD\�FRQWURO�WKLV�EHKDYLRU�H[SOLFLWO\�E\�PHDQV�RI OHYHO�

7KH�FRPPDQG KROG�REMHFW� LV�VLPLODU�WR OHYHO�REMHFW��� DQG�SUHYHQWV�WKHHYDOXDWLRQ�RI REMHFW�� 7KLV�PD\�EH�GHVLUDEOH�LQ�PDQ\�VLWXDWLRQV� 7KH�IROORZLQJIXQFWLRQ� ZKLFK�FDQQRW�EH�H[HFXWHG�IRU�V\PEROLF�DUJXPHQWV� \LHOGV�DQ�H[DPSOHZKHUH�WKH��SUHPDWXUH��HYDOXDWLRQ�LV�XQGHVLUDEOH�

DEV9DOXH � ; �! �LI ; ! � WKHQ ; HOVH �; HQGBLI��DEV9DOXH�;�

(UURU� &DQW HYDOXDWH WR ERROHDQ >BOHHTXDO@�GXULQJ HYDOXDWLRQ RI DEV9DOXH

,I�\RX�ZDQW�WR�QXPHULFDOO\�LQWHJUDWH�WKLV�IXQFWLRQ�XVLQJ QXPHULF��LQW� WKHREYLRXV�LQSXW�UHWXUQV�WKH�VDPH�HUURU�

QXPHULF��LQW�DEV9DOXH�;�� ; ������(UURU� &DQW HYDOXDWH WR ERROHDQ >BOHHTXDO@�GXULQJ HYDOXDWLRQ RI DEV9DOXH

,I�\RX�GHOD\�WKH�HYDOXDWLRQ�RI DEV9DOXH�;� E\ KROG� WKH�YDOXH�LV�FRPSXWHG�

QXPHULF��LQW�KROG�DEV9DOXH��;�� ; ������1.0

7KH�UHDVRQ�LV�WKDW QXPHULF��LQW LQWHUQDOO\�VXEVWLWXWHV�QXPHULFDO�YDOXHV�IRU ;��IRUZKLFK DEV9DOXH FDQ�EH�HYDOXDWHG�ZLWKRXW�SUREOHPV�

�6HH "KROG IRU�WKH�H[DFW�GLIIHUHQFH�EHWZHHQ KROG�REMHFW� DQG OHYHO�REMHFW����

���

Page 167: Mupad Tutorial

<hfie^m^% Bg\hfie^m^% Zg]�>g_hk\^]�>oZenZmbhg

+HUH�LV�DQRWKHU�H[DPSOH� OLNH�PRVW�0X3$' IXQFWLRQV� WKH�IXQFWLRQ GRPW\SH ILUVWHYDOXDWHV�LWV�DUJXPHQW� VR�WKDW�WKH�FRPPDQG GRPW\SH�REMHFW� UHWXUQV�WKHGRPDLQ�W\SH�RI�WKH HYDOXDWLRQ RI REMHFW�

[ � �� \ � �� [� [ � \� VLQ���� VLQ�����1, 2, 0, 0.09983341665

GRPW\SH�[�� GRPW\SH�[ � \�� GRPW\SH�VLQ�����GRPW\SH�VLQ������

'20B,17, '20B,17, '20B,17, '20B)/2$7

8VLQJ KROG� \RX�REWDLQ�WKH�GRPDLQ�W\SHV�RI�WKH�REMHFWV�WKHPVHOYHV� [ LV�DQLGHQWLILHU� [�\ LV�DQ�H[SUHVVLRQ� DQG VLQ��� DQG VLQ����� DUH�IXQFWLRQ�FDOOV�DQGKHQFH�H[SUHVVLRQV�DV�ZHOO�

GRPW\SH�KROG�[��� GRPW\SH�KROG�[ � \���GRPW\SH�KROG�VLQ������ GRPW\SH�KROG�VLQ�������

'20B,'(17, '20B(;35, '20B(;35, '20B(;35

7KH�FRPPDQGV "OHYHO DQG "KROG SURYLGH�IXUWKHU�LQIRUPDWLRQ�IURP�WKHFRUUHVSRQGLQJ�KHOS�SDJHV�

([HUFLVH����� :KDW�DUH�WKH YDOXHV RI�WKH�LGHQWLILHUV [� \� DQG ] DIWHU�WKH�IROORZLQJVWDWHPHQWV" :KDW�LV�WKH HYDOXDWLRQ RI�WKH�ODVW�VWDWHPHQW�LQ�HDFK�FDVH"

GHOHWH D�� E�� F�� [�[ � D�� D� � E�� D� � F�� [GHOHWH D�� E�� F�� \�D� � E�� \ � D�� E� � F�� \GHOHWH D�� E�� ]�E� � D�� ] � E�� D� � ��� ]

3UHGLFW�WKH�UHVXOWV�RI�WKH�IROORZLQJ�VWDWHPHQW�VHTXHQFHV�

GHOHWH X�� Y�� Z��X� � Y�� Y� � Z�� Z� � X�� X�GHOHWH X�� Y��X� � Y�� X� � X�A� � �� X�

���

Page 168: Mupad Tutorial

. >oZenZmbhg�Zg]�LbfiebÛ\Zmbhg

9mlgeYla[�KaehdaÚ[Ylagf

0X3$'� DXWRPDWLFDOO\�VLPSOLILHV�PDQ\�REMHFWV�VXFK�DV�FHUWDLQ�IXQFWLRQ�FDOOV�RUDULWKPHWLFDO�H[SUHVVLRQV�ZLWK�QXPEHUV�

VLQ��� 3,�� H[S���� �� � ,� �� � ,�0, 1, 2

7KH�VDPH�KROGV�WUXH�IRU�DULWKPHWLF�H[SUHVVLRQV�FRQWDLQLQJ LQILQLW\�

� LQILQLW\ � �∞

6XFK�DXWRPDWLF�VLPSOLILFDWLRQV�VHUYH�IRU�UHGXFLQJ�WKH�FRPSOH[LW\�RI�H[SUHVVLRQVZLWKRXW�DQ�H[SOLFLW�UHTXHVW�E\�WKH�XVHU�

FRV�� � H[S�����A����� 3,��1

7KH�XVHU�FDQ�QHLWKHU�FRQWURO�QRU�H[WHQG�WKH�DXWRPDWLF�VLPSOLILHU�

,Q�PRVW�FDVHV� KRZHYHU� 0X3$' GRHV QRW DXWRPDWLFDOO\�VLPSOLI\�H[SUHVVLRQV� 7KHUHDVRQ�LV�WKDW�WKH�V\VWHP�JHQHUDOO\�FDQQRW�GHFLGH�ZKLFK�LV�WKH�PRVW�UHDVRQDEOHZD\�RI�VLPSOLILFDWLRQ� )RU�H[DPSOH� FRQVLGHU�WKH�IROORZLQJ�H[SUHVVLRQ�ZKLFK�LV�QRWVLPSOLILHG�

\ � ��� [ � [A� � [A� � �� �� [ � � [A� � [A� � ��−(x3 − 5x2 + 7x− 3

) (−x3 − x2 + 4x+ 4

)

1DWXUDOO\� \RX�FDQ�H[SDQG�WKLV�H[SUHVVLRQ� ZKLFK�PD\�EH�UHDVRQDEOH� IRU�H[DPSOH�EHIRUH�FRPSXWLQJ�LWV�V\PEROLF�LQWHJUDO�

H[SDQG�\�x6 − 4x5 − 2x4 + 20x3 − 11x2 − 16x+ 12

����

Page 169: Mupad Tutorial

:nmhfZmb\�LbfiebÛ\Zmbhg

+RZHYHU� LI�\RX�DUH�LQWHUHVWHG�LQ�WKH�URRWV�RI�WKH�SRO\QRPLDO� LW�PDNHV�PRUH�VHQVHWR�FRPSXWH�LWV�OLQHDU�IDFWRUV�

IDFWRU�\�

(x− 3) · (x− 1)2 · (x− 2) · (x+ 2) · (x+ 1)

1R�XQLYHUVDO�DQVZHU�LV�SRVVLEOH�WR�WKH�TXHVWLRQ�ZKLFK�RI�WKH�WZR�UHSUHVHQWDWLRQV�LV³VLPSOHU�´ 'HSHQGLQJ�RQ�WKH�LQWHQGHG�DSSOLFDWLRQ� \RX�FDQ�VHOHFWLYHO\�DSSO\DSSURSULDWH�V\VWHP�IXQFWLRQV��VXFK�DV H[SDQG RU IDFWRU��WR�VLPSOLI\�DQH[SUHVVLRQ�

7KHUH�LV�DQRWKHU�DUJXPHQW�DJDLQVW�DXWRPDWLF�VLPSOLILFDWLRQ� 7KH�V\PERO f � IRUH[DPSOH� PLJKW�UHSUHVHQW�D�ERXQGHG�IXQFWLRQ� IRU�ZKLFK�WKH�OLPLW OLPx→0 x f(x) LV0� +RZHYHU� VLPSOLI\LQJ�WKLV�H[SUHVVLRQ�WR 0 FDQ�EH�ZURQJ�IRU�IXQFWLRQV f ZLWK�DVLQJXODULW\�DW�WKH�RULJLQ�VXFK�DV f(x) = 1/x� 7KXV� DXWRPDWLF�VLPSOLILFDWLRQV�VXFKDV 0 · f(0) = 0 DUH�TXHVWLRQDEOH�DV�ORQJ�DV�WKH�V\VWHP�KDV�QR�DGGLWLRQDO�NQRZOHGJHDERXW�WKH�V\PEROV�LQYROYHG� ,Q�JHQHUDO� 0X3$' FDQQRW�NQRZ�ZKLFK�UXOH�FDQ�EHDSSOLHG�DQG�ZKLFK�PXVW�QRW� 1RZ�\RX�PLJKW�REMHFW�WKDW�0X3$' VKRXOG�SHUIRUPQR�VLPSOLILFDWLRQV�DW�DOO�LQVWHDG�RI�ZURQJ�RQHV� 8QIRUWXQDWHO\� WKLV�LV�QRWUHDVRQDEOH�HLWKHU� VLQFH�LQ�V\PEROLF�FRPSXWDWLRQV� H[SUHVVLRQV�WHQG�WR�JURZ�YHU\TXLFNO\� DQG�WKLV�VHULRXVO\�DIIHFWV�WKH�SHUIRUPDQFH�RI�WKH�V\VWHP� ,Q�IDFW� LQ�PRVWFDVHV�0X3$' VLPSOLILHV�DQ�H[SUHVVLRQ�RI�WKH�IRUP � \ WR �� H[FHSW� H�J�� ZKHQ�WKHYDOXH�RI \ LV LQILQLW\� )$,/��RU XQGHILQHG� <RX�VKRXOG�DOZD\V�NHHS�LQ�PLQG�WKDWVXFK�D�VLPSOLILHG�UHVXOW�PD\�EH�ZURQJ�LQ�H[WUHPH�FDVHV�

)RU�WKDW�UHDVRQ� 0X3$' SHUIRUPV�RQO\�VRPH�VLPSOLILFDWLRQV�DXWRPDWLFDOO\� DQG\RX�PXVW�H[SOLFLWO\�UHTXHVW�RWKHU�VLPSOLILFDWLRQV�\RXUVHOI� )RU�WKLV�SXUSRVH0X3$' SURYLGHV�D�YDULHW\�RI�IXQFWLRQV� VRPH�RI�ZKLFK�DUH�GHVFULEHG�LQ6HFWLRQ ����

:KLOH�WKH�DXWRPDWLF�VLPSOLILFDWLRQV�KDYH�EHHQ�FDUHIXOO\�FKRVHQ� WKHUH�DUH�VWLOOH[DPSOHV�ZKHUH�WKH\�OHDG�WR�XQH[SHFWHG�UHVXOWV� XVXDOO\�EHFDXVH�RI�DPELJXLWLHV�LQPDWKHPDWLFDO�QRWDWLRQ� $Q�H[DPSOH�LV�WKH�VROXWLRQ�RI�WKH�HTXDWLRQ x/x = 1 IRUx ̸= 0�

VROYH�[�[ �� [�C

����

Page 170: Mupad Tutorial

. >oZenZmbhg�Zg]�LbfiebÛ\Zmbhg

7KH�0X3$' HTXDWLRQ�VROYHU VROYH �&KDSWHU ���UHWXUQV�WKH�VHW C RI�DOO�FRPSOH[QXPEHUV�� 7KXV� VROYH FODLPV�WKDW DUELWUDU\ FRPSOH[�YDOXHV�RI x \LHOG�D�VROXWLRQRI�WKH�HTXDWLRQ x/x = 1� 7KH�UHDVRQ�LV�WKDW�WKH�V\VWHP�ILUVW�DXWRPDWLFDOO\VLPSOLILHV�WKH�H[SUHVVLRQ [�[ WR �� DQG�WKHQ�LQ�IDFW�VROYHV�WKH�HTXDWLRQ 1 = 1� 7KHH[FHSWLRQDO�FDVH x = 0� IRU�ZKLFK�WKH�RULJLQDO�SUREOHP�PDNHV�QR�VHQVH� LVFRPSOHWHO\�LJQRUHG�LQ�WKH�VLPSOLILHG�RXWSXW�

�7R�HQWHU�WKLV�VHW� W\SH�³&B�´

����

Page 171: Mupad Tutorial

>oZenZmbhg�Zm�Z�Ihbgm

=nYdmYlagf�Yl�Y�Hgafl

7KHUH�LV�DOVR�DQRWKHU�FRQFHSW�FRPPRQO\�NQRZQ�DV�³HYDOXDWLRQ�´ QDPHO\�UHSODFLQJVRPH�SODFHKROGHU��IUHH�YDULDEOH��E\�D�VSHFLILF�YDOXH�DQG�VLPSOLI\LQJ�WKH�UHVXOWLQJH[SUHVVLRQ� 7KLV�RSHUDWLRQ�LV�LPSOHPHQWHG�E\�WKH�IXQFWLRQ HYDO$W� ZKLFK�LV�DOVRDFFHVVLEOH�DV�WKH _ RSHUDWRU� �7KH _ RSHUDWRU�PLPLFV�WKH�QRWDWLRQ f(x)|x=1

FRPPRQO\�XVHG�LQ�PDWK�WH[WV��

I � VLQ�[��I _ [ �� I _ [ �

0, VLQ(1)

1RWH�WKDW� XQOLNH VXEV� HYDO$W UHVSHFWV�WKH�PDWKHPDWLFDO�PHDQLQJ�DQG�RQO\VXEVWLWXWHV�IUHH�YDULDEOHV� QRW�ERXQG�RQHV�

GHOHWH I�) � LQW�I�[�� [ ���LQILQLW\� � VLQ�[��VXEV�)� [ ��� ) _ [ �

VLQ(1) +∫ ∞

0f(1) G1, VLQ(1) +

∫ ∞

0f(x) Gx

����

Page 172: Mupad Tutorial
Page 173: Mupad Tutorial

.

6XEVWLWXWLRQ� VXEV� VXEVH[� DQGVXEVRS

$OO�0X3$'� REMHFWV�FRQVLVW�RI�RSHUDQGV �6HFWLRQ ����� $Q�LPSRUWDQW�IHDWXUH�RI�DFRPSXWHU�DOJHEUD�V\VWHP�LV�WKDW�LW�FDQ�UHSODFH�WKHVH�EXLOGLQJ�EORFNV�E\�QHZYDOXHV� )RU�WKDW�SXUSRVH� 0X3$' SURYLGHV�WKH�IXQFWLRQV VXEV� VXEVH[ �VKRUW�IRU�VXEVWLWXWH�H[SUHVVLRQ�� DQG VXEVRS �VKRUW�IRU� VXEVWLWXWH�RSHUDQG��

7KH�FRPPDQG VXEV�REMHFW� 2OG 1HZ� UHSODFHV�DOO�RFFXUUHQFHV�RI�WKHVXEH[SUHVVLRQ 2OG LQ REMHFW E\�WKH�YDOXH 1HZ�

I � D � E FAE� J � VXEV�I� E ��� I� Ja+ b cb, 2 c2 + a

<RX�VHH�WKDW VXEV UHWXUQV�WKH�UHVXOW�RI�WKH�VXEVWLWXWLRQ� EXW�WKH�LGHQWLILHU IUHPDLQV�XQFKDQJHG� ,I�\RX�UHSUHVHQW�D�PDS F E\�WKH�H[SUHVVLRQ f = F (x)� WKHQ\RX�PD\�XVH VXEV WR�HYDOXDWH�WKH�IXQFWLRQ�DW�VRPH�SRLQW�

I � � � [ � [A��VXEV�I� [ ��� VXEV�I� [ ���VXEV�I� [ ��� VXEV�I� [ ��

1, 3, 7, 13

1RWH� KRZHYHU� WKDW VXEV �DV�ZHOO�DV�WKH�FRPPDQGV VXEVH[ DQG VXEVRS WR�EHKDQGOHG�VKRUWO\��SHUIRUPV�D�SXUHO\�V\QWDFWLFDO�UHSODFHPHQW� ZKLFK�PD\�OHDG�WRLQFRUUHFW�UHVXOWV�ZKHQ�HYDOXDWLQJ�D�IXQFWLRQ�LQ�WKLV�ZD\�

I � [�VLQ�[�� VXEV�I� [ ��0

Page 174: Mupad Tutorial

/ Ln[lmbmnmbhg3 VXEV% VXEVH[% Zg] VXEVRS

GHOHWH J� * � LQW�J�[�� [ ���LQILQLW\��VXEV�*� [ ��∫ ∞

0g(0) G0

7R�JHW�VXEVWLWXWLRQV�UHVSHFWLQJ�PDWKHPDWLFDO�PHDQLQJ� XVH�WKH _ RSHUDWRU�RU�LWVHTXLYDOHQW�IXQFWLRQ HYDO$W LQVWHDG� ZKLFK�KDV�DOUHDG\�EHHQ�GHVFULEHG�DW�WKH�HQGRI�&KDSWHU ��

I _ [ �(UURU� 'LYLVLRQ E\ ]HUR >BSRZHU@�GXULQJ HYDOXDWLRQ RI HYDO$W

* _ [ �∫ ∞

0g(x) Gx

7KH�RXWSXW�RI�WKH VXEV FRPPDQG�LV�VXEMHFWHG�WR�WKH�XVXDO�VLPSOLILFDWLRQV�RI�WKHLQWHUQDO�VLPSOLILHU� ,Q�WKH�DERYH�H[DPSOH� WKH�FDOO VXEV�I�[ �� SURGXFHV�WKHREMHFW �����A�� ZKLFK�LV�DXWRPDWLFDOO\�VLPSOLILHG�WR 1� <RX�PXVW�QRW�FRQIXVH�WKLVZLWK HYDOXDWLRQ �&KDSWHU ��� ZKHUH�LQ�DGGLWLRQ�DOO�LGHQWLILHUV�LQ�DQ�H[SUHVVLRQ�DUHUHSODFHG�E\�WKHLU�YDOXHV�

7KH�IXQFWLRQ VXEV SHUIRUPV�D�VXEVWLWXWLRQ� 7KH�V\VWHP�RQO\�VLPSOLILHVWKH�UHVXOWLQJ�REMHFW� EXW�GRHV�QRW�HYDOXDWH�LW�XSRQ�UHWXUQ�

,Q�WKH�IROORZLQJ�H[DPSOH

I � [ � VLQ�[�� J � VXEV�I� [ ��VLQ(0)

WKH�LGHQWLILHU VLQ IRU�WKH�VLQH�IXQFWLRQ�LV�QRW�UHSODFHG�E\�WKH�FRUUHVSRQGLQJ0X3$' IXQFWLRQ� ZKLFK�ZRXOG�UHWXUQ VLQ��� �� 2QO\�WKH�QH[W�FDOO�WR J SHUIRUPVD�FRPSOHWH�HYDOXDWLRQ�

J0

���

Page 175: Mupad Tutorial

Ln[lmbmnmbhg3 VXEV% VXEVH[% Zg] VXEVRS

<RX�FDQ�HQIRUFH�HYDOXDWLRQ�E\�XVLQJ HYDO�

HYDO�VXEV�I� [ ���0

<RX�PD\�UHSODFH�DUELWUDU\�0X3$' REMHFWV�E\�VXEVWLWXWLRQ� ,Q�SDUWLFXODU� \RX�FDQVXEVWLWXWH�IXQFWLRQV�RU�SURFHGXUHV�DV�QHZ�YDOXHV�

HYDO�VXEV�K�D � E�� K �[ �! � � [A����

(a+ b)2 + 1

,I�\RX�ZDQW�WR�UHSODFH�D�V\VWHP�IXQFWLRQ� HQFORVH�LWV�QDPH�LQ�D KROG FRPPDQG�

HYDO�VXEV�VLQ�D � E�� KROG�VLQ� �[ �! [ � [A������

a+ b− (a+ b)3

3

<RX�FDQ�DOVR�UHSODFH�PRUH�FRPSOH[�VXEH[SUHVVLRQV�

VXEV�VLQ�[���VLQ�[� � FRV�[��� VLQ�[� � FRV�[� ��VLQ(x)

<RX�VKRXOG�EH�FDUHIXO�ZLWK�VXFK�VXEVWLWXWLRQV� WKH�FRPPDQGVXEV�REMHFW�2OG 1HZ� UHSODFHV�DOO�WKRVH�RFFXUUHQFHV�RI�WKH�H[SUHVVLRQ 2OG WKDWFDQ�EH�IRXQG�E\�PHDQV�RI RS� 7KLV�H[SODLQV�ZK\�QRWKLQJ�KDSSHQV�LQ�WKH�IROORZLQJH[DPSOH�

VXEV�D � E � F� D � E ��� VXEV�D E F� D E ��a+ b+ c, a b c

+HUH�WKH�VXP D�E DQG�WKH�SURGXFW D E DUH QRW RSHUDQGV�RI�WKH�FRUUHVSRQGLQJH[SUHVVLRQV� (YHQ�ZRUVH� ZH�ILQG�

I � D � E � VLQ�D � E�� VXEV�I� D � E ��a+ b+ VLQ(1)

$JDLQ� \RX�FDQQRW�REWDLQ�WKH�VXEH[SUHVVLRQ D�E RI�WKH�RXWHU�VXP�E\�PHDQV�RI RS�+RZHYHU� WKH�DUJXPHQW�RI�WKH�VLQH�LV�WKH�VXE�RSHUDQG RS�I�>���@� �VHH6HFWLRQV ��� DQG ����� DQG�KHQFH�LW�LV�UHSODFHG�E\ ��

���

Page 176: Mupad Tutorial

/ Ln[lmbmnmbhg3 VXEV% VXEVH[% Zg] VXEVRS

,Q�FRQWUDVW�WR VXEV� WKH�IXQFWLRQ VXEVH[ DOVR�UHSODFHV�VXEH[SUHVVLRQV�LQ�VXPV�DQGSURGXFWV�

VXEVH[�I� D � E [ � \�� VXEVH[�D E F� D E [ � \�x+ y + VLQ(x+ y) , c (x+ y)

7KLV�NLQG�RI�VXEVWLWXWLRQ�UHTXLUHV�D�FORVHU�DQDO\VLV�RI�WKH�H[SUHVVLRQ�WUHH� DQGKHQFH VXEVH[ LV�PXFK�VORZHU�WKDQ VXEV IRU�ODUJH�REMHFWV� :KHQ�UHSODFLQJ�PRUHFRPSOH[�VXEH[SUHVVLRQV� \RX�VKRXOG�QRW�EH�PLVOHG�E\�WKH�VFUHHQ�RXWSXW�RIH[SUHVVLRQV�

I � D��E F�a

b c

VXEV�I� E F 1HZ�� VXEVH[�I� E F 1HZ�a

b c,a

1HZ

,I�\RX�ORRN�DW�WKH�RSHUDQGV�RI I� \RX�VHH�WKDW�WKH�H[SUHVVLRQ�WUHH�GRHV�QRW�FRQWDLQWKH�SURGXFW E F� 7KLV�H[SODLQV�ZK\�QR�VXEVWLWXWLRQ�WRRN�SODFH�LQ�WKH VXEV FDOO�

RS�I�

a,1b,

1c

<RX�FDQ�SHUIRUP�VHYHUDO�VXEVWLWXWLRQV�ZLWK�D�VLQJOH�FDOO�RI VXEV�

VXEV�D � E � F� D $� E %� F &�A+B + C

7KLV�LV�HTXLYDOHQW�WR�WKH�QHVWHG�FDOO

VXEV�VXEV�VXEV�D � E � F� D $�� E %�� F &��

7KXV�ZH�REWDLQ�

VXEV�D � EA�� D E� E D�a2 + a

)LUVW� 0X3$' UHSODFHV D E\ E� \LHOGLQJ E�EA�� 7KHQ�LW�VXEVWLWXWHV D IRU E LQ�WKLVQHZ�H[SUHVVLRQ�DQG�UHWXUQV�WKH�DERYH�UHVXOW� ,Q�FRQWUDVW� \RX�PD\�DFKLHYH�D

���

Page 177: Mupad Tutorial

Ln[lmbmnmbhg3 VXEV% VXEVH[% Zg] VXEVRS

VLPXOWDQHRXV�VXEVWLWXWLRQ E\�VSHFLI\LQJ�WKH�VXEVWLWXWLRQ�HTXDWLRQV�LQ�IRUP�RI�DOLVW�RU�D�VHW�

VXEV�D � EA�� >D E� E D@��VXEV�D � EA�� ^D E� E D`�a2 + b, a2 + b

7KH�RXWSXW�RI�WKH�HTXDWLRQ�VROYHU VROYH �&KDSWHU ���VXSSRUWV�WKH�IXQFWLRQDOLW\�RIVXEV� ,Q�JHQHUDO� VROYH UHWXUQV�OLVWV�RI�HTXDWLRQV� ZKLFK�PD\�EH�XVHG�LQ VXEV�

HTXDWLRQV � ^[ � \ �� [ � \ �`�VROXWLRQ � VROYH�HTXDWLRQV� ^[� \`�{[x =

32, y =

12

]}

VXEV�HTXDWLRQV� RS�VROXWLRQ� ���{1 = 1, 2 = 2}

7KH�IXQFWLRQ VXEVRS SURYLGHV�DQRWKHU�YDULDQW�RI�VXEVWLWXWLRQ�VXEVRS�REMHFW� L 1HZ� VHOHFWLYHO\�UHSODFHV�WKH i�WK�RSHUDQG�RI�WKH�REMHFW�E\�WKHYDOXH 1HZ�VXEVRS�� F � DA�� � GA��d5 + 2 c

+HUH� ZH�KDYH�UHSODFHG�WKH�VHFRQG�RSHUDQG DA� RI�WKH�VXP�E\ GA�� ,Q�WKHIROORZLQJ�H[DPSOH� ZH�ILUVW�UHSODFH�WKH�H[SRQHQW�RI�WKH�VHFRQG�WHUP��WKLV�LV�WKHRSHUDQG >���@ RI�WKH�VXP�� DQG�WKHQ�WKH�ILUVW�WHUP�

VXEVRS�� F � DA�� >�� �@ �� � [ \�a4 + x y

,Q�WKH�IROORZLQJ�H[SUHVVLRQ� ZH�ILUVW�UHSODFH�WKH�ILUVW�WHUP� \LHOGLQJ�WKH�H[SUHVVLRQ[ \�FA�� 7KHQ�ZH�VXEVWLWXWH ] IRU�WKH�VHFRQG�IDFWRU�RI�WKH�ILUVW�WHUP��ZKLFK�QRZLV \��VXEVRS�D E � FA�� � [ \� >�� �@ ]�c2 + x z

���

Page 178: Mupad Tutorial

/ Ln[lmbmnmbhg3 VXEV% VXEVH[% Zg] VXEVRS

7KH�H[SUHVVLRQ D�� LV�D�V\PEROLF�VXP� ZKLFK�KDV�D 0�WK RSHUDQG� QDPHO\� WKHV\VWHP�IXQFWLRQ BSOXV IRU�JHQHUDWLQJ�VXPV�

RS�D � �� ��BSOXV

<RX�FDQ�UHSODFH�WKLV�RSHUDQG�E\�DQ\�RWKHU�IXQFWLRQ��IRU�H[DPSOH� E\�WKH�V\VWHPIXQFWLRQ BPXOW ZKLFK�PXOWLSOLHV�LWV�DUJXPHQWV��

VXEVRS�D � �� � BPXOW�2 a

:KHQ�XVLQJ VXEVRS� \RX�QHHG�WR�NQRZ�WKH�SRVLWLRQ�RI�WKH�RSHUDQG�WKDW�\RX�ZDQWWR�UHSODFH� 1RQHWKHOHVV� \RX�VKRXOG�EH�FDXWLRXV� VLQFH�WKH�V\VWHP�PD\�FKDQJH�WKHRUGHU�RI�WKH�RSHUDQGV�ZKHQ�WKLV�LV�PDWKHPDWLFDOO\�YDOLG��IRU�H[DPSOH� LQ�VXPV�SURGXFWV� RU�VHWV��

VHW � ^VLQ�� � D�� D� E� FA�`{a, b, VLQ(a+ 1) , c2

}

,I�\RX�XVH VXEV� \RX�QHHG�QRW�NQRZ�WKH�SRVLWLRQ�RI�WKH�VXEH[SUHVVLRQ� $QRWKHUGLIIHUHQFH�EHWZHHQ VXEV DQG VXEVRS LV�WKDW VXEV WUDYHUVHV�WKH�H[SUHVVLRQ�WUHH�RIWKH�REMHFW UHFXUVLYHO\� DQG�WKXV�DOVR�UHSODFHV�VXERSHUDQGV�

VXEV�VHW� D DA��{b, VLQ

(a2 + 1

), a2, c2

}

���

Page 179: Mupad Tutorial

Ln[lmbmnmbhg3 VXEV% VXEVH[% Zg] VXEVRS

([HUFLVH����� 'RHV�WKH�FRPPDQG VXEVRS�E�D� � F� UHSODFHWKH�LGHQWLILHU E E\ F"

([HUFLVH����� 7KH�FRPPDQGV

GHOHWH I� J � GLII�I�[��GLII�I�[�� [�� [ � ���� GLII�I�[�� [� [� GLII�I�[�� [� [� [� [������������������������������������������� �

�GLII�I�[�� [�

� GLII�I�[�� [� [� [� [� [���������������������������� � ���

GLII�I�[�� [�

JHQHUDWH�D�OHQJWK\�H[SUHVVLRQ�FRQWDLQLQJ�V\PEROLF�GHULYDWLYHV� 0DNH�WKLVH[SUHVVLRQ�PRUH�UHDGDEOH�E\�UHSODFLQJ�WKHVH�GHULYDWLYHV�E\�VLPSOHU�QDPHVf0 = f(x)� f1 = f ′(x)� HWF�

���

Page 180: Mupad Tutorial
Page 181: Mupad Tutorial

/

'LIIHUHQWLDWLRQ�DQG�,QWHJUDWLRQ

:H�KDYH�DOUHDG\�XVHG�WKH�0X3$'� FRPPDQGV�IRU�GLIIHUHQWLDWLRQ�DQG�LQWHJUDWLRQ�6LQFH�WKH\�DUH�LPSRUWDQW� ZH�UHFDSLWXODWH�WKH�XVDJH�RI�WKHVH�URXWLQHV�KHUH�

Page 182: Mupad Tutorial

0 =b__^k^gmbZmbhg�Zg]�Bgm^`kZmbhg

<a^^]j]flaYlagf

7KH�FDOO GLII�H[SUHVVLRQ�[� FRPSXWHV�WKH�GHULYDWLYH RI�WKH�H[SUHVVLRQ�ZLWKUHVSHFW�WR�WKH�XQNQRZQ [�

GLII�VLQ�[A��� [�2x FRV

(x2)

,I�WKH�H[SUHVVLRQ�FRQWDLQV�V\PEROLF�FDOOV�WR�IXQFWLRQV�ZKRVH�GHULYDWLYH�LV�QRWNQRZQ� WKHQ GLII UHWXUQV�LWVHOI�V\PEROLFDOO\�

GLII�[ I�[�� [�

f(x) + x∂

∂xf(x)

<RX�PD\�FRPSXWH�KLJKHU�GHULYDWLYHV�YLD GLII�H[SUHVVLRQ�[�[������ 7KHVHTXHQFH [�[���� RI�LGHQWLILHUV�PD\�EH�JHQHUDWHG�FRQYHQLHQWO\�YLD�WKH�VHTXHQFHRSHUDWRU � �6HFWLRQ �����

GLII�VLQ�[A��� [� [� [� GLII�VLQ�[A��� [ � ��−12x VLQ

(x2)− 8x3 FRV

(x2)

= −12x VLQ(x2)− 8x3 FRV

(x2)

<RX�FDQ�FRPSXWH�SDUWLDO�GHULYDWLYHV�LQ�WKH�VDPH�ZD\� 0X3$'� DVVXPHV�WKDWPL[HG�SDUWLDO�GHULYDWLYHV�RI�V\PEROLF�H[SUHVVLRQV�DUH�V\PPHWULF�

GLII�I�[�\�� [� \� � GLII�I�[�\�� \� [�0

,I�D�PDWKHPDWLFDO�PDS�LV�UHSUHVHQWHG�E\�D�IXQFWLRQ�LQVWHDG�RI�DQ�H[SUHVVLRQ� WKHQWKH�GLIIHUHQWLDO�RSHUDWRU ' FRPSXWHV�WKH�GHULYDWLYH�DV�D�IXQFWLRQ�

'�VLQ�� '�H[S�� '�OQ�� '�VLQ FRV�� '�VLQ#OQ�� '�I�J�

FRV, H[S,1LG, FRV2 − VLQ2,

FRV ◦ OQLG, f ′ + g′

I � [ �! �VLQ�OQ�[���� '�I�

x→ FRV(OQ(x))x

���

Page 183: Mupad Tutorial

=b__^k^gmbZmbhg

+HUH� LG GHQRWHV�WKH�LGHQWLW\�PDS x→ x� 7KH�H[SUHVVLRQ '�I��[� UHWXUQV�WKHYDOXH�RI�WKH�GHULYDWLYH�DW�D�SRLQW�

'�I����� '�I��\A��� '�J����

1,FRV(OQ(y2))

y2, g′(0)

7KH�V\VWHP�FRQYHUWV�WKH�SULPH IRU�WKH�GHULYDWLYH�WR�D�FDOO�RI '�

I���� I�\A��� J���

1,FRV(OQ(y2))

y2, g′(0)

)RU�D�IXQFWLRQ�ZLWK�PRUH�WKDQ�RQH�DUJXPHQW� '�>L@�I� LV�WKH�SDUWLDO�GHULYDWLYHZLWK�UHVSHFW�WR�WKH i�WK�DUJXPHQW� DQG '�>L�M����@�I� LV�HTXLYDOHQW�WR'�>L@�'�>M@������� IRU�KLJKHU�SDUWLDO�GHULYDWLYHV�

([HUFLVH����� &RQVLGHU�WKH�IXQFWLRQ f : x→ VLQ(x)/x� &RPSXWH�ILUVW�WKH�YDOXH�RIf DW�WKH�SRLQW x = 1.23� DQG�WKHQ�WKH�GHULYDWLYH f ′(x)� :K\�GRHV�WKH�IROORZLQJLQSXW�QRW�\LHOG�WKH�GHVLUHG�UHVXOW"

I � VLQ�[��[� [ � ����� GLII�I� [�

([HUFLVH����� 'H�O¶+RVSLWDO¶V�UXOH�VWDWHV�WKDW

OLPx→x0

f(x)g(x)

= OLPx→x0

f ′(x)g′(x)

= · · · = OLPx→x0

f (k−1)(x)g(k−1)(x)

=f (k)(x0)g(k)(x0)

,

LI f(x0) = g(x0) = · · · = f (k−1)(x0) = g(k−1)(x0) = 0 DQG g(k)(x0) ̸= 0� &RPSXWH

OLPx→0

x3 VLQ(x)(1− FRV(x))2 E\�DSSO\LQJ�WKLV�UXOH�LQWHUDFWLYHO\� 8VH�WKH�IXQFWLRQ OLPLW WR

FKHFN�\RXU�UHVXOW�

([HUFLVH����� 'HWHUPLQH�WKH�ILUVW�DQG�VHFRQG�RUGHU�SDUWLDO�GHULYDWLYHV�RIf1(x1, x2) = VLQ(x1 x2) � /HW x = x(t) = VLQ(t)� y = y(t) = FRV(t)� DQGf2(x, y) = x2 y2� &RPSXWH�WKH�GHULYDWLYH�RI f2(x(t), y(t)) ZLWK�UHVSHFW�WR t�

���

Page 184: Mupad Tutorial

0 =b__^k^gmbZmbhg�Zg]�Bgm^`kZmbhg

Afl]_jYlagf

7KH�IXQFWLRQ LQW IHDWXUHV�ERWK�GHILQLWH�DQG�LQGHILQLWH�LQWHJUDWLRQ�

LQW�VLQ�[�� [�� LQW�VLQ�[�� [ ���3,���− FRV(x) , 1

,I LQW LV�XQDEOH�WR�FRPSXWH�D�UHVXOW� LW�UHWXUQV�LWVHOI�V\PEROLFDOO\� ,Q�WKH�IROORZLQJH[DPSOH� WKH�LQWHJUDQG�LV�VSOLW�LQWR�WZR�WHUPV� 2QO\�RQH�RI�WKHVH�KDV�DQ�LQWHJUDOWKDW�FDQ�EH�UHVSUHVHQWHG�E\�HOHPHQWDU\�IXQFWLRQV�

LQW��[ � ����[ VTUW�[A� � ���� [�∫

x− 1x√x3 + 1

Gx

7KH�IXQFWLRQ 6L(x) =∫ x

0VLQ(t)/tGt LV�LPSOHPHQWHG�DV�D�VSHFLDO�IXQFWLRQ�LQ

0X3$'�

LQW�VLQ�D [��[� [�6L(a x)

:KLOH�LQWHJUDWLQJ� WKH�LQWHJUDWLRQ�YDULDEOH�LWVHOI�LV�DVVXPHG�WR�EH�UHDO� 2WKHU�WKDQWKLV� DOO�FRPSXWDWLRQV�WDNH�SODFH�RYHU�WKH�FRPSOH[�QXPEHUV� DQG�WKH�V\VWHPUHJDUGV�HYHU\�V\PEROLF�SDUDPHWHU�LQ�WKH�LQWHJUDQG�DV�D�FRPSOH[�QXPEHU�XQOHVVWROG�RWKHUZLVH� ,Q�WKH�IROORZLQJ�H[DPSOH� WKH�GHILQLWH�LQWHJUDO�H[LVWV�RQO\�IRUVSHFLILF�UDQJHV�RI�WKH�SDUDPHWHU a� DQG�WKH�V\VWHP�UHWXUQV�D�V\PEROLF OLPLW FDOO�

LQW�VLQ�D [��[� [ ���LQILQLW\�OLPx→∞

6L(x a)

<RX�FDQ�WHOO�WKH�V\VWHP�WKDW�DQ�LGHQWLILHU�KDV�FHUWDLQ�SURSHUWLHV�E\�XVLQJ�WKHRSHUDWRU DVVXPLQJ �6HFWLRQ ����� 7KH�IROORZLQJ�FDOO�VWLSXODWHV�WKDW a EH D�SRVLWLYHUHDO�QXPEHU�

LQW�VLQ�D [��[� [ ���LQILQLW\� DVVXPLQJ D ! �π

2

���

Page 185: Mupad Tutorial

Bgm^`kZmbhg

%HVLGHV�WKH�H[DFW�FRPSXWDWLRQ�RI�GHILQLWH�LQWHJUDOV� 0X3$' DOVR�SURYLGHV�VHYHUDOQXPHULFDO�PHWKRGV�

IORDW�LQW�VLQ�[��[� [ ������1.605412977

,Q�WKH�SUHYLRXV�FRPSXWDWLRQ� LQW ILUVW�UHWXUQV�D�V\PEROLF�UHVXOW��WKH�HUURUIXQFWLRQ HUI�� ZKLFK�LV�WKHQ�DSSUR[LPDWHG�E\ IORDW� ,I�\RX�ZDQW�WR�FRPSXWHQXPHULFDOO\�IURP�WKH�EHJLQQLQJ� WKHQ�\RX�FDQ�VXSSUHVV�WKH�V\PEROLF�FRPSXWDWLRQYLD LQW E\�XVLQJ KROG �6HFWLRQ �����

IORDW�KROG�LQW��VLQ�[��[� [ ������1.605412977

$OWHUQDWLYHO\� WKH�IXQFWLRQ QXPHULF��LQW IURP�WKH QXPHULF OLEUDU\�FDQ�EH�XVHG�

QXPHULF��LQW�VLQ�[��[� [ �����1.605412977

7KLV�IXQFWLRQ�DOORZV�\RX�WR�FKRRVH�GLIIHUHQW�QXPHULFDO�PHWKRGV�IRU�FRPSXWLQJ�WKHLQWHJUDO� <RX�ILQG�PRUH�GHWDLOHG�LQIRUPDWLRQ�XVLQJ "QXPHULF��LQW� ,W�ZRUNV�LQ�DSXUHO\�QXPHULFDO�IDVKLRQ�ZLWKRXW�DQ\�V\PEROLF�SUHSURFHVVLQJ�RI�WKH�LQWHJUDQG�)RU�VPRRWK�LQWHJUDQGV�ZLWKRXW�VLQJXODULWLHV� QXPHULF��LQW LV�TXLWH�HIILFLHQW�

([HUFLVH����� &RPSXWH�WKH�IROORZLQJ�LQWHJUDOV�

∫ π/2

0VLQ(x) FRV(x)Gx ,

∫ 1

0

Gx√1− x2

,

∫ 1

0x DUFWDQ(x)Gx .

8VH�0X3$' WR�YHULI\�WKH�IROORZLQJ�HTXDOLW\�∫ −1

−2

Gxx

= − OQ(2) �

([HUFLVH����� 8VH�0X3$' WR�GHWHUPLQH�WKH�IROORZLQJ�LQGHILQLWH�LQWHJUDOV�∫ x tGt√

(2 a t− t2)3,

∫ x√t2 − a2 Gt ,

∫ x Gt

t√

1 + t2.

���

Page 186: Mupad Tutorial

0 =b__^k^gmbZmbhg�Zg]�Bgm^`kZmbhg

([HUFLVH����� 7KH�IXQFWLRQ LQWOLE��FKDQJHYDU SHUIRUPV�D�FKDQJH�RI�YDULDEOH LQD�V\PEROLF�LQWHJUDO� 5HDG�WKH�FRUUHVSRQGLQJ�KHOS�SDJH� 0X3$' FDQQRW�FRPSXWHWKH�LQWHJUDO ∫ π/2

−π/2VLQ(x)

√1 + VLQ(x) Gx.

$VVLVW�WKH�V\VWHP�E\�XVLQJ�WKH�VXEVWLWXWLRQ W VLQ�[�� &RPSDUH�WKH�YDOXH�WKDW�\RXJHW�WR�WKH�QXPHULFDO�UHVXOW�UHWXUQHG�E\�WKH�IXQFWLRQ QXPHULF��LQW�

���

Page 187: Mupad Tutorial

0

6ROYLQJ�(TXDWLRQV� VROYH

7KH�IXQFWLRQ VROYH VROYHV�HTXDWLRQV� LQHTXDOLWLHV� V\VWHPV�RI�WKHVH� GLIIHUHQWLDOHTXDWLRQV� UHFXUUHQFH�HTXDWLRQV�HWF� 7KLV�URXWLQH�FDQ�KDQGOH�YDULRXV�GLIIHUHQWW\SHV�RI�HTXDWLRQV� %HVLGHV�³DOJHEUDLF´�HTXDWLRQV� DOVR�FHUWDLQ�FODVVHV�RIGLIIHUHQWLDO�DQG�UHFXUUHQFH�HTXDWLRQV�FDQ�EH�VROYHG� )XUWKHU� WKHUH�LV�D�YDULHW\�RIVSHFLDOL]HG�VROYHUV�WKDW�RQO\�KDQGOH�VSHFLDO�FODVVHV�RI�HTXDWLRQV� 0DQ\�RI�WKHVHDOJRULWKPV�DUH�DFWXDOO\�FDOOHG�E\�WKH�³XQLYHUVDO´ VROYH RQFH�LW�KDV�LGHQWLILHG�WKHW\SH�RI�WKH�HTXDWLRQV� 7KH�XVHU�FDQ�DOVR�FDOO�WKH�VSHFLDOL]HG�VROYHUV�GLUHFWO\� <RXFDQ�ILQG�D�VXUYH\�RI�DOO�DYDLODEOH�0X3$'� VROYHUV�LQ�LQ�WKH�³4XLFN�5HIHUHQFH´�LQWKH�RQOLQH�GRFXPHQWDWLRQ�

Page 188: Mupad Tutorial

1 Lheobg`�>jnZmbhgl3 VROYH

HgdqfgeaYd�=imYlagfk

<RX�FDQ�VXSSO\�DQ�LQGLYLGXDO�HTXDWLRQ�DV�ILUVW�DUJXPHQW�WR VROYH� 7KH�XQNQRZQIRU�ZKLFK�\RX�ZDQW�WR�VROYH�LV�WKH�VHFRQG�DUJXPHQW�

VROYH�[A� � [ \��� [�� VROYH�[A� � [ � \�� �� \�{−√y + 12− 1

2,

√y + 12− 1

2

},{4x2 + 4x

}

,Q�WKLV�FDVH� WKH�V\VWHP�UHWXUQV�D�VHW�RI�VROXWLRQV� ,I�\RX�VSHFLI\�DQ�H[SUHVVLRQLQVWHDG�RI�DQ�HTXDWLRQ� WKHQ VROYH DVVXPHV�WKH�HTXDWLRQ H[SUHVVLRQ ��

VROYH�[A� � [ � \��� \�{4x2 + 4x

}

)RU�SRO\QRPLDOV�RI�KLJKHU�GHJUHH� LW�LV�SURYDEO\�LPSRVVLEOH�WR�DOZD\V�ILQG�D�FORVHGIRUP�IRU�WKH�VROXWLRQV�E\�PHDQV�RI�UDGLFDOV�HWF� ,Q�VXFK�FDVHV� 0X3$'� XVHV�WKH5RRW2I REMHFW�

VROYH�[A� � [A� � [� [�{0} ∪ 5RRW2I(;�6 + ;�+ 1,;�)

7KH�DERYH 5RRW2I REMHFW�UHSUHVHQWV�DOO�VROXWLRQV�RI�WKH�HTXDWLRQ [A��[�� ��<RX�FDQ�XVH IORDW WR�DSSUR[LPDWH�VXFK�REMHFWV�E\�IORDWLQJ�SRLQW�QXPEHUV� 7KHV\VWHP�LQWHUQDOO\�HPSOR\V�D�QXPHULFDO�SURFHGXUH�WR�GHWHUPLQH�DOO��FRPSOH[��URRWVRI�WKH�SRO\QRPLDO�

IORDW���^���� � ������������ � ����������� ,�

� ������������ � ������������ ,�

������������ � ������������ ,�

������������ � ������������ ,�

� ������������ � ������������ ,�

� ������������ � ����������� ,`

���

Page 189: Mupad Tutorial

IherghfbZe�>jnZmbhgl

,I�\RX�ZDQW�WR�VROYH�D�FROOHFWLRQ�RI�HTXDWLRQV�IRU�SRVVLEO\�VHYHUDO�XQNQRZQV�VSHFLI\�ERWK�WKH�HTXDWLRQV�DQG�WKH�XQNQRZQV�DV�VHWV�RU�OLVWV� ,Q�WKH�IROORZLQJH[DPSOH� ZH�VROYH�WZR OLQHDU HTXDWLRQV LQ�WKUHH�XQNQRZQV�

GHOHWH [� \� ]�HTXDWLRQV � ^[ � \ � ] �� [ � \ �`�VROXWLRQ � VROYH�HTXDWLRQV� ^[� \� ]`�{[x = 2− ]�, y = ]�, z = 1]}

,I�\RX�VROYH�HTXDWLRQV�IRU�VHYHUDO�YDULDEOHV� WKHQ�0X3$' UHWXUQV�D�VHW�RI�³VROYHGHTXDWLRQV´�HTXLYDOHQW�WR�WKH�RULJLQDO�V\VWHP�RI�HTXDWLRQV� <RX�FDQ�QRZ�UHDG�RIIWKH�VROXWLRQV�LPPHGLDWHO\� WKH�XQNQRZQ z KDV�WKH�YDOXH 1� WKH�XQNQRZQ y PD\�EHDUELWUDU\��LQGLFDWHG�E\�WKH�QHZ�LQGHWHUPLQDWH ]���DQG� IRU�DQ\�JLYHQ�YDOXH�RI y� ZHKDYH x = 2− y� ,W�LV�SRVVLEOH�WR�JHW�D�FOHDUHU�LQGLFDWLRQ�RI�WKH�QHZ�SDUDPHWHU�QRWSUHVHQW�LQ�WKH�LQSXW�E\�JLYLQJ�WKH�RSWLRQ 9HFWRU)RUPDW WR�WKH VROYH FRPPDQG�

VROYH�HTXDWLRQV� ^[� \� ]`� 9HFWRU)RUPDW�⎛

⎜⎝x

y

z

⎟⎠ ∈

⎧⎪⎨

⎪⎩

⎜⎝2− ]�]�

1

⎟⎠

∣∣∣∣∣∣∣]� ∈ C

⎫⎪⎬

⎪⎭

+RZHYHU� WKLV�UHWXUQ�IRUPDW�LV�PRUH�GLIILFXOW�WR�XVH�LQ�VXEVHTXHQW�FRPPDQGV�

7KH�FDOO�WR VROYH GRHV�QRW�DVVLJQ�WKH�YDOXHV�WR�WKH�XQNQRZQV� x� y� DQG z DUH�VWLOOXQNQRZQV� +RZHYHU� WKH�IRUP�RI�WKH�RXWSXW�DV�D�OLVW�RI�VROYHG�HTXDWLRQV�LV�FKRVHQLQ�VXFK�D�ZD\�WKDW�\RX�FDQ�FRQYHQLHQWO\�XVH _ �&KDSWHU �����WR�VXEVWLWXWH�WKHVHYDOXHV�LQ�RWKHU�REMHFWV� )RU�H[DPSOH� \RX�PD\�VXEVWLWXWH�WKH�VROXWLRQ�LQ�WKHRULJLQDO�HTXDWLRQV�WR�YHULI\�WKH�UHVXOW�

HTXDWLRQV _ VROXWLRQ>�@{2 = 2, 3 = 3}

<RX�FDQ�XVH DVVLJQ�VROXWLRQ>�@� WR�DVVLJQ�WKH�VROXWLRQ�YDOXHV�WR�WKH�LGHQWLILHUV[� \� DQG ]�

���

Page 190: Mupad Tutorial

1 Lheobg`�>jnZmbhgl3 VROYH

,Q�WKH�QH[W�H[DPSOH� ZH�VROYH�WZR QRQOLQHDU SRO\QRPLDO�HTXDWLRQV�LQ�VHYHUDOXQNQRZQV�

HTXDWLRQV � ^[A� � \ �� [ � \ �`�VROXWLRQV � VROYH�HTXDWLRQV� ^[� \`�{[x = −

√132− 1

2, y = −

√132− 5

2

],

[x =√

132− 1

2, y =

√132− 5

2

]}

0X3$' IRXQG�WZR�GLVWLQFW�VROXWLRQV� $JDLQ� \RX�FDQ�XVH _ WR�VXEVWLWXWH�WKHVROXWLRQV�LQ�RWKHU�H[SUHVVLRQV�

PDS�HTXDWLRQV _ VROXWLRQV>�@� H[SDQG��PDS�HTXDWLRQV _ VROXWLRQV>�@� H[SDQG�{1 = 1, 2 = 2} , {1 = 1, 2 = 2}

2IWHQ� VROXWLRQV�DUH�UHSUHVHQWHG�E\ 5RRW2I H[SUHVVLRQV�

VROYH�^[A� � [A� � � [ \� \A� [A�`� ^[� \`��� �� ^ �� �� `_ [ _ ^ _ � _ `_ _ LQ ^ _ _ ` XQLRQ_ \ _ ^ _ � _ `�� �� ^ �� �� `

^ �� �� _^ _ � _ _^ _ � ] � � ] � � _ _^ _ _ _ ] LQ^ _ ] _ _^ �� �� _

``

� � � `5RRW2I�]� � � ]� � � ]� � � ]� � �� ]�� `

``

���

Page 191: Mupad Tutorial

IherghfbZe�>jnZmbhgl

,I�\RX�XVH�WKH�RSWLRQ 0D['HJUHH Q� WKHQ 5RRW2I H[SUHVVLRQV�IRU�SRO\QRPLDOV�RIGHJUHH�XS�WR n DUH�UHSODFHG�E\�UHSUHVHQWDWLRQV�LQ�WHUPV�RI�UDGLFDOV�LI�WKLV�LVSRVVLEOH� 1RWH�WKDW�VXFK�VROXWLRQV�WHQG�WR�EH�UDWKHU�FRPSOLFDWHG�

VROYH�^[A� � [A� � � [ \� \A� [A�`� ^[� \`�0D['HJUHH ��

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎢⎢⎢⎢⎣x = 7 +

⎜⎜⎜⎜⎝

√129(

348554 +

√3√

73 5 L18

) 13

4 + 9(

348554 +

√3√

73 5 L18

) 23

+ 145

6(

348554 +

√3√

73 5 L18

) 16

+ . . .

6SHFLI\LQJ�WKH�XQNQRZQV�WR�VROYH�IRU�LV�RSWLRQDO� )RU�WKH�VSHFLDO�FDVH�ZKHUH�\RXKDYH�RQH�XQNQRZQ� KRZHYHU� WKH�IRUPDW�RI�WKH�RXWSXW�GHSHQGV�RQ�ZKHWKHU�\RXVSHFLI\�WKH�XQNQRZQ�RU�QRW� 7KH�JHQHUDO�UXOH�LV�DV�IROORZV�

$ FDOO�RI�WKH�IRUP VROYH�HTXDWLRQ�XQNQRZQ�� ZKHUH XQNQRZQ LV�DQLGHQWLILHU� UHWXUQV�D�0X3$' REMHFW�UHSUHVHQWLQJ�D�VHW�RI�QXPEHUV� $OORWKHU�IRUPV�RI VROYH IRU��RQH�RU�VHYHUDO��SRO\QRPLDO�HTXDWLRQV�ZLWKRXW�VSHFLDO�RSWLRQV�OLNH 9HFWRU)RUPDW� UHWXUQ�D�VHW�RI�OLVWV�RIHTXDWLRQV�RU�DQ�H[SUHVVLRQ�LQYROYLQJ 5RRW2I�

+HUH�DUH�VRPH�H[DPSOHV�

VROYH�[A� � � [ � � �� [�� VROYH�[A� � � [ � �� [�{1, 2} , {1, 2}

VROYH�[A� � � [ � � ��� VROYH�[A� � � [ � ��{[x = 1] , [x = 2]} , {[x = 1] , [x = 2]}

VROYH�^[A� � � [ � � �`� [��VROYH�^[A� � � [ � �`� [�{1, 2} , {1, 2}

VROYH�^[A� � � [ � \ �� \ � � [ �`� ^[� \`�{[x = 0, y = 0] , [x = 1, y = 2]}

VROYH�^[A� � � [ � \ �� \ � � [ �`�{[x = 0, y = 0] , [x = 1, y = 2]}

���

Page 192: Mupad Tutorial

1 Lheobg`�>jnZmbhgl3 VROYH

,I�\RX�GR�QRW�VXSSO\�XQNQRZQV�WR�VROYH�IRU� VROYH LQWHUQDOO\�XVHV�WKH�V\VWHPIXQFWLRQ LQGHWV WR�ILQG�WKH�V\PEROLF�LGHQWLILHUV�LQ�WKH�HTXDWLRQV�DQG�UHJDUGV�DOO�RIWKHP�DV�XQNQRZQV�

VROYH�^[ � \A� �� [A� � \ �`�(x

y

)∈{(

1− z2

z

)∣∣∣∣∣ z ∈ 5RRW2I(]�4 − 2 ]�2 − ]�+ 1, ]�)

}

%\�GHIDXOW� VROYH WULHV�WR�ILQG�DOO FRPSOH[ VROXWLRQV�RI�WKH�JLYHQ�HTXDWLRQ�V�� ,I\RX�ZDQW�WR�ILQG�RQO\�WKH�UHDO�VROXWLRQV�RI�D�VLQJOH�HTXDWLRQ� XVH�WKH�RSWLRQ'RPDLQ 'RP��5HDO RU�DVVXPH�[�WR�EH�UHDO�VROYH�[A� � [ �� [�{0,−L, L}

VROYH�[A� � [ �� [� DVVXPLQJ [ LQ 5B{0}

2WKHU�DVVXPSWLRQV� VXFK�DV [!�� [LQ4B� [LQ=BDQG[!�� HWF�� DUH�KDQGOHG�DV�ZHOO�

,Q�VRPH��XVXDOO\�H[RWLF��FDVHV� WKH�³UHDO�VROXWLRQV´�PD\�LQFOXGH�XQH[SHFWHG�YDOXHV�

VROYH�OQ�[�A� 3,A�� [� DVVXPLQJ [ LQ 5B{−1, H−π, Hπ

}

:KHQ�UHSODFLQJ�WKH x LQ�RXU�LQSXW�E\ −1� ZH�JHW�DQ OQ(−1)� ZKLFK�LV π L� D�FRPSOH[YDOXH� ,Q�WKH�HQG� ZH�JHW�D�UHDO�YDOXHG�UHVXOW� EXW�D�FRPPRQ�UHTXHVW�LV�WR�ILQGVROXWLRQV�IRU�D�SUREOHP�UHVWULFWHG�WR�WKH�UHDOV� LQFOXGLQJ�DOO�LQWHUPHGLDWH�YDOXHV�³VROYLQJ�RYHU�WKH�UHDOV´�� 7R�PDNH VROYH WDFNOH�WKLV�SUREOHP� XVH�WKH�RSWLRQ 5HDOLQVWHDG�RI�WKH�DVVXPSWLRQ�

VROYH�OQ�[�A� 3,A�� [� 5HDO�{H−π, Hπ

}

2I�FRXUVH� XVLQJ�WKH�RSWLRQ�VDYHV�D�ELW�RI�W\SLQJ� WRR�

���

Page 193: Mupad Tutorial

IherghfbZe�>jnZmbhgl

,I�DOO�FRPSOH[�QXPEHUV�VDWLVI\�D�JLYHQ�HTXDWLRQ� WKHQ VROYH UHWXUQV C� WKH�VHW�RIDOO�FRPSOH[�QXPEHUV�

VROYH�VLQ�[� FRV�[ � 3,���� [�C

GRPW\SH���VROYHOLE::%DVLF6HW

7KHUH�DUH�IRXU�VXFK�³EDVLF�VHWV´� WKH�LQWHJHUV Z �HQWHUHG�DV =B LQ�0X3$'�LQSXW��WKH�UDWLRQDO�QXPEHUV Q �4B�� WKH�UHDO�QXPEHUV R �5B�� DQG�WKH�FRPSOH[�QXPEHUV C�&B��

<RX�FDQ�XVH�WKH�IXQFWLRQ IORDW WR�ILQG QXPHULFDO VROXWLRQV� +RZHYHU� ZLWK�DVWDWHPHQW�RI�WKH�IRUP IORDW�VROYH�HTXDWLRQV� XQNQRZQV��� VROYH ILUVW�WULHV�WRVROYH�WKH�HTXDWLRQV�V\PEROLFDOO\� ,I�DQ�H[DFW�VROXWLRQ�LV�IRXQG� IORDW FRQYHUWV�WKHUHVXOW�WR�IORDWLQJ�SRLQW�DSSUR[LPDWLRQV� ,I�\RX�ZDQW�WR�FRPSXWH�LQ�D�SXUHO\QXPHULFDO�ZD\� \RX�FDQ�XVH KROG �6HFWLRQ �����WR�DYRLG�V\PEROLF�SUHSURFHVVLQJ�

IORDW�KROG�VROYH��^[A� � [A� � � [ \� \A� [A�`�^[� \`��

^>[ � ��� � ����������� ,� \ ��� � ����������� ,@�

>[ � ��� � ����������� ,� \ ��� � ����������� ,

@� >[ � ��� � ������������ ,�

\ � ��� � ������������ ,@� >[ ���� \ ���@�

>[ � ��� � ������������ ,�

\ � ��� � ������������ ,@`

$OWHUQDWLYHO\� WKH�OLEUDU\ QXPHULF SURYLGHV�YDULRXV�IXQFWLRQV�IRU�QXPHULFDOHTXDWLRQ�VROYLQJ�VXFK�DV QXPHULF��VROYH�� QXPHULF��IVROYH� RUQXPHULF��UHDOURRWV� 7KH�KHOS�V\VWHP�JLYHV�GHWDLOV�DERXW�WKHVH�URXWLQHV� FDOO"VROYHUV IRU�D�VXUYH\�RU�VHH "QXPHULF��VROYH HWF�

�,Q�IDFW� WKH�FDOOV IORDW�KROG�VROYH������� DQG QXPHULF��VROYH����� DUH�HTXLYDOHQW�

���

Page 194: Mupad Tutorial

1 Lheobg`�>jnZmbhgl3 VROYH

([HUFLVH����� &RPSXWH�WKH�JHQHUDO�VROXWLRQ�RI�WKH�V\VWHP�RI�OLQHDU�HTXDWLRQV

a + b + c + d + e = 1 ,a + 2 b + 3 c + 4 d + 5 e = 2 ,a − 2 b − 3 c − 4 d − 5 e = 2 ,a − b − c − d − e = 3 .

+RZ�PDQ\�IUHH�SDUDPHWHUV�GRHV�WKH�VROXWLRQ�KDYH"

���

Page 195: Mupad Tutorial

@^g^kZe�>jnZmbhgl�Zg]�Bg^jnZebmb^l

?]f]jYd�=imYlagfk�Yf\�Af]imYdala]k

VROYH FDQ�KDQGOH�D�YDULHW\�RI��QRQ�SRO\QRPLDO��HTXDWLRQV� )RU�H[DPSOH� WKHHTXDWLRQ H[S(x) = 8 KDV�LQILQLWHO\�PDQ\�VROXWLRQV�RI�WKH�IRUP OQ(8) + 2 Lπ k ZLWKk ∈ Z LQ�WKH�FRPSOH[�SODQH�

6 � VROYH�H[S�[� �� [�{ OQ(8) + π k 2 L| k ∈ Z}

7KH�GDWD�W\SH�RI�WKH�UHWXUQHG�UHVXOW�LV�D�VR�FDOOHG�³LPDJH�VHW�´ ,W�UHSUHVHQWV�DPDWKHPDWLFDO�VHW�RI�WKH�IRUP {f(x) | x ∈ A}� ZKHUH A LV�VRPH�RWKHU�VHW�

GRPW\SH�6�'RP::,PDJH6HW

7KLV�GDWD�W\SH�LV�FDSDEOH�RI�UHSUHVHQWLQJ�LQILQLWHO\�PDQ\�HOHPHQWV�

,I�\RX�RPLW�WKH�YDULDEOH�WR�VROYH�IRU� WKH�V\VWHP�UHWXUQV�D�ORJLFDO�IRUPXOD� XVLQJ�WKHRSHUDWRU LQ�

VROYH�H[S�[� ��(x)∈{(

OQ(8) + π k 2 L)∣∣∣ k ∈ Z

}

<RX�FDQ�XVH PDS WR�DSSO\�D�IXQFWLRQ�WR�DQ�LPDJH�VHW�

PDS�6� BSOXV� �OQ����{π k 2 L| k ∈ Z}

$OWHUQDWLYHO\� PRVW�DULWKPHWLFDO�RSHUDWLRQV�FDQ�EH�DSSOLHG�GLUHFWO\�

H[S�6�{8}

7KH�IXQFWLRQ LV �6HFWLRQ �����FDQ�KDQGOH�LPDJH�VHWV�

6 � VROYH�VLQ�3, [��� �� [�{2 k| k ∈ Z}

���

Page 196: Mupad Tutorial

1 Lheobg`�>jnZmbhgl3 VROYH

LV�� LQ 6�� LV�� LQ 6�)$/6(, 758(

7KH�HTXDWLRQ H[S�[� VLQ�[� DOVR�KDV�LQILQLWHO\�PDQ\�VROXWLRQV� ZKLFK� KRZHYHU�0X3$'� FDQQRW�UHSUHVHQW�H[DFWO\� ,Q�WKLV�FDVH� LW�UHWXUQV�WKH�FDOO�WR VROYHV\PEROLFDOO\�

VROXWLRQV � VROYH�H[S�[� VLQ�[�� [�VROYH(Hx − VLQ(x) = 0, x)

:DUQLQJ� ,Q�FRQWUDVW�WR�SRO\QRPLDO�HTXDWLRQV� WKH�QXPHULFDO�VROYHU�FRPSXWHV�DWPRVW�RQH�VROXWLRQ�RI�D�QRQ�SRO\QRPLDO�HTXDWLRQ�

IORDW�VROXWLRQV�{−226.1946711}

+RZHYHU� \RX�FDQ�VSHFLI\�D�VHDUFK�UDQJH WR�VHOHFW�D�SDUWLFXODU�QXPHULFDO�VROXWLRQ�

QXPHULF��VROYH�H[S�[� VLQ�[�� [ ��������{−9.424858654}

8VLQJ QXPHULF��UHDOURRWV� \RX�FDQ�ILQG�HQFORVXUHV�IRU DOO UHDO�URRWV�LQ�D�JLYHQLQWHUYDO�

QXPHULF��UHDOURRWV�H[S�[� VLQ�[�� [ ��������[[−9.43359375,−9.423828125] , [−6.2890625,−6.279296875]]

0X3$' KDV�D�VSHFLDO�GDWD�W\SH�IRU�WKH�VROXWLRQ�RI�SDUDPHWULF�HTXDWLRQV�SLHFHZLVH� )RU�H[DPSOH� WKH VHW�RI�VROXWLRQV x ∈ C RI�WKH�HTXDWLRQ(a x2 − 4) (x− b) = 0 WDNHV�RQ�GLIIHUHQW�IRUPV� GHSHQGLQJ�RQ�WKH�YDOXH�RI�WKHSDUDPHWHU a�

GHOHWH D� S � VROYH��D [A� � �� �[ � E�� [�{

{b} LI a = 0{b,− 2√

a, 2√a

}LI a ̸= 0

GRPW\SH�S�SLHFHZLVH

����

Page 197: Mupad Tutorial

@^g^kZe�>jnZmbhgl�Zg]�Bg^jnZebmb^l

7KH�IXQFWLRQ PDS DSSOLHV�D�IXQFWLRQ�WR�DOO�EUDQFKHV�RI�D SLHFHZLVH REMHFW�

PDS�S� BSRZHU� ��{ {

b2}

LI a = 0{

4a , b

2}

LI a ̸= 0

$IWHU�WKH�IROORZLQJ�VXEVWLWXWLRQ� WKH SLHFHZLVH REMHFW�LV�VLPSOLILHG�WR�D�VHW�

� _ >D �� E �@{1, 4}

7KH�IXQFWLRQ VROYH FDQ�DOVR�KDQGOH�LQHTXDOLWLHV� ,W�WKHQ�UHWXUQV�DQ�LQWHUYDO�RU�DXQLRQ�RI�LQWHUYDOV� RI�GRPDLQ�W\SH 'RP��,QWHUYDO�

VROYH�[A� � �� [�(−1, 1) ∪ {x L|x ∈ (−∞,∞)}

GRPW\SH���'20B(;35

6 � VROYH�[A� ! �� [�[1,∞) ∪ (−∞,−1]

LV��� LQ 6�� LV�� LQ 6�758(, )$/6(

����

Page 198: Mupad Tutorial

1 Lheobg`�>jnZmbhgl3 VROYH

<a^^]j]flaYd�=imYlagfk

7KH�IXQFWLRQ RGH GHILQHV�DQ�RUGLQDU\�GLIIHUHQWLDO�HTXDWLRQ� 6XFK�DQ�REMHFW�KDV�WZRFRPSRQHQWV� DQ�HTXDWLRQ�DQG�WKH�IXQFWLRQ�WR�VROYH�IRU�

GLII(TXDWLRQ � RGH�\�[� \�[�A�� \�[��

RGH

(∂

∂xy(x)− y(x)2, y(x)

)

7KH�IROORZLQJ�FDOO�WR VROYH ILQGV�WKH�JHQHUDO�VROXWLRQ�FRQWDLQLQJ�DQ�DUELWUDU\FRQVWDQW C3�

VROYH�GLII(TXDWLRQ�{

0,− 1&�+ x

}

'LIIHUHQWLDO�HTXDWLRQV�RI�KLJKHU�RUGHU�FDQ�EH�KDQGOHG�DV�ZHOO�

VROYH�RGH�\�[� \�[�� \�[���{&� Hx +

&�Hx

}

<RX�FDQ�VSHFLI\�LQLWLDO�FRQGLWLRQV�RU�ERXQGDU\�YDOXHV�E\�SDVVLQJ�WKH�GLIIHUHQWLDOHTXDWLRQ�WRJHWKHU�ZLWK�WKH�LQLWLDO�ERXQGDU\�FRQGLWLRQV�DV�D�VHW�ZKHQ�FDOOLQJ RGH�

GLII(TXDWLRQ � RGH�^\�[� \�[�� \��� �� \��� �`� \�[���

0X3$' QRZ�DGMXVWV�WKH�IUHH�FRQVWDQWV�LQ�WKH�JHQHUDO�VROXWLRQ�DFFRUGLQJ�WR�WKHLQLWLDO�ERXQGDU\�FRQGLWLRQV�

VROYH�GLII(TXDWLRQ�{

12 Hx

+Hx

2

}

����

Page 199: Mupad Tutorial

=b__^k^gmbZe�>jnZmbhgl

<RX�FDQ�VSHFLI\�V\VWHPV�RI�HTXDWLRQV�ZLWK�VHYHUDO�IXQFWLRQV�LQ�IRUP�RI�D�VHW�

VROYH�RGH�^\�[� \�[� � � ]�[�� ]�[� \�[�`�^\�[�� ]�[�`��

{[z(x) =

&�� H2 x

2− &��

Hx, y(x) =

&��Hx

+ &�� H2 x]}

7KH�IXQFWLRQ QXPHULF��RGHVROYH RI�WKH QXPHULF OLEUDU\ VROYHV�WKH�RUGLQDU\GLIIHUHQWLDO�HTXDWLRQ Y ′(x) = f(x, Y (x)) ZLWK�WKH�LQLWLDO�FRQGLWLRQ Y (x0) = Y0QXPHULFDOO\� <RX�PXVW�VXSSO\�WKH�ULJKW�KDQG�VLGH�RI�WKH�GLIIHUHQWLDO�HTXDWLRQ�DV�DIXQFWLRQ f(x, Y ) RI�WZR�DUJXPHQWV� ZKHUH x LV�D�VFDODU�DQG Y LV�D�YHFWRU� ,I�\RXFRPELQH�WKH�FRPSRQHQWV y DQG z LQ�WKH�SUHYLRXV�H[DPSOH�WR�D�YHFWRU Y = (y, z)�\RX�FDQ�GHILQH�WKH�ULJKW�KDQG�VLGH�RI�WKH�HTXDWLRQ

GGxY =

GGx

(yz

)=(y + 2 zy

)=(

<[�] + � · <[�]<[�]

)=: I([, <)

LQ�WKH�IRUP

I � �[� <� �! ><>�@ � � <>�@� <>�@@�

1RWH�WKDW f(x, Y )PXVW�EH�D�YHFWRU� +HUH� WKLV�LV�UHDOL]HG�E\�PHDQV�RI�D�OLVWFRQWDLQLQJ�WKH�FRPSRQHQWV�RQ�WKH�ULJKW�KDQG�VLGH�RI�WKH�GLIIHUHQWLDO�HTXDWLRQ�7KH�FDOO

QXPHULF��RGHVROYH������ I� >�� �@�[9.729448318, 5.04866388]

LQWHJUDWHV�WKH�V\VWHP�RI�GLIIHUHQWLDO�HTXDWLRQV�ZLWK�WKH�LQLWLDO�YDOXHVY (0) = (y(0), z(0)) = (1, 1) RYHU�WKH�LQWHUYDO x ∈ [0, 1]� WKH�LQLWLDO�FRQGLWLRQV�DUHVSHFLILHG�E\�WKH�OLVW >���@� 7KH�QXPHULFDO�VROYHU�UHWXUQV�WKH�QXPHULFDO�VROXWLRQYHFWRU Y (1) = (y(1), z(1)) DV�D�OLVW�

([HUFLVH����� &KHFN�WKH�QXPHULFDO�VROXWLRQV y(1) = 9.729 . . . DQGz(1) = 5.048 . . . RI�WKH�V\VWHP�RI�GLIIHUHQWLDO�HTXDWLRQV

y′(x) = y(x) + 2 z(x) , z′(x) = y(x)

FRPSXWHG�DERYH�E\�VXEVWLWXWLQJ�WKH�LQLWLDO�YDOXHV y(0) = 1� z(0) = 1 LQ�WKH�JHQHUDOV\PEROLF�VROXWLRQ� GHWHUPLQLQJ�WKH�YDOXHV�IRU�WKH�IUHH�FRQVWDQWV� DQG�HYDOXDWLQJWKH�V\PEROLF�VROXWLRQ�DW x = 1�

����

Page 200: Mupad Tutorial

1 Lheobg`�>jnZmbhgl3 VROYH

([HUFLVH�����

�� &RPSXWH�WKH�JHQHUDO�VROXWLRQ y(x) RI�WKH�GLIIHUHQWLDO�HTXDWLRQ y′ = y2/x .

�� 'HWHUPLQH�WKH�VROXWLRQ y(x) IRU�HDFK�RI�WKH�IROORZLQJ�LQLWLDO�YDOXH�SUREOHPV�

a) y′ − y VLQ(x) = 0 , y′(1) = 1,

b) 2 y′ +y

x= 0 , y′(1) = π .

�� )LQG�WKH�JHQHUDO�VROXWLRQ�RI�WKH�IROORZLQJ�V\VWHP�RI�RUGLQDU\�GLIIHUHQWLDOHTXDWLRQV�LQ x(t), y(t), z(t)�

x′ = y z, y′ = x z, z′ = t z.

����

Page 201: Mupad Tutorial

K^\nkk^g\^�>jnZmbhgl

J][mjj]f[]�=imYlagfk

5HFXUUHQFH�HTXDWLRQV�DUH�HTXDWLRQV�IRU�IXQFWLRQV�GHSHQGLQJ�RQ�D�GLVFUHWHSDUDPHWHU��DQ�³LQGH[´�� <RX�FDQ�JHQHUDWH�VXFK�DQ�REMHFW�ZLWK�WKH�IXQFWLRQ UHF�ZKRVH�DUJXPHQWV�DUH�DQ�HTXDWLRQ� WKH�IXQFWLRQ�WR�EH�GHWHUPLQHG�DQG� RSWLRQDOO\�D�VHW�RI�LQLWLDO�FRQGLWLRQV�

HTXDWLRQ � UHF�\�Q � �� \�Q � �� � � \�Q�� \�Q���VROYH�HTXDWLRQ�{(−1)n &�+ 2n &�}

7KH�JHQHUDO�VROXWLRQ�FRQWDLQV�WZR�DUELWUDU\�FRQVWDQWV��C1, C2 LQ�WKLV�FDVH�� ZKLFKDUH�VXLWDEO\�DGMXVWHG�ZKHQ�\RX�VSHFLI\�LQLWLDO�FRQGLWLRQV�

VROYH�UHF�\�Q � �� � \�Q� � \�Q � ��� \�Q��^\��� �`��

{2n &�− (−1)n (&�− 1)}

VROYH�UHF�\�Q � �� � \�Q� � \�Q � ��� \�Q��^\��� �� \��� �`��

{(−1)n

3+

2 2n

3

}

([HUFLVH����� 7KH�)LERQDFFL�QXPEHUV�DUH�GHILQHG E\�WKH�UHFXUUHQFHFn = Fn−1 + Fn−2 ZLWK�WKH�LQLWLDO�YDOXHV F0 = 0, F1 = 1� 8VH VROYH WR�ILQG�DQH[SOLFLW�UHSUHVHQWDWLRQ�IRU Fn�

����

Page 202: Mupad Tutorial
Page 203: Mupad Tutorial

1

0DQLSXODWLQJ�([SUHVVLRQV

:KHQ�HYDOXDWLQJ�REMHFWV� 0X3$'� DXWRPDWLFDOO\�SHUIRUPV�D�YDULHW\�RIVLPSOLILFDWLRQV� )RU�H[DPSOH� DULWKPHWLF�RSHUDWLRQV�EHWZHHQ�LQWHJHUV�DUH�H[HFXWHGRU H[S(OQ(x)) LV�VLPSOLILHG�WR x� 2WKHU�VLPSOLILFDWLRQV�VXFK�DVVLQ(x)2 + FRV(x)2 = 1� OQ(H[S(x)) = x� (x2 − 1)/(x− 1) = x+ 1� RU

√x2 = x GR�QRW

KDSSHQ�DXWRPDWLFDOO\� 2QH�UHDVRQ�LV�WKDW�PDQ\�RI�WKHVH�UXOHV�DUH�QRW�XQLYHUVDOO\YDOLG� IRU�H[DPSOH�

√x2 = x LV�ZURQJ�IRU x = −2� 2WKHU�VLPSOLILFDWLRQV�VXFK�DV

VLQ(x)2 + FRV(x)2 = 1 DUH�YDOLG�XQLYHUVDOO\� EXW�WKHUH�ZRXOG�EH�D�VLJQLILFDQW�ORVV�RIHIILFLHQF\�LI�0X3$' ZRXOG�DOZD\V�VFDQ�H[SUHVVLRQV�IRU�WKH�RFFXUUHQFH�RI VLQ DQGFRV WHUPV�

0RUHRYHU� LW�LV�QRW�FOHDU�LQ�JHQHUDO�ZKLFK�RI�VHYHUDO�PDWKHPDWLFDOO\�HTXLYDOHQWUHSUHVHQWDWLRQV�LV�WKH�PRVW�DSSURSULDWH� )RU�H[DPSOH� LW�PLJKW�EH�UHDVRQDEOH�WRUHSODFH�DQ�H[SUHVVLRQ�VXFK�DV VLQ(x) E\�LWV�FRPSOH[�H[SRQHQWLDO�UHSUHVHQWDWLRQ

VLQ(x) = − L2H[S(x L) +

L2H[S(−x L).

,Q�VXFK�D�VLWXDWLRQ� \RX�FDQ�FRQWURO�WKH�PDQLSXODWLRQ�DQG�VLPSOLILFDWLRQ�RIH[SUHVVLRQV�E\�H[SOLFLWO\�DSSO\LQJ�DSSURSULDWH�V\VWHP�IXQFWLRQV� 0X3$' SURYLGHVWKH�IROORZLQJ�IXQFWLRQV� ZKLFK�ZH�KDYH�SDUWO\�GLVFXVVHG�LQ�6HFWLRQ ����

Page 204: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

FROOHFW � FROOHFWLQJ�FRHIILFLHQWVFRPELQH � FRPELQLQJ�VXEH[SUHVVLRQVH[SDQG � H[SDQVLRQIDFWRU � IDFWRUL]DWLRQQRUPDO � QRUPDOL]DWLRQ�RI�UDWLRQDO�H[SUHVVLRQVSDUWIUDF � SDUWLDO�IUDFWLRQ�GHFRPSRVLWLRQUDGVLPS � VLPSOLILFDWLRQ�RI�UDGLFDOVUHFWIRUP � &DUWHVLDQ�UHSUHVHQWDWLRQ�RI�FRPSOH[�YDOXHVUHZULWH � DSSO\LQJ�PDWKHPDWLFDO�LGHQWLWLHVVLPSOLI\ � XQLYHUVDO�VLPSOLILHU6LPSOLI\ � XQLYHUVDO�VLPSOLILHU

���

Page 205: Mupad Tutorial

MkZgl_hkfbg`�>qik^llbhgl

LjYfk^gjeaf_�=phj]kkagfk

,I�\RX�HQWHU�WKH�FRPPDQG FROOHFW�H[SUHVVLRQ�XQNQRZQ�� WKH�V\VWHP�UHJDUGVWKH�H[SUHVVLRQ�DV�D�SRO\QRPLDO�LQ�WKH�VSHFLILHG�XQNQRZQ�DQG�JURXSV�WKHFRHIILFLHQWV�RI�HTXDO�SRZHUV�

[A� � D [ � VTUW��� [ � E [A� � VLQ�[� � D VLQ�[��FROOHFW��� [�

(b+ 1) x2 +(a+√

2)x+ (VLQ(x) + a VLQ(x))

<RX�FDQ�VSHFLI\�VHYHUDO�³XQNQRZQV�´ ZKLFK�PD\�WKHPVHOYHV�EH�H[SUHVVLRQV� DV�DOLVW�

FROOHFW��� >[� VLQ�[�@�

(b+ 1) x2 +(a+√

2)x+ (a+ 1) VLQ(x)

7KH�IXQFWLRQ FRPELQH�H[SUHVVLRQ�RSWLRQ� FRPELQHV�VXEH[SUHVVLRQV�XVLQJPDWKHPDWLFDO�LGHQWLWLHV�EHWZHHQ�IXQFWLRQV�LQGLFDWHG�E\ RSWLRQ� 7KH�LPSOHPHQWHGRSWLRQV�DUH DUFWDQ� H[S� OQ� ORJ� VLQFRV� VLQKFRVK� DQG JDPPD�

%\�GHIDXOW� FRPELQH RQO\�HPSOR\V�WKH�LGHQWLWLHV ab ac = ab+c� ac bc = (a b)c� DQG(ab)c = abc IRU�SRZHUV� ZKHUH�WKH\�DUH�YDOLG��

I � [A�Q � �� [A3,�[A�� I FRPELQH�I�xπ xn+1

x2= xπ+n−1

I � DA[ �A\��A[��A\� I FRPELQH�I�3y ax

2x 9y=(

13

)y (a2

)x

FRPELQH�VTUW��� VTUW��� VTUW�[��√

42x

�$Q�H[DPSOH�ZKHUH�WKH\�DUH�QRW�LV ((−1)2)1/2 ̸= −1�

���

Page 206: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

I � �3,A������A[� I FRPELQH�I�√πx = π

x2

,I |xy| < 1� WKH�LQYHUVH DUFWDQ RI�WKH�WDQJHQW�IXQFWLRQ�VDWLVILHV�WKH�IROORZLQJLGHQWLW\�

I � DUFWDQ�[� � DUFWDQ�\��I FRPELQH�I� DUFWDQ� DVVXPLQJ � � [ \ � �

DUFWDQ(x) + DUFWDQ(y) = − DUFWDQ(x+ yx y − 1

)

)RU�WKH�H[SRQHQWLDO�IXQFWLRQ� ZH�KDYH H[S(x) H[S(y) = H[S(x+ y)�

FRPELQH�H[S�[� H[S�\�A��H[S��]�� H[S�Hx+2 y+z

1RWH� KRZHYHU� WKDW�WKH�LGHQWLW\ H[S(x)y = H[S(x y) NQRZQ�WR�KROG�IRU�UHDO�YDOXHVRI x GRHV�QRW�KROG�WKURXJKRXW�WKH�FRPSOH[�SODQH� &RQVHTXHQWO\� ZLWKRXWDGGLWLRQDO�DVVXPSWLRQV�RQ x� 0X3$'� GRHV�QRW�FRPELQH�VXFK�WHUPV�

FRPELQH�H[S�[�A\� H[S�(Hx)y

FRPELQH�H[S�[�A\� H[S� DVVXPLQJ [ LQ 5BHx y

:LWK�FHUWDLQ�DVVXPSWLRQV�DERXW x DQG y� WKH�ORJDULWKP�VDWLVILHV�WKH�UXOHVOQ(x) + OQ(y) = OQ(x y) DQG x OQ(y) = OQ(yx)�

FRPELQH�OQ�[� � OQ��� � � OQ������ OQ�

OQ

(27x4

)

���

Page 207: Mupad Tutorial

MkZgl_hkfbg`�>qik^llbhgl

7KH�WULJRQRPHWULF�IXQFWLRQV�VDWLVI\�D�YDULHW\�RI�LGHQWLWLHV�WKDW�WKH�V\VWHP�HPSOR\VWR�FRPELQH�SURGXFWV�

FRPELQH�VLQ�[� FRV�\�� VLQFRV��FRPELQH�VLQ�[�A�� VLQFRV�

VLQ(x− y)2

+VLQ(x+ y)

2,

12− FRV(2x)

2

6LPLODU�UXOHV�DUH�DSSOLHG�WR�WKH�K\SHUEROLF�IXQFWLRQV�

FRPELQH�VLQK�[� FRVK�\�� VLQKFRVK��FRPELQH�VLQK�[�A�� VLQKFRVK�

VLQK(x− y)2

+VLQK(x+ y)

2,FRVK(2x)

2− 1

2

7KH�IXQFWLRQ H[SDQG DSSOLHV�WKH�LGHQWLWLHV XVHG�E\ FRPELQH LQ�WKH�UHYHUVHGLUHFWLRQ� LW�WUDQVIRUPV�VSHFLDO�IXQFWLRQ�FDOOV�ZLWK�FRPSRVLWH�DUJXPHQWV�WR�VXPVRU�SURGXFWV�RI�IXQFWLRQ�FDOOV�ZLWK�VLPSOHU�DUJXPHQWV�YLD�³DGGLWLRQ�WKHRUHPV�´

H[SDQG�[A�\ � ]��� H[SDQG�H[S�[ � \ � ] � ����H[SDQG�OQ�� 3, [ \��

xy xz,H4 Hx Hy

Hz, OQ(2) + OQ(π) + OQ(x y)

H[SDQG�VLQ�[ � \��� H[SDQG�FRVK�[ � \��FRV(x) VLQ(y) + FRV(y) VLQ(x) , FRVK(x) FRVK(y) + VLQK(x) VLQK(y)

H[SDQG�VTUW��� [ \��√

42√x y

+HUH� WKH�V\VWHP�GRHV�QRW�SHUIRUP�FHUWDLQ�³H[SDQVLRQV´�VXFK�DVOQ(x y) = OQ(x) + OQ(y)� VLQFH�VXFK�DQ�LGHQWLW\�KROGV�RQO\�XQGHU�DGGLWLRQDODVVXPSWLRQV��IRU�H[DPSOH� IRU�SRVLWLYH�UHDO x DQG y��

7KH�PRVW�IUHTXHQW�XVH�RI H[SDQG LV�IRU�WUDQVIRUPLQJ�D�SURGXFW�RI�VXPV�LQWR�D�VXPRI�SURGXFWV�

H[SDQG��[ � \�A� �[ � \�A��x4 − 2x2 y2 + y4

���

Page 208: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

7KH�IXQFWLRQ H[SDQG ZRUNV�UHFXUVLYHO\�IRU�DOO�VXEH[SUHVVLRQV�

H[SDQG��[ � \� �[ � \� VLQ�H[S�[ � \ � ]���x2 VLQ(Hx Hy Hz)− y2 VLQ(Hx Hy Hz)

<RX�FDQ�VXSSO\�H[SUHVVLRQV�DV�DGGLWLRQDO�DUJXPHQWV�WR H[SDQG� 7KHVHVXEH[SUHVVLRQV�DUH QRW H[SDQGHG�

H[SDQG��[ � \� �[ � \� VLQ�H[S�[ � \ � ]���[ � \� [ � \ � ]�

x VLQ(Hx+y+z

)(x− y) + y VLQ

(Hx+y+z

)(x− y)

7KH�IXQFWLRQ IDFWRU IDFWRUV�SRO\QRPLDOV DQG�H[SUHVVLRQV�

IDFWRU�[A� � � [A� � � [ � ��

(x+ 1)3

+HUH�WKH�V\VWHP�IDFWRUV�³RYHU�WKH�UDWLRQDO�QXPEHUV�´ LW�ORRNV�IRU�SRO\QRPLDOIDFWRUV�ZLWK�UDWLRQDO�QXPEHU�FRHIILFLHQWV� ,Q�HIIHFW� 0X3$' GRHV�QRW�UHWXUQ�WKHIDFWRUL]DWLRQ x2 − 2 = (x−

√2) (x+

√2)�

IDFWRU�[A� � ��(x2 − 2

)

7R�H[WHQG�WKH�ULQJ�RI�FRQVWDQWV�RYHU�ZKLFK�WR�IDFWRU� \RX�FDQ�H[SOLFLWO\�DGMRLQFRQVWDQWV�WR�WKH�ILHOG�RI�UDWLRQDO�QXPEHUV�

IDFWRU�[A� � �� $GMRLQ VTUW����(x−√

2)·(x+√

2)

)RU�VXPV�RI�UDWLRQDO�H[SUHVVLRQV� IDFWRU ILUVW�FRPSXWHV�D�FRPPRQ�GHQRPLQDWRUDQG�WKHQ�IDFWRUV�ERWK�WKH�QXPHUDWRU�DQG�WKH�GHQRPLQDWRU�

I � �[A� � � \A����[A� � \A�� � �� I IDFWRU�I�x3 + 3 y2

x2 − y2 + 3 =x2 · (x+ 3)

(x− y) · (x+ y)

0X3$' FDQ�IDFWRU�QRW�RQO\�SRO\QRPLDOV�DQG�UDWLRQDO�IXQFWLRQV� )RU�PRUH�JHQHUDOH[SUHVVLRQV� WKH�V\VWHP�LQWHUQDOO\�UHSODFHV�VXEH[SUHVVLRQV�VXFK�DV�V\PEROLF

���

Page 209: Mupad Tutorial

MkZgl_hkfbg`�>qik^llbhgl

IXQFWLRQ�FDOOV�E\�LGHQWLILHUV� IDFWRUV�WKH�FRUUHVSRQGLQJ�SRO\QRPLDO�RU�UDWLRQDOIXQFWLRQ� DQG�UH�VXEVWLWXWHV�WKH�WHPSRUDU\�LGHQWLILHUV�

IDFWRU��H[S�[�A� � ����FRV�[�A� � VLQ�[�A���(H2 x − 1

)

(FRV(x)− VLQ(x)) · (FRV(x) + VLQ(x))

7KH�IXQFWLRQ QRUPDO FRPSXWHV�D�³QRUPDO�IRUP´�IRU�UDWLRQDO�H[SUHVVLRQV� /LNHIDFWRU� LW�ILUVW�FRPSXWHV�D�FRPPRQ�GHQRPLQDWRU�IRU�VXPV�RI�UDWLRQDO�H[SUHVVLRQV�EXW�LW�WKHQ�H[SDQGV�QXPHUDWRU�DQG�GHQRPLQDWRU�LQVWHDG�RI�IDFWRULQJ�WKHP�

I � ��[ � ��A� � �����[ � ����[ � �� � ��I� IDFWRU�I�� QRUPDO�I�

(x+ 6)2 − 17(x− 1) (x+ 1)

+ 1,2 · (x+ 3)2

(x− 1) · (x+ 1),

2x2 + 12x+ 18x2 − 1

1HYHUWKHOHVV� QRUPDO FDQFHOV�FRPPRQ�IDFWRUV�LQ�QXPHUDWRU�DQG�GHQRPLQDWRU�I � [A���[ � \� � \A���[ � \�� I QRUPDO�I�

x2

x+ y− y2

x+ y= x− y

/LNH IDFWRU� QRUPDO FDQ�KDQGOH�DUELWUDU\�H[SUHVVLRQV�

I � �H[S�[�A��H[S�\�A����H[S�[�A� � H[S�\�A���I QRUPDO�I�

H2 x − H2 y

H3 x − H3 y =Hx + Hy

H2 x + H2 y + Hx Hy

7KH�IXQFWLRQ SDUWIUDF GHFRPSRVHV�D�UDWLRQDO�H[SUHVVLRQ�LQWR�D�SRO\QRPLDO�SDUWSOXV�D�VXP�RI�UDWLRQDO�WHUPV�LQ�ZKLFK�WKH�GHJUHH�RI�WKH�QXPHUDWRU�LV�VPDOOHU�WKDQWKH�GHJUHH�RI�WKH�FRUUHVSRQGLQJ�GHQRPLQDWRU��SDUWLDO�IUDFWLRQ�GHFRPSRVLWLRQ��

I � [A���[A� � ��� I SDUWIUDF�I� [�x2

x2 − 1=

12 (x− 1)

− 12 (x+ 1)

+ 1

7KH�GHQRPLQDWRUV�RI�WKH�WHUPV�DUH�WKH�IDFWRUV�WKDW�0X3$' ILQGV�ZKHQ�IDFWRULQJ

���

Page 210: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

WKH�FRPPRQ�GHQRPLQDWRU�

GHQRPLQDWRU � [A� � [A� � � [A� � �� [A� � � [ � ���IDFWRU�GHQRPLQDWRU�

(x− 3) ·(x2 + 1

)· (x+ 2)2

SDUWIUDF���GHQRPLQDWRU� [�9 x250 −

13250

x2 + 1− 1

25 (x+ 2)− 1

25 (x+ 2)2+

1250 (x− 3)

$QRWKHU�IXQFWLRQ�IRU�PDQLSXODWLQJ�H[SUHVVLRQV�LV UHZULWH� ,W�HPSOR\V�LGHQWLWLHVWR�HOLPLQDWH�FHUWDLQ�IXQFWLRQV�FRPSOHWHO\�IURP�DQ�H[SUHVVLRQ�DQG�UHSODFHV�WKHPE\�RWKHU�IXQFWLRQV� )RU�H[DPSOH� \RX�FDQ�DOZD\V�H[SUHVV VLQ DQG FRV E\ WDQ ZLWKWKH�KDOI�DUJXPHQW�

VLQ(x) =2 WDQ(x/2)

1 + WDQ(x/2)2, FRV(x) =

1− WDQ(x/2)2

1 + WDQ(x/2)2.

7KH�WULJRQRPHWULF�IXQFWLRQV�DUH�DOVR�UHODWHG�WR�WKH�FRPSOH[�H[SRQHQWLDO�IXQFWLRQ�

VLQ(x) = − L2H[S(Lx) +

L2H[S(−Lx) ,

FRV(x) =12H[S(Lx) +

12H[S(−Lx) .

<RX�FDQ�H[SUHVV�WKH�K\SHUEROLF�IXQFWLRQV�DQG�WKHLU�LQYHUVH�IXQFWLRQV�LQ�WHUPV�RIWKH�H[SRQHQWLDO�IXQFWLRQ�DQG�WKH�ORJDULWKP�

VLQK(x) =H[S(x)− H[S(−x)

2, FRVK(x) =

H[S(x) + H[S(−x)2

,

DUFVLQK(x) = OQ(x+√x2 + 1) , DUFFRVK(x) = OQ(x+

√x2 − 1) .

$ FDOO�RI�WKH�IRUP UHZULWH�H[SUHVVLRQ�WDUJHW� HPSOR\V�WKHVH�LGHQWLWLHV�7DEOH ��� OLVWV�WKH�UXOHV�LPSOHPHQWHG�LQ�0X3$'�

UHZULWH�'�'�X���[�� GLII�∂2

∂x2u(x)

���

Page 211: Mupad Tutorial

MkZgl_hkfbg`�>qik^llbhgl

WDUJHW : IXQFWLRQ�V� → UHZULWWHQLQ�WHUPV�RI

DQGRU : ORJLFDO�RSHUDWRUV [RU� !� � ! → DQG� RU� QRWDUFFRV : LQYHUVH�WULJ��IXQFWLRQV → DUFFRVDUFFRVK : LQYHUVH�WULJ��IXQFWLRQV� OQ → DUFFRVKDUFFRW : LQYHUVH�WULJ��IXQFWLRQV → DUFFRWDUFFRWK : LQYHUVH�WULJ��IXQFWLRQV� OQ → DUFFRWKDUFVLQ : LQYHUVH�WULJ��IXQFWLRQV → DUFVLQDUFVLQK : LQYHUVH�WULJ��IXQFWLRQV� OQ → DUFVLQKDUFWDQ : LQYHUVH�WULJ��IXQFWLRQV → DUFWDQDUFWDQK : LQYHUVH�WULJ��IXQFWLRQV� OQ → DUFWDQKEHUQRXOOL : HXOHU → EHUQRXOOLFRV : H[SRQHQWLDO�IXQFWLRQ H[S�

WULJ��DQG�K\SHUEROLF�IXQFWLRQV→ FRV

FRVK : H[SRQHQWLDO�IXQFWLRQ H[S�WULJ��DQG�K\SHUEROLF�IXQFWLRQV

→ FRVK

FRW : H[SRQHQWLDO�IXQFWLRQ H[S�WULJ��DQG�K\SHUEROLF�IXQFWLRQV

→ FRW

FRWK : H[SRQHQWLDO�IXQFWLRQ H[S�WULJ��DQG�K\SHUEROLF�IXQFWLRQV

→ FRWK

GLII : GLIIHUHQWLDO�RSHUDWRU ' → GLII' : GLIIHUHQWLDWLQJ�IXQFWLRQ GLII → 'H[S : SRZHUV��A�� WULJ��DQG�K\SHUEROLF

IXQFWLRQV�DQG�WKHLU�LQYHUVHV�SRODU�DQJOH DUJ� GDZVRQ

→ H[S� OQ

HUI : GDZVRQ → HUIIDFW : Γ�IXQFWLRQ JDPPD�

GRXEOH�IDFWRULDO IDFW��ELQRPLDO�FRHIILFLHQWV ELQRPLDO�β�IXQFWLRQ EHWD� SRFKKDPPHU

→ IDFW

JDPPD : IDFWRULDO IDFW�GRXEOH�IDFWRULDO IDFW��ELQRPLDO�FRHIILFLHQWV ELQRPLDO�β�IXQFWLRQ EHWD� SRFKKDPPHU

→ JDPPD

7DEOH ���D� 7DUJHWV�RI UHZULWH

���

Page 212: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

WDUJHW : IXQFWLRQ�V� → UHZULWWHQLQ�WHUPV�RI

KDUPRQLF : SVL → KDUPRQLFKHDYLVLGH : VLJQ → KHDYLVLGH,P : 5H → ,POQ : LQYHUVH�WULJ��DQG�LQYHUVH�K\SHUEROLF

IXQFWLRQV� SRODU�DQJOH DUJ� ORJ→ OQ

ODPEHUW: : ZULJKW2PHJD → ODPEHUW:PD[ : PLQ� DEV → PD[PLQ : PD[� DEV → PLQSLHFHZLVH : VLJQ� DEVROXWH�YDOXH DEV�

VWHS�IXQFWLRQ KHDYLVLGH�PD[LPXP PD[� PLQLPXP PLQ�NURQHFNHU'HOWD

→ SLHFHZLVH

SVL : KDUPRQLF → SVL5H : ,P → 5HVLJQ : VWHS�IXQFWLRQ KHDYLVLGH�

DEVROXWH�YDOXH DEV→ VLJQ

VLQ : H[SRQHQWLDO�IXQFWLRQ H[S�WULJ��DQG�K\SHUEROLF�IXQFWLRQV

→ VLQ

VLQFRV : H[SRQHQWLDO�IXQFWLRQ H[S�WULJ��DQG�K\SHUEROLF�IXQFWLRQV

→ VLQ� FRV

VLQK : H[SRQHQWLDO�IXQFWLRQ H[S�WULJ��DQG�K\SHUEROLF�IXQFWLRQV

→ VLQK

VLQKFRVK : H[SRQHQWLDO�IXQFWLRQ H[S�WULJ��DQG�K\SHUEROLF�IXQFWLRQV

→ VLQK� FRVK

WDQ : H[SRQHQWLDO�IXQFWLRQ H[S�WULJ��DQG�K\SHUEROLF�IXQFWLRQV

→ WDQ

WDQK : H[SRQHQWLDO�IXQFWLRQ H[S�WULJ��DQG�K\SHUEROLF�IXQFWLRQV

→ WDQK

7DEOH ���E� 7DUJHWV�RI UHZULWH �FRQW��

����

Page 213: Mupad Tutorial

MkZgl_hkfbg`�>qik^llbhgl

UHZULWH�VLQ�[��FRV�[�� H[S� UHZULWH�WDQ�[�� H[S�H−x L L

2 − Hx L L2

H−x L2 + Hx L

2

= −Hx 2 L L− LHx 2 L + 1

UHZULWH�DUFVLQK�[� � DUFFRVK�[�� OQ�

OQ(x+

√x2 + 1

)− OQ

(x+

√x2 − 1

)

)RU�H[SUHVVLRQV�UHSUHVHQWLQJ�FRPSOH[ QXPEHUV� \RX�FDQ�HDVLO\�FRPSXWH�UHDO�DQGLPDJLQDU\�SDUWV�E\�XVLQJ 5H DQG ,P�

] � � � � ,� 5H�]�� ,P�]�2, 3

] � VLQ�� ,� � OQ����� 5H�]�� ,P�]�0, VLQK(2)− π

:KHQ�DQ�H[SUHVVLRQ�FRQWDLQV�V\PEROLF�LGHQWLILHUV� 0X3$' DVVXPHV�DOO�VXFKXQNQRZQV�WR�EH�FRPSOH[�YDOXHV� 5H DQG ,P DUH�UHWXUQHG�V\PEROLFDOO\�

5H�D E � ,�� ,P�D E � ,�ℜ(a b) , ℑ(a b) + 1

,Q�VXFK�D�FDVH� \RX�FDQ�XVH�WKH�IXQFWLRQ UHFWIRUP �VKRUW�IRU� UHFWDQJXODU�IRUP��WRGHFRPSRVH�WKH�H[SUHVVLRQ�LQWR�UHDO�DQG�LPDJLQDU\�SDUW� 7KH�QDPH�RI�WKLV�IXQFWLRQLV�GHULYHG�IURP�WKH�IDFW�WKDW�LW�FRPSXWHV�WKH�FRRUGLQDWHV�RI�WKH�XVXDO�UHFWDQJXODU�&DUWHVLDQ��FRRUGLQDWH�V\VWHP� 0X3$' GHFRPSRVHV�WKH�V\PEROV�FRQWDLQHG�LQ�WKHH[SUHVVLRQ�LQWR�WKHLU�UHDO�DQG�LPDJLQDU\�SDUWV�DQG�H[SUHVVHV�WKH�ILQDO�UHVXOWDFFRUGLQJO\�

UHFWIRUP�D E � ,�ℜ(a) ℜ(b)−ℑ(a) ℑ(b) + ℑ(a) ℜ(b) L+ ℑ(b) ℜ(a) L+ L

UHFWIRUP�H[S�[��

FRV(ℑ(x)) Hℜ(x) + VLQ(ℑ(x)) Hℜ(x) L

����

Page 214: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

$JDLQ� \RX�FDQ�H[WUDFW�WKH�UHDO�DQG�LPDJLQDU\�SDUWV�RI�WKH�UHVXOW�ZLWK 5H DQG ,P�UHVSHFWLYHO\�

5H���� ,P���

FRV(ℑ(x)) Hℜ(x), VLQ(ℑ(x)) Hℜ(x)

$V�D�EDVLF�SULQFLSOH� UHFWIRUP UHJDUGV�DOO�V\PEROLF�LGHQWLILHUV�DV�UHSUHVHQWDWLYHVRI�FRPSOH[�QXPEHUV� +RZHYHU� \RX�FDQ�XVH DVVXPH �6HFWLRQ �����WR�VSHFLI\�WKDW�DQLGHQWLILHU�UHSUHVHQWV�RQO\�UHDO�QXPEHUV�

DVVXPH�D LQ 5B��] � UHFWIRUP�D E � ,�aℜ(b) + aℑ(b) L+ L

7KH�UHVXOWV�RI UHFWIRUP KDYH�D�VSHFLDO�GDWD�W\SH�

GRPW\SH�]�UHFWIRUP

<RX�FDQ�XVH�WKH�IXQFWLRQ H[SU WR�FRQYHUW�VXFK�DQ�REMHFW�WR�D�³QRUPDO´�H[SUHVVLRQRI�GRPDLQ�W\SH '20B(;35�

H[SU�]�aℑ(b) L+ aℜ(b) + L

����

Page 215: Mupad Tutorial

Lbfieb_rbg`�>qik^llbhgl

Kaehda^qaf_�=phj]kkagfk

,Q�VRPH�FDVHV� D�WUDQVIRUPDWLRQ�OHDGV�WR�D�VLPSOHU�H[SUHVVLRQ�

I � �A[ �A[��A[��A[� I FRPELQH�I�2x 3x

8x 9x=(

112

)x

I � [��[ � \� � \��[ � \�� I QRUPDO�I�x

x+ y+y

x+ y= 1

7R�WKLV�HQG� KRZHYHU� \RX�PXVW�LQVSHFW�WKH�H[SUHVVLRQ�DQG�GHFLGH�\RXUVHOI�ZKLFKIXQFWLRQ�WR�XVH�IRU�VLPSOLILFDWLRQ� 7KHUH�DUH�WRROV�IRU�DSSO\LQJ�YDULRXVVLPSOLILFDWLRQ�DOJRULWKPV�WR�DQ�H[SUHVVLRQ DXWRPDWLFDOO\� WKH�IXQFWLRQV VLPSOLI\DQG 6LPSOLI\� 7KHVH�DUH�XQLYHUVDO�VLPSOLILHUV�ZKLFK�0X3$'� XVHV�WR�DFKLHYH�DUHSUHVHQWDWLRQ�RI�DQ�H[SUHVVLRQ�WKDW�LV�DV�³VLPSOH´�DV�SRVVLEOH�

I � �A[ �A[��A[��A[� I VLPSOLI\�I�2x 3x

8x 9x=(

112

)x

I � �� � �VLQ�[�A� � FRV�[�A��A���VLQ�[��I VLPSOLI\�I�

(FRV(x)2 + VLQ(x)2

)2+ 1

VLQ(x)=

2VLQ(x)

I � [��[ � \� � \��[ � \� � VLQ�[�A� � FRV�[�A��I VLPSOLI\�I�

x

x+ y− VLQ(x)2 − FRV(x)2 +

y

x+ y= 0

I � �H[S�[� � ����H[S�[��� � ��� I VLPSOLI\�I�Hx − 1Hx2 + 1

= Hx2 − 1

����

Page 216: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

I � VTUW����� � ����A��A������ I VLPSOLI\�I�√

997− 99102697316 = 0

1RWH�WKDW VLPSOLI\ RSHUDWHV�LQ�D�SXUHO\�KHXULVWLF�ZD\�VLQFH�WKHUH�LV�QR�JHQHUDODQVZHU�ZKDW�³VLPSOH´�PHDQV� <RX�FDQ�FRQWURO�WKH�VLPSOLILFDWLRQ�SURFHVV�E\VXSSO\LQJ�DGGLWLRQDO�DUJXPHQWV� $V�LQ�WKH�FDVH�RI FRPELQH� \RX�FDQ�UHTXHVWSDUWLFXODU�VLPSOLILFDWLRQV�E\�PHDQV�RI�RSWLRQV� )RU�H[DPSOH� \RX�FDQ�WHOO�WKHVLPSOLILHU�H[SOLFLWO\�WR�VLPSOLI\�H[SUHVVLRQV�FRQWDLQLQJ�VTXDUH�URRWV�

I � VTUW�� � � VTUW�����I VLPSOLI\�I� VTUW�√

2√√

3 + 2 =√

3 + 1

7KH�SRVVLEOH�RSWLRQV�DUH H[S� OQ� ORJ� FRV� VLQ� FRVK� VLQK� VTUW� JDPPD� XQLW�FRQGLWLRQ� ORJLF� DQG UHODWLRQ� ,QWHUQDOO\� VLPSOLI\ WKHQ�FRQILQHV�LWVHOI�WRWKRVH�VLPSOLILFDWLRQ�UXOHV�WKDW�DUH�YDOLG�IRU�WKH�IXQFWLRQ�JLYHQ�DV�RSWLRQ� 7KHRSWLRQV ORJLF DQG UHODWLRQ DUH�IRU�VLPSOLI\LQJ�ORJLFDO�H[SUHVVLRQV�DQG�HTXDWLRQVDQG�LQHTXDOLWLHV� UHVSHFWLYHO\��VHH�DOVR�WKH�FRUUHVSRQGLQJ�KHOS�SDJH� "VLPSOLI\��

,QVWHDG�RI VLPSOLI\�H[SUHVVLRQ�VTUW�� \RX�PD\�DOVR�XVH�WKH�IXQFWLRQ UDGVLPSWR�VLPSOLI\�QXPHULFDO�H[SUHVVLRQV�FRQWDLQLQJ�VTXDUH�URRWV�RU�RWKHU�UDGLFDOV�

I UDGVLPS�I�√

2√√

3 + 2 =√

3 + 1

I � �A����� � � �A����� � VTUW�� � � �A�������I UDGVLPS�I�

2 214 −√

2√

3√

2 + 4 + 234 = 0

,Q�PDQ\�FDVHV� XVLQJ VLPSOLI\ ZLWKRXW�RSWLRQV�LV�DSSURSULDWH� +RZHYHU� VXFK�DFDOO�LV�RIWHQ�YHU\�WLPH�FRQVXPLQJ� VLQFH�WKH�VLPSOLILFDWLRQ�DOJRULWKP�LV�TXLWHFRPSOH[� ,W�PD\�EH�XVHIXO�WR�VSHFLI\�DGGLWLRQDO�RSWLRQV�WR�VDYH�FRPSXWLQJ�WLPH�VLQFH�WKHQ�VLPSOLILFDWLRQV�DUH�SHUIRUPHG�RQO\�IRU�VSHFLDO�IXQFWLRQV�

����

Page 217: Mupad Tutorial

Lbfieb_rbg`�>qik^llbhgl

7KH�VHFRQG�JHQHUDO�VLPSOLILHU� 6LPSOLI\ �QRWH�WKH�FDSLWDO 6���LV�RIWHQ�VORZHU�WKDQVLPSOLI\� EXW�PXFK�PRUH�SRZHUIXO�DQG�DOORZV�PXFK�ILQHU�WXQLQJ�

6LPSOLI\����[�\� � ���[�\���6LPSOLI\����[�\� � ���[�\�� 9DOXDWLRQ OHQJWK�

2xx2 − y2 ,

1x− y +

1x+ y

I � JDPPD�Q � Q��JDPPD�Q�����I� VLPSOLI\�I�� 6LPSOLI\�I�� 6LPSOLI\�I� 6WHSV ����

Γ(n+

n!Γ (n+ 1)

), Γ(n! + nΓ (n+ 1)

Γ (n+ 1)

), Γ(n+

n!Γ (n+ 1)

), Γ (n+ 1)

+HUH� ZH�KDYH�XVHG�WKH�RSWLRQ 6WHSV WR�WHOO 6LPSOLI\ KRZ�PDQ\�³HOHPHQWDU\´VWHSV�LW�PD\�WU\�EHIRUH�JLYLQJ�XS�WKH�VHDUFK�IRU�D�VLPSOHU�H[SUHVVLRQ� ,Q�WKH�HQG�DOPRVW�DOO�RI�WKH�VWHSV�KDYH�OHG�QRZKHUH� ZH�FDQ�DVN 6LPSOLI\ IRU�D�OLVW�RI�VWHSV�LWSHUIRUPHG�DQG�VHH�WKDW�RQO\�WKUHH�VWHSV�ZHUH�DFWXDOO\�XVHG�

6LPSOLI\�I� 6WHSV ���� 2XWSXW7\SH �3URRI��,QSXW ZDV JDPPD�Q � ��JDPPD�Q � �� Q���/HPPD� Q � ��JDPPD�Q � �� Q� Q � �,QSXW ZDV Q � ��JDPPD�Q � �� Q��$SSO\LQJ WKH UXOH SURF 6LPSOLI\��H[SDQG�D� ��� HQG�;�?

6WUDWHJ\� WDEOH?��'HIDXOW� ����WR Q � ��JDPPD�Q � �� Q� JLYHV Q � ��Q Q��JDPPD�Q�

/HPPD� ��Q Q��JDPPD�Q� �,QSXW ZDV ��Q Q��JDPPD�Q��$SSO\LQJ WKH UXOH ; �! FRPELQH�;� JDPPD��;�WR ��Q Q��JDPPD�Q� JLYHV ��JDPPD�Q � �� Q�

/HPPD� Q� JDPPD�Q � ��,QSXW ZDV Q��$SSO\LQJ WKH UXOH ; �! JDPPD�RS�;� �� � ���;�WR Q� JLYHV JDPPD�Q � ��

(QG RI OHPPDVXEVWLWXWLQJ JLYHV �(QG RI OHPPDVXEVWLWXWLQJ JLYHV Q � �(QG RI OHPPDVXEVWLWXWLQJ JLYHV JDPPD�Q � ��(1' 2) 3522)

����

Page 218: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

$V�WKLV��DGPLWWHGO\�GLIILFXOW�WR�UHDG��RXWSXW�VXJJHVWV� 6LPSOLI\ ZRUNV�E\�IROORZLQJD UXOH�EDVH� 7KH�GRFXPHQWDWLRQ� DFFHVVLEOH�DW "6LPSOLI\��GHWDLOV� �QRWH�WKHVSDFH�DQG�WKH�SDUHQWKHVHV��� VKRZV�H[DPSOHV�ZLWK�DSSOLFDWLRQ�VSHFLILF�UXOH�VHWV�

$V�ZH�KDYH�VHHQ�DERYH�LQ�WKH�ILUVW�H[DPSOH� WKH�XVHU�FDQ�DOVR�FRQWURO 6LPSOLI\¶VLGHD�RI�ZKDW�LV�³VLPSOH�´ ([HUFLVH ��� VKRZV�DQ�DSSOLFDWLRQ�

([HUFLVH����� ,W�LV�SRVVLEOH�WR�UHZULWH�SURGXFWV�RI�WULJRQRPHWULF�IXQFWLRQV�DVOLQHDU�FRPELQDWLRQV�RI VLQ DQG FRV WHUPV�ZKRVH�DUJXPHQWV�DUH�LQWHJUDO�PXOWLSOHVRI�WKH�RULJLQDO�DUJXPHQWV��)RXULHU�H[SDQVLRQ�� )LQG�FRQVWDQWV a, b, c, d� DQG e VXFKWKDW

FRV(x)2 + VLQ(x) FRV(x)= a+ b VLQ(x) + c FRV(x) + d VLQ(2x) + e FRV(2x)

KROGV�

([HUFLVH����� 8VH�0X3$' WR�SURYH�WKH�IROORZLQJ�LGHQWLWLHV�

��FRV(5x)

VLQ(2x) FRV2(x)= −5 VLQ(x) +

FRV2(x)2 VLQ(x)

+5 VLQ3(x)2 FRV2(x)

,

��VLQ2(x)− e2 x

VLQ2(x) + 2 VLQ(x) ex + e2 x=VLQ(x)− ex

VLQ(x) + ex,

��VLQ(2x)− 5 VLQ(x) FRV(x)

VLQ(x) (1 + WDQ2(x))= −9 FRV(x)

4− 3 FRV(3x)

4,

��

√√√√14 + 3

3 + 2

5− 12√

3− 2√

2 =√

2 + 3 .

([HUFLVH����� 0X3$' FRPSXWHV�WKH�IROORZLQJ�LQWHJUDO�IRU I�

I � VTUW�VLQ�[� � ��� LQW�I� [�

2 (VLQ(x)− 1)√VLQ(x) + 1

FRV(x)

����

Page 219: Mupad Tutorial

Lbfieb_rbg`�>qik^llbhgl

,WV�GHULYDWLYH�LV�QRW�OLWHUDOO\�LGHQWLFDO�WR�WKH�LQWHJUDQG�

GLII��� [�

VLQ(x)− 1√VLQ(x) + 1

+ 2√VLQ(x) + 1 +

2 VLQ(x) (VLQ(x)− 1)√VLQ(x) + 1

FRV(x)2

6LPSOLI\�WKLV�H[SUHVVLRQ�

([HUFLVH����� 8VLQJ�WKH�RSWLRQ 9DOXDWLRQ� ZH�PD\�SDVV�D�IXQFWLRQ�WR 6LPSOLI\WR�GHWHUPLQH�ZKLFK�H[SUHVVLRQV�DUH�³VLPSOH�´ WKH�IXQFWLRQ�LV�FDOOHG�ZLWK�DQH[SUHVVLRQ�DV�LWV�DUJXPHQW�DQG�UHWXUQV�D�QXPEHU� KLJKHU�QXPEHUV�IRU�PRUHFRPSOH[�H[SUHVVLRQV�

/HW�XV�WU\�WR�ZULWH WDQ(x)− FRW(x) ZLWKRXW�WKH�WDQJHQW�DQG�FRWDQJHQW�IXQFWLRQV�7KH�ILUVW� VRPHZKDW�QDLYH�DSSURDFK�LV�WR�XVH�D�YDOXDWLRQ�IXQFWLRQ�WKDW�VLPSO\ORRNV�IRU�WKH�SUHVHQFH�RI WDQ DQG FRW��

QR7DQJHQW � [ �! LI KDV�[� >KROG�WDQ�� KROG�FRW�@�WKHQ ��HOVH � HQGBLI�

6LPSOLI\�WDQ�[� � FRW�[�� 9DOXDWLRQ QR7DQJHQW�

2 VLQ(x)2 − 1

VLQ(x)− 2 VLQ(x2

)2VLQ(x)

1RZ� WKLV�H[SUHVVLRQ�DFWXDOO\�GRHV�QRW�FRQWDLQ WDQ� EXW�LV�KDUGO\�³VLPSOH�´,PSURYH QR7DQJHQW VXFK�WKDW�WKH�FDOO�DERYH�UHWXUQV�D�VLPSOH�H[SUHVVLRQ�ZLWKRXWWDQ DQG FRW� <RX�PLJKW�ZDQW�WR�LQIRUP�\RXUVHOI�DERXW OHQJWK ILUVW�RU�XVH6LPSOLI\��GHIDXOW9DOXDWLRQ LQVWHDG� ZKLFK�LV�WKH�IXQFWLRQ�XVHG�E\ 6LPSOLI\ LIQRWKLQJ�LV�VSHFLILHG�

�6HH�&KDSWHU �� IRU�DQ�H[SODQDWLRQ�RI�³LI�´

����

Page 220: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

9kkmehlagfk�YZgml�EYl`]eYla[Yd�Hjgh]jla]k

0X3$'� SHUIRUPV�WUDQVIRUPDWLRQV�RU�VLPSOLILFDWLRQV�IRU�REMHFWV�FRQWDLQLQJV\PEROLF�LGHQWLILHUV�RQO\�LI�WKH�FRUUHVSRQGLQJ�UXOHV�DSSO\�LQ�WKH�HQWLUH�FRPSOH[SODQH� +RZHYHU� VRPH�IDPLOLDU�UXOHV�IRU�FRPSXWLQJ�ZLWK�UHDO�QXPEHUV�DUH�QRWJHQHUDOO\�YDOLG�IRU�FRPSOH[�QXPEHUV� )RU�H[DPSOH� WKH�VTXDUH�URRW�DQG�WKHORJDULWKP�DUH�EUDQFKHV�RI�PXOWL�YDOXHG�FRPSOH[�IXQFWLRQV� DQG�WKH�0X3$'IXQFWLRQV�LQWHUQDOO\�PDNH�FHUWDLQ�DVVXPSWLRQV�DERXW�WKH�EUDQFK�FXWV�

OQ(ex) = x IRU UHDO x

OQ(xn) = n OQ(x) IRU UHDO x > 0

OQ(x y) = OQ(x) + OQ(y) IRU UHDO x > 0 RU�UHDO y > 0√x2 = |x| IRU UHDO x

ex/2 = (ex)1/2 IRU UHDO x

<RX�FDQ�XVH�WKH�IXQFWLRQ DVVXPH WR�WHOO�WKH�V\VWHP�IXQFWLRQV�VXFK�DV H[SDQG�VLPSOLI\� OLPLW� VROYH� DQG LQW WKDW�WKH\�PD\�PDNH�FHUWDLQ�DVVXPSWLRQV�DERXWWKH�PHDQLQJ�RI�FHUWDLQ�H[SUHVVLRQV� :H�RQO\�GHPRQVWUDWH�VRPH�VLPSOH�H[DPSOHVKHUH� <RX�ILQG�PRUH�LQIRUPDWLRQ�RQ�WKH�FRUUHVSRQGLQJ�KHOS�SDJH� "DVVXPH�

<RX�FDQ�XVH�DQ�H[SUHVVLRQ�VXFK�DV�DQ�LQHTXDOLW\�RU�FRQWDLQPHQW�LQ�D�VHW� LQFOXGLQJWKH�EDVLF�VHWV�RI�WKH�VROYHU��FI��SDJH �����WR�WHOO�0X3$' WKDW�D�V\PEROLF�LGHQWLILHURU�H[SUHVVLRQ�UHSUHVHQWV�RQO\�YDOXHV�FRUUHVSRQGLQJ�WR�WKH�PDWKHPDWLFDO�PHDQLQJRI�WKH�W\SH� )RU�H[DPSOH� WKH�FRPPDQGV

DVVXPH�[ LQ 5B�� DVVXPH�\ LQ 5B�� DVVXPH�Q LQ =B��

UHVWULFW x DQG y WR�EH�UHDO�QXPEHUV�DQG n WR�EH�DQ�LQWHJHU� 1RZ VLPSOLI\ FDQDSSO\�DGGLWLRQDO�UXOHV�

VLPSOLI\�OQ�H[S�[���� VLPSOLI\�VTUW�[A���x, x VLJQ(x)

7KH�FRPPDQG

DVVXPH�[ ! ���

����

Page 221: Mupad Tutorial

:llnfimbhgl�Z[hnm�FZma^fZmb\Ze�Ikhi^kmb^l

UHVWULFWV x WR�SRVLWLYH�UHDO�QXPEHUV� $IWHU�WKLV�DVVXPSWLRQ� RQH�REWDLQV�

VLPSOLI\�OQ�[AQ��� VLPSOLI\�OQ�[ \� � OQ�[� � OQ�\���VLPSOLI\�VTUW�[A���n OQ(x) , 0, x

7UDQVIRUPDWLRQV�DQG�VLPSOLILFDWLRQV�ZLWK�FRQVWDQWV�DUH�H[HFXWHG�ZLWKRXWDGGLWLRQDO�DVVXPSWLRQV�VLQFH�WKHLU�PDWKHPDWLFDO�PHDQLQJ�LV�NQRZQ�

H[SDQG�OQ�� 3, ]��� VTUW��� 3, ]�A��

OQ(2) + OQ(π) + OQ(z) , 2π√z2

7KH�DULWKPHWLF�RSHUDWRUV�WDNH�LQWR�DFFRXQW�FHUWDLQ�PDWKHPDWLFDO�SURSHUWLHVDXWRPDWLFDOO\�

�D E�AP(a b)m

DVVXPH�P LQ =B�� �D E�APam bm

7KH�IXQFWLRQ LV FKHFNV�ZKHWKHU�D�0X3$' H[SUHVVLRQ�LV�NQRZQ�WR�EH�WUXH�

LV�� LQ =B�� LV�3, � � LQ 5B�758(, 758(

,Q�DGGLWLRQ� LV WDNHV�LQWR�DFFRXQW�WKH�PDWKHPDWLFDO�SURSHUWLHV�VHW�E\ DVVXPH�

GHOHWH [� LV�[ LQ =B�81.12:1

DVVXPH�[ LQ =B��LV�[ LQ =B�� LV�[ LQ 5B�

758(, 758(

+HUH� WKH�%RROHDQ�YDOXH 81.12:1 H[SUHVVHV�WKH�IDFW�WKDW�WKH�V\VWHP�FDQQRW�GHFLGHZKHWKHU [ UHSUHVHQWV�DQ�LQWHJHU�RU�QRW�

����

Page 222: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

,Q�FRQWUDVW�WR LV� WKH�IXQFWLRQ WHVWW\SH SUHVHQWHG�LQ�6HFWLRQ ���� FKHFNV�WKHWHFKQLFDO W\SH�RI�D�0X3$' REMHFW�

WHVWW\SH�[� 7\SH��,QWHJHU�� WHVWW\SH�[� '20B,'(17�)$/6(, 758(

4XHULHV�RI�WKH�IROORZLQJ�IRUP�DUH�DOVR�SRVVLEOH�

DVVXPH�\ ! ��� LV�\ � � ! ��758(

7R�VHW�DQ�DVVXPSWLRQ�RQO\�WHPSRUDULO\�GXULQJ�D�VLQJOH�FRPPDQG� XVH�WKHDVVXPLQJ RSHUDWRU�

OLPLW�H[S�D [�� [ LQILQLW\��OLPLW�H[S�D [�� [ LQILQLW\� DVVXPLQJ D � �⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 LI a = 00 LI ℜ(a) < 0∞ LI 0 < ℜ(a) ∧ 0 < VLJQ(ℜ(a)) + VLJQ(ℑ(a)) L

OLPx→∞ Ha x LI ℜ(a) = 0 ∧ a ̸= 0 ∨ 0 < ℜ(a) ∧ . . .

0

7KH�IXQFWLRQ JHWSURS UHWXUQV�D�PDWKHPDWLFDO�SURSHUW\�RI�DQ�LGHQWLILHU�RU�DQH[SUHVVLRQ� L�H�� D�VHW�VXFK�WKDW x ∈ JHWSURS(x)�

JHWSURS�\�� JHWSURS�\A� � ��(5,∞), (26,∞)

<RX�FDQ�GHOHWH�WKH�SURSHUWLHV�RI�DQ�LGHQWLILHU�XVLQJ XQDVVXPH RU�WKH�NH\ZRUGGHOHWH�GHOHWH \� LV�\ ! ��

81.12:1

����

Page 223: Mupad Tutorial

:llnfimbhgl�Z[hnm�FZma^fZmb\Ze�Ikhi^kmb^l

,I�QRQH�RI�WKH�VXE�H[SUHVVLRQV�LQ�DQ�H[SUHVVLRQ�KDV�DQ\�SURSHUWLHV�DWWDFKHG�WR�LW�WKHQ JHWSURSPD\�UHWXUQ�WKH�VHW�RI�FRPSOH[�QXPEHUV�

JHWSURS�\�� JHWSURS�\A� � ��� JHWSURS�DEV�\��C, C, [0,∞)

JHWSURS���� JHWSURS�VTUW��� � ��

{3} ,{√

2 + 1}

$VVXPLQJ�VRPH�SURSHUW\�GHOHWHV�DOO�SUHYLRXV�DVVXPSWLRQV�XVLQJ�WKH�VDPHLGHQWLILHUV�

DVVXPH�VLQ�[� ! \��JHWSURS�\�

(−∞, VLQ(x))

DVVXPH�DEV�[� ! ���JHWSURS�\�

C

7R�PDNH�DQ�DVVXPSWLRQ�LQ�DGGLWLRQ�WR�H[LVWLQJ�DVVXPSWLRQV� XVH DVVXPH$OVRLQVWHDG�RI DVVXPH� DQG DVVXPLQJ$OVR LQVWHDG�RI DVVXPLQJ�DVVXPH�VLQ�[� ! \��JHWSURS�\�

(−∞, VLQ(x))

DVVXPH$OVR�DEV�[� ! ���JHWSURS�\�

(−∞, VLQ(x))

����

Page 224: Mupad Tutorial

2 FZgbineZmbg`�>qik^llbhgl

:H�QRZ�LOOXVWUDWH�VRPH�RI�WKH�YDULRXV�W\SHV�RI�SURSHUWLHV�ZLWK�D�VKRUW�H[DPSOH�7KH�HTXDWLRQ (xa)b = xab LV�QRW�JHQHUDOO\�YDOLG� DV�WKH�H[DPSOH x = −1� a = 2� DQGb = 1/2 VKRZV� +RZHYHU� LW�LV�YDOLG�LI b LV�DQ�LQWHJHU�

DVVXPH�E LQ =B�� �[AD�AExa b

XQDVVXPH�E�� �[AD�AE

(xa)b

7KH�IXQFWLRQ OLQDOJ��LV3RV'HI FKHFNV�ZKHWKHU�D�PDWUL[�LV�SRVLWLYH�GHILQLWH� )RUD�PDWUL[�ZLWK�V\PEROLF�HQWULHV� LW�PD\�QRW�EH�SRVVLEOH�WR�GHFLGH�WKLV�FRUUHFWO\�

$ � PDWUL[�>>��D� �@� >�� ��D@@�(

1a 11 1a

)

OLQDOJ��LV3RV'HI�$�(UURU� FDQQRW FKHFN ZKHWKHU PDWUL[ FRPSRQHQW LV SRVLWLYH>OLQDOJ��IDFWRU&KROHVN\@

:LWK�WKH�DGGLWLRQDO�DVVXPSWLRQ�WKDW�WKH�SDUDPHWHU D EH�SRVLWLYH�DQG�OHVV�WKDQ 1�0X3$' FDQ�GHFLGH�WKDW�WKH�PDWUL[�LV�SRVLWLYH�GHILQLWH�

DVVXPH�D� 7\SH��,QWHUYDO��� ���OLQDOJ��LV3RV'HI�$�

758(

3URSHUWLHV�RI�WKLV�W\SH�FDQ�EH�VSHFLILHG�LQ�WKH�IROORZLQJ�PRUH�LQWXLWLYH�ZD\�DV�ZHOO�

DVVXPH�� � D � ��

7KH�IXQFWLRQ VLPSOLI\ UHDFWV�WR�SURSHUWLHV�

DVVXPH�N� 7\SH��5HVLGXH������VLQ�N 3,���

VLQ

(π k

2

)

����

Page 225: Mupad Tutorial

:llnfimbhgl�Z[hnm�FZma^fZmb\Ze�Ikhi^kmb^l

VLPSOLI\���−1

7KH�SURSHUW\�DERYH�FDQ�DOVR�EH�VSHFLILHG�LQ�DQ\�RI�WKH�IROORZLQJ�HTXLYDOHQW�IRUPV�

DVVXPH�N LQ � =B � ��DVVXPH��N � ���� LQ =B�

7KH�IXQFWLRQV 5H� ,P� VLJQ� DQG DEV WDNH�SURSHUWLHV�LQWR�DFFRXQW�

DVVXPH�[ ! ���5H�[ �[ � ���� VLJQ�[ �[ � ���� DEV�[ �[ � ���x (x− 1) , 1, x (x− 1)

6LQFH�RQO\�D�OLPLWHG�QXPEHU�RI�PDWKHPDWLFDO�SURSHUWLHV�DQG�GHULYDWLRQ�UXOHV�DUHLPSOHPHQWHG�LQ�0X3$'��WKH�V\VWHP�SHUIRUPV�VRPH�VLPSOLILFDWLRQV�ZKHQHYDOXDWLQJ�WKH�SURSHUWLHV�RI�D�QRQWULYLDO�H[SUHVVLRQ� 7KXV�WKH�DQVZHU�RI JHWSURSRU LV LV�VRPHWLPHV�QRW�³DV�FORVH�DV�SRVVLEOH�´ )RU�H[DPSOH� LI x LV�D�UHDO�QXPEHU�WKHQ x2 − x ≥ −1/4� EXW JHWSURS \LHOGV�WKH�IROORZLQJ�OHVV�DFFXUDWH�DQVZHU�

DVVXPH�[� 7\SH��5HDO�� JHWSURS�[A� � [�R

([HUFLVH����� 8VH�0X3$' WR�VKRZ�

OLPx→∞

(L(ax) =

{∞ IRU a > 0 ,0 IRU a < 0 .

+LQW� 8VH�WKH�IXQFWLRQ DVVXPH WR�GLVWLQJXLVK�WKH�FDVHV� 7KH�H[SRQHQWLDO�LQWHJUDO(L(x) =

∫ x−∞

Ht

t Gt LV�DYDLODEOH�DV (L�

����

Page 226: Mupad Tutorial
Page 227: Mupad Tutorial

)(

&KDQFH�DQG�3UREDELOLW\

<RX�FDQ�XVH�0X3$'�¶V�UDQGRP�QXPEHU�JHQHUDWRUV UDQGRP� IUDQGRP DQGVWDWV��[[[5DQGRP WR�SHUIRUP�PDQ\�H[SHULPHQWV�

7KH�FDOO UDQGRP�� JHQHUDWHV�D�UDQGRP�QRQQHJDWLYH 12 GLJLW�LQWHJHU� <RX�REWDLQ�DVHTXHQFH�RI 4 VXFK�UDQGRP�QXPEHUV�DV�IROORZV�

UDQGRP��� UDQGRP��� UDQGRP��� UDQGRP��427419669081, 321110693270, 343633073697, 474256143563

,I�\RX�ZDQW�WR�JHQHUDWH�UDQGRP�LQWHJHUV�LQ�D�GLIIHUHQW�UDQJH� \RX�FDQ�FRQVWUXFW�DUDQGRP�QXPEHU�JHQHUDWRU JHQHUDWRU� UDQGRP�P��Q�� <RX�FDOO�WKLV�JHQHUDWRUZLWKRXW�DUJXPHQWV�� DQG�LW�UHWXUQV�LQWHJHUV�EHWZHHQm DQG n� 7KH�FDOO UDQGRP�Q�LV�HTXLYDOHQW�WR UDQGRP����Q���� 7KXV�\RX�FDQ�VLPXODWH 15 UROOV�RI�D�GLH�DVIROORZV�

GLH � UDQGRP�������GLH([SHULPHQW � >GLH�� � L �����@

[5, 3, 6, 3, 2, 2, 2, 4, 4, 3, 3, 2, 1, 4, 4]

:H�VWUHVV�WKDW�\RX�PXVW�VSHFLI\�D�ORRS�YDULDEOH�ZKHQ�XVLQJ�WKH�VHTXHQFH�JHQHUDWRU�� VLQFH�RWKHUZLVH GLH�� LV�FDOOHG�RQO\�RQFH�DQG�D�VHTXHQFH�RI�FRSLHV�RI�WKLV�YDOXHLV�JHQHUDWHG�

GLH�� � ��6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

�,Q�IDFW� \RX�FDQ�FDOO JHQHUDWRU ZLWK�DUELWUDU\�DUJXPHQWV� ZKLFK�DUH�LJQRUHG�ZKHQ�JHQHUDWLQJ�UDQ�GRP�QXPEHUV�

Page 228: Mupad Tutorial

*) <aZg\^�Zg]�Ikh[Z[bebmr

+HUH�LV�D�VLPXODWLRQ�RI 8 FRLQ�WRVVHV�FRLQ � UDQGRP����FRLQ7RVVHV � >FRLQ�� � L ����@

[0, 0, 0, 1, 1, 1, 0, 0]

VXEV�FRLQ7RVVHV� >� KHDG� � WDLO@�[KHDG,KHDG,KHDG, WDLO, WDLO, WDLO,KHDG,KHDG]

7KH�IROORZLQJ�H[DPSOH�JHQHUDWHV�XQLIRUPO\�GLVWULEXWHG�IORDWLQJ�SRLQW�QXPEHUVIURP�WKH�LQWHUYDO [0, 1]� ,W�XVHV�WKH�UDQGRP�JHQHUDWRU IUDQGRP�

UDQGRP1XPEHUV � >IUDQGRP�� � L �����@>������������� ������������� ������������

������������� ������������� �������������

������������ ������������� �������������

������������@

7KH�OLEUDU\ VWDWV FRPSULVHV�IXQFWLRQV IRU�VWDWLVWLFDO�DQDO\VLV� <RX�REWDLQLQIRUPDWLRQ�E\�HQWHULQJ LQIR�VWDWV� RU "VWDWV� )RU�H[DPSOH� WKH�IXQFWLRQVWDWV��PHDQ FRPSXWHV�WKH�PHDQ�YDOXH X = 1

n

∑ni=1 xi RI�D�OLVW�RI�QXPEHUV

[x1, . . . , xn]�

VWDWV��PHDQ�GLH([SHULPHQW��VWDWV��PHDQ�FRLQ7RVVHV��VWDWV��PHDQ�UDQGRP1XPEHUV�

165,

38, 0.4863510522

7KH�IXQFWLRQ VWDWV��YDULDQFH UHWXUQV�WKH�YDULDQFH

V ar =1n− 1

n∑

i=1

(xi −X)2 :

����

Page 229: Mupad Tutorial

<aZg\^�Zg]�Ikh[Z[bebmr

VWDWV��YDULDQFH�GLH([SHULPHQW��VWDWV��YDULDQFH�FRLQ7RVVHV��VWDWV��YDULDQFH�UDQGRP1XPEHUV�

6135,

1556, 0.08565360412

<RX�FDQ�FRPSXWH�WKH�VWDQGDUG�GHYLDWLRQ√V ar ZLWK VWDWV��VWGHY�

VWDWV��VWGHY�GLH([SHULPHQW��VWDWV��VWGHY�FRLQ7RVVHV��VWDWV��VWGHY�UDQGRP1XPEHUV�√

35√

6135

,

√14√

1528

, 0.29266637

,I�\RX�VSHFLI\�WKH�RSWLRQ 3RSXODWLRQ� WKH�V\VWHP�UHWXUQV√n−1n V ar LQVWHDG�

VWDWV��VWGHY�GLH([SHULPHQW� 3RSXODWLRQ��VWDWV��VWGHY�FRLQ7RVVHV� 3RSXODWLRQ��VWDWV��VWGHY�UDQGRP1XPEHUV� 3RSXODWLRQ�√

3√

12215

,

√158, 0.2776476971

7KH�GDWD�VWUXFWXUH 'RP��0XOWLVHW �VHH "'RP��0XOWLVHW� SURYLGHV�D�VLPSOH�PHDQVRI�GHWHUPLQLQJ�IUHTXHQFLHV�LQ�VHTXHQFHV� 7KH�FDOO 'RP��0XOWLVHW�D�E�����UHWXUQV�D�PXOWLVHW� ZKLFK�LV�SULQWHG�DV�D�VHW�RI�OLVWV� 7KH�ILUVW�HQWU\�RI�HDFK�VXFK�OLVWLV�RQH�RI�WKH�DUJXPHQWV� WKH�VHFRQG�HQWU\�FRXQWV�WKH�QXPEHU�RI�RFFXUUHQFHV�LQ�WKHDUJXPHQW�VHTXHQFH�

'RP��0XOWLVHW�D� E� D� F� E� E� D� D� F� G� H� G�{[a, 4] , [b, 3] , [c, 2] , [d, 2] , [e, 1]}

,I�\RX�VLPXODWH 1000 UROOV�RI�D�GLH� \RX�PLJKW�REWDLQ�WKH�IROORZLQJ�IUHTXHQFLHV�

UROOV � GLH�� � L ��������'RP��0XOWLVHW�UROOV�{[1, 158] , [2, 152] , [3, 164] , [4, 188] , [5, 176] , [6, 162]}

,Q�WKLV�FDVH� \RX�ZRXOG�KDYH�UROOHG 158 WLPHV�D 1� 152 WLPHV�D 2 HWF�

����

Page 230: Mupad Tutorial

*) <aZg\^�Zg]�Ikh[Z[bebmr

$Q�H[DPSOH�IURP�QXPEHU�WKHRU\�LV�WKH�GLVWULEXWLRQ�RI�WKH�JUHDWHVW�FRPPRQGLYLVRUV��JFG��RI�UDQGRP�SDLUV�RI�LQWHJHUV� :H�XVH�WKH�IXQFWLRQ ]LS �6HFWLRQ ����WR�FRPELQH�WZR UDQGRP�OLVWV�YLD�WKH�IXQFWLRQ LJFG� ZKLFK�FRPSXWHV�WKH�JFG�RILQWHJHUV�

OLVW� � >UDQGRP�� � L �������@�OLVW� � >UDQGRP�� � L �������@�JFGOLVW � ]LS�OLVW�� OLVW�� LJFG�

>�� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���@

1RZ� ZH�XVH 'RP��0XOWLVHW WR�FRXQW�WKH�IUHTXHQFLHV�RI�WKH�LQGLYLGXDO�JFGV�

IUHTXHQFLHV � 'RP��0XOWLVHW�RS�JFGOLVW��^>��� �@� >��� �@� >��� �@� ���� >���� �@� >�� ���@`

7KH�UHVXOW�EHFRPHV�PXFK�PRUH�UHDGDEOH�LI�ZH�VRUW�LW�E\�WKH�ILUVW�HQWU\�RI�WKHVXEOLVWV� :H�HPSOR\�WKH�IXQFWLRQ VRUW ZKLFK�WDNHV�D�IXQFWLRQ�UHSUHVHQWLQJ�DVRUWLQJ�RUGHU�DV�VHFRQG�DUJXPHQW� 7KLV�IXQFWLRQ�GHFLGHV�ZKLFK�RI�WZR�HOHPHQWVx, y VKDOO�SUHFHGH�WKH�RWKHU� :H�UHIHU�WR�WKH�FRUUHVSRQGLQJ�KHOS�SDJH� "VRUW� ,QWKLV�FDVH� x, y DUH�OLVWV�ZLWK�WZR�HQWULHV� DQG�ZH�ZDQW x WR�DSSHDU�EHIRUH y LI�ZHKDYH x[1] < y[1] �L�H�� ZH�VRUW�QXPHULFDOO\�ZLWK�UHVSHFW�WR�WKH�ILUVW�HQWULHV��

VRUWLQJ2UGHU � �[� \� �! �[>�@ � \>�@��VRUW�>RS�IUHTXHQFLHV�@� VRUWLQJ2UGHU�

>>�� ���@� >�� ���@� >�� ��@� ���� >���� �@� >���� �@@

,Q�WKLV�H[SHULPHQW� 596 RXW�RI�WKH 1000 FKRVHQ�UDQGRP�SDLUV�KDYH�D�JFG�RI 1 DQGKHQFH�DUH�FRSULPH� 7KXV� ZH�IRXQG 59.6� DV�DQ�DSSUR[LPDWLRQ�RI�WKH�SUREDELOLW\WKDW�WZR�UDQGRPO\�FKRVHQ�LQWHJHUV�DUH�FRSULPH� 7KH�WKHRUHWLFDO�YDOXH�RI�WKLVSUREDELOLW\�LV 6/π2 ≈ 0.6079.. =̂ 60.79��

7KH VWDWV OLEUDU\�SURYLGHV�PDQ\�VWRFKDVWLF�GLVWULEXWLRQ�IXQFWLRQV� (DFK�VXFKGLVWULEXWLRQ [[[� VD\� FRQVLVWV�RI�IRXU�IXQFWLRQV� D�FXPXODWLYH�GLVWULEXWLRQIXQFWLRQ [[[&')� D�SUREDELOLW\�GHQVLW\�IXQFWLRQ [[[3') �RU�D�GLVFUHWH�SUREDELOLW\IXQFWLRQ [[[3)��UHVSHFWLYHO\�� D�TXDQWLOH�IXQFWLRQ [[[4XDQWLOH� DQG D�UDQGRPQXPEHU�JHQHUDWRU [[[5DQGRP� )RU�H[DPSOH� WKH�IROORZLQJ�FRPPDQG�FUHDWHV�D�OLVWRI�UDQGRP�QXPEHUV�GLVWULEXWHG�DFFRUGLQJ�WR�WKH�VWDQGDUG�QRUPDO�GLVWULEXWLRQZLWK�PHDQ 0 DQG�YDULDQFH 1�

����

Page 231: Mupad Tutorial

<aZg\^�Zg]�Ikh[Z[bebmr

UDQGRP*HQHUDWRU � VWDWV��QRUPDO5DQGRP��� ���GDWD � >UDQGRP*HQHUDWRU�� � L �������@�

7KH VWDWV OLEUDU\�LQFOXGHV�DQ�LPSOHPHQWDWLRQ�RI�WKH�FODVVLFDO χ2�WHVW� :H�XVH�LWKHUH�WR�WHVW�ZKHWKHU�WKH�UDQGRP�GDWD�JHQHUDWHG�DERYH�GR�LQGHHG�VDWLVI\�D�QRUPDOGLVWULEXWLRQ� 3UHWHQGLQJ�WKDW�ZH�GR�QRW�NQRZ�WKH�PHDQ�DQG�WKH�YDULDQFH�RI�WKHGDWD� ZH�FRPSXWH�VWDWLVWLFDO�HVWLPDWHV�RI�WKHVH�SDUDPHWHUV�

P � VWDWV��PHDQ�GDWD�−0.03440235553

9 � VWDWV��YDULDQFH�GDWD�0.9962289589

7KH χ2�WHVW�UHTXLUHV�WR�VSHFLI\�D�³FHOO�SDUWLWLRQLQJ´�RI�WKH�UHDO�OLQH�WR�FRPSDUH�WKHREVHUYHG�IUHTXHQFLHV�RI�WKH�GDWD�IDOOLQJ�LQWR�WKH�FHOOV�ZLWK�WKH�H[SHFWHGIUHTXHQFLHV�JLYHQ�D�K\SRWKHVL]HG�GLVWULEXWLRQ�RI�WKH�GDWD� 7KH�IXQFWLRQVWDWV��HTXLSUREDEOH&HOOV LV�D�FRQYHQLHQW�XWLOLW\�IXQFWLRQ�WR�FRPSXWH�D�FHOOSDUWLWLRQLQJ�FRQVLVWLQJ�RI�FHOOV�WKDW�DUH�HTXLSUREDEOH�ZLWK�UHVSHFW�WR�D�JLYHQGLVWULEXWLRQ� 7KH�IROORZLQJ�FDOO�SDUWLWLRQV�WKH�UHDO�OLQH�LQWR 32 FHOOV�ZKLFK�DUHHTXLSUREDEOH�ZLWK�UHVSHFW�WR�WKH�QRUPDO�GLVWULEXWLRQ�ZLWK�WKH�HPSLULFDO�PHDQ�DQGYDULDQFH�FRPSXWHG�DERYH�

FHOOV � VWDWV��HTXLSUREDEOH&HOOV����VWDWV��QRUPDO4XDQWLOH�P� 9��

>>�LQILQLW\� �����������@� >������������ ������������@�

���� >������������ LQILQLW\@@

7KH χ2�JRRGQHVV�RI�ILW�WHVW�LV�LPSOHPHQWHG�E\ VWDWV��FV*2)7��:H�XVH�LW�WR�WHVWKRZ�JRRG�WKH�UDQGRP�GDWD�ILW�D�QRUPDO�GLVWULEXWLRQ�ZLWK�PHDQ�DQG�YDULDQFH�JLYHQE\�WKH�HPSLULFDO�YDOXHV P DQG 9�

VWDWV��FV*2)7�GDWD� FHOOV�&') VWDWV��QRUPDO&')�P� 9��

>39DOXH ������������� 6WDW9DOXH ������

0LQLPDO([SHFWHG&HOO)UHTXHQF\ �����@

7KH�ILUVW�YDOXH�UHWXUQHG�E\ VWDWV��FV*2)7 LV�WKH�VLJQLILFDQFH�OHYHO�DWWDLQHG�E\�WKHGDWD� 6LQFH�WKLV�YDOXH�LV�QRW�VPDOO� WKH�JLYHQ�GDWD�SDVV�WKH�WHVW�ZHOO�

����

Page 232: Mupad Tutorial

*) <aZg\^�Zg]�Ikh[Z[bebmr

)LQDOO\� ZH�GRWH�WKH�GDWD�E\�DGGLQJ 35 ]HURHV�

GDWD � DSSHQG�GDWD� � � ����

:H�FKHFN�ZKHWKHU�WKH�QHZ�GDWD�VWLOO�PD\�EH�UHJDUGHG�DV�D�QRUPDOO\�GLVWULEXWHGVDPSOH�

P � VWDWV��PHDQ�GDWD�� 9 � VWDWV��YDULDQFH�GDWD��FHOOV � VWDWV��HTXLSUREDEOH&HOOV����

VWDWV��QRUPDO4XDQWLOH�P� 9���VWDWV��FV*2)7�GDWD� FHOOV�

&') VWDWV��QRUPDO&')�P� 9��>39DOXH ������������������ 6WDW9DOXH ������������

0LQLPDO([SHFWHG&HOO)UHTXHQF\ ��������@

1RZ� WKH�DWWDLQHG�VLJQLILFDQFH�OHYHO 0.0010... LQGLFDWHV�WKDW�WKH�K\SRWKHVLV�RI�DQRUPDO�GLVWULEXWLRQ�RI�WKH�GDWD�PXVW�EH�UHMHFWHG�DW�VLJQLILFDQFH�OHYHOV�DV�ORZ�DV0.001�

6HH�WKH�KHOS�SDJH�RI VWDWV��FV*2)7 IRU�IXUWKHU�GHWDLOV�

([HUFLVH������ 7KUHH�GLFH�DUH�WKURZQ�VLPXOWDQHRXVO\� )RU�HDFK�YDOXH�EHWZHHQ 3DQG 18� WKH�IROORZLQJ�WDEOH�FRQWDLQV�WKH�H[SHFWHG�IUHTXHQFLHV�RI�WKH�WRWDO�GLFHVFRUH�ZKHQ�UROOLQJ�WKH�GLFH 216 WLPHV�

VFRUH

� � � � � � � �� �� �� �� �� �� �� �� ��� � � �� �� �� �� �� �� �� �� �� �� � � �

IUHTXHQF\

6LPXODWH 216 UROOV�DQG�FRPSDUH�WKH�IUHTXHQFLHV�WKDW�\RX�REVHUYH�ZLWK�WKRVH�IURPWKH�WDEOH�

([HUFLVH������ 7KH�0RQWH�&DUOR�PHWKRG�IRU�DSSUR[LPDWLQJ�WKH�DUHD�RI�D�UHJLRQA ⊂ R2 ZRUNV�DV�IROORZV� )LUVW� ZH�FKRRVH�D��SUHIHUDEO\�VPDOO��UHFWDQJOH QHQFORVLQJ A� 7KHQ�ZH�UDQGRPO\�FKRRVH n SRLQWV�LQ Q� ,Im RI�WKHVH�SRLQWV�OLH�LQ A�WKH�IROORZLQJ�HVWLPDWH�KROGV�IRU�VXIILFLHQWO\�ODUJH n�

DUHD�RI A ≈ mn× DUHD�RI Q.

����

Page 233: Mupad Tutorial

<aZg\^�Zg]�Ikh[Z[bebmr

/HW U�� EH�D�FDOO�WR�D�JHQHUDWRU U RI�XQLIRUPO\�GLVWULEXWHG�UDQGRP�QXPEHUV�LQ�WKHLQWHUYDO [0, 1]� :H�FDQ�XVH U WR�JHQHUDWH�XQLIRUPO\�GLVWULEXWHG�UDQGRP�YHFWRUV�LQWKH�UHFWDQJOH Q = [0, a]× [0, b] YLD >D U���E U��@�

D� &RQVLGHU�WKH�ULJKW�XSSHU�TXDGUDQW�RI�WKH�XQLW�FLUFOH�DURXQG�WKH�RULJLQ� 8VH0RQWH�&DUOR�VLPXODWLRQ�ZLWK Q = [0, 1]× [0, 1] WR�DSSUR[LPDWH�LWV�DUHD� 7KLVZD\� RQH�JHWV�VWRFKDVWLF�DSSUR[LPDWLRQV�RI π�

E� /HW f : x #→ x VLQ(x) + FRV(x) H[S(x)� 'HWHUPLQH�DQ�DSSUR[LPDWLRQ�IRU∫ 10 f(x) dx� )RU�WKDW�SXUSRVH� ILQG�DQ�XSSHU�ERXQGM IRU f RQ�WKH�LQWHUYDO

[0, 1] DQG�DSSO\�WKH�VLPXODWLRQ�WR Q = [0, 1]× [0,M ]� &RPSDUH�\RXU�UHVXOW�WRWKH�H[DFW�LQWHJUDO�

����

Page 234: Mupad Tutorial
Page 235: Mupad Tutorial

))

*UDSKLFV

Afljg\m[lagf

6LQFH�H[SORULQJ�PDWKHPDWLFDO�REMHFWV�LV�SUREDEO\�RQH�RI�WKH�PRVW�LPSRUWDQW�XVHVRI�FRPSXWHU�DOJHEUD�V\VWHPV� JUDSKLFV�DUH�D�UDWKHU�FHQWUDO�SDUW�RI�PRGHUQ�&$6�7KH�0X3$'� V\VWHP�DOORZV�WR�HDVLO\�FUHDWH�WZR��DQG�WKUHH�GLPHQVLRQDO�JUDSKLFVDQG�DQLPDWLRQV�ZLWK�KLJK�TXDOLW\�RXWSXW�DQG�DGYDQFHG�LQWHUDFWLYH�PDQLSXODWLRQFDSDELOLWLHV�

7R�DLG�WKH�XVHU�LQ�ILQH�WXQLQJ�D�SORW� DQ�REMHFW�EURZVHU�LV�SURYLGHG� ,W�DOORZV�WRLQVSHFW�DQG�DQDO\]H�WKH�WUHH�VWUXFWXUH�RI�JUDSKLFDO�VFHQHV�LQWHUDFWLYHO\� JLYLQJ�HDV\DFFHVV�WR�DOO�SDUWV�RI�WKH�SORW�E\�D�PRXVH�FOLFN� $IWHU�VHOHFWLQJ�DQ�REMHFW� DOO�SORWDWWULEXWHV�VXFK�DV�FRORU� OLQH�ZLGWK� DQQRWDWLRQV�HWF��DVVRFLDWHG�ZLWK�WKH�REMHFWEHFRPH�YLVLEOH� 2QH�PD\�VHOHFW�WKH�DWWULEXWHV�ZLWK�WKH�PRXVH�DQG�FKDQJH�WKHLUYDOXHV�LQWHUDFWLYHO\��VHH�SDJH ������� 7KXV� DOPRVW�DQ\�GHWDLO�RI�D�SORW�JHQHUDWHGE\�D�0X3$'�FDOO�FDQ�EH�FKDQJHG�LQWHUDFWLYHO\�PDNLQJ�LW�SRVVLEOH�WR�ILQH�WXQH�ERWKWKH�FRQWHQWV�DV�ZHOO�DV�WKH�ORRNV�E\�YLVXDO�LQVSHFWLRQ�

7KH�IHDWXUHV�RI�WKH�UHQGHUHU�DUH�IXOO\�VXSSRUWHG�E\�WKH SORW OLEUDU\� ,W�SURYLGHV�DOLVW�RI�JUDSKLFDO�SULPLWLYHV�VXFK�DV�SRLQWV� OLQHV� SRO\JRQV�HWF��WKDW�RQH�PD\�XVH�WREXLOG�XS�KLJKO\�FRPSOH[�JUDSKLFDO�VFHQHV� 7KH�SULPLWLYHV�DOVR�LQFOXGH�JUDSKLFDOREMHFWV�WKDW�DUH�QRW�DW�DOO�³SULPLWLYH�´ EXW�FRPH�DORQJ�ZLWK�DGYDQFHG�DOJRULWKPVDQG�PXFK�EXLOW�LQ�LQWHOOLJHQFH� ([DPSOHV�DUH�IXQFWLRQ�JUDSKV�LQ��' DQG��' �ZLWKDXWRPDWLF�FOLSSLQJ�IRU�VLQJXODU�IXQFWLRQV�� LPSOLFLW�SORWV�LQ��' DQG��'��YHFWRU�ILHOGSORWV� JUDSKLFDO�VROXWLRQV�RI�RUGLQDU\�GLIIHUHQWLDO�HTXDWLRQV�LQ��' DQG��' HWF�

7KLV�FKDSWHU�JLYHV�DQ�LQWURGXFWLRQ�WR�WKH�QHZ�JUDSKLFV�V\VWHP� ,W�H[SODLQV�WKHEDVLF�FRQFHSWV�DQG�LGHDV�DQG�SURYLGHV�PDQ\�H[DPSOHV�

Page 236: Mupad Tutorial

** @kZiab\l

=Ykq�Hdgllaf_2 ?jYh`k�g^�>mf[lagfk

3UREDEO\�WKH�PRVW�LPSRUWDQW�JUDSKLFDO�WDVN�LQ�D�PDWKHPDWLFDO�FRQWH[W�LV�WRYLVXDOL]H�IXQFWLRQ�JUDSKV� L�H�� WR�SORW�IXQFWLRQV� 7KH SORW FRPPDQG�DOORZV�WRFUHDWH��' SORWV�RI�IXQFWLRQV�ZLWK�RQH�DUJXPHQW��VXFK�DV f(x) = VLQ(x)��RU��' SORWVRI�IXQFWLRQV�ZLWK�WZR�DUJXPHQWV��VXFK�DV f(x, y) = VLQ(x2 + y2)�� 7KH�FDOOLQJV\QWD[�LV�VLPSOH� MXVW�SDVV�WKH�H[SUHVVLRQ�WKDW�GHILQHV�WKH�IXQFWLRQ�DQG�RSWLRQDOO\� D�UDQJH�IRU�WKH�LQGHSHQGHQW�YDULDEOH�V�� 7R�GLVWLQJXLVK��' SORWV�IURP�' DQLPDWLRQV� DGG ��' WR�WKH�FDOO�

�' )XQFWLRQ�*UDSKV

:H�FRQVLGHU��' H[DPSOHV� L�H�� SORWV�RI�XQLYDULDWH�IXQFWLRQV f(x)� +HUH�LV�RQHSHULRG�RI�WKH�VLQH�IXQFWLRQ�

SORW�VLQ�[�� [ ���� 3,�

,I�VHYHUDO�IXQFWLRQV�DUH�WR�EH�SORWWHG�LQ�WKH�VDPH�JUDSKLFDO�VFHQH� MXVW�SDVV�DVHTXHQFH�RI�IXQFWLRQ�H[SUHVVLRQV� $OO�IXQFWLRQV�DUH�SORWWHG�RYHU�WKH�VSHFLILHGFRPPRQ�UDQJH�

����

Page 237: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

SORW�VLQ�[��[� [ FRV�[�� WDQ�[�� [ ������

7R�JHW�D�OHJHQG�H[SODLQLQJ�WKH�PDSSLQJ�IURP�FRORU�WR�IXQFWLRQ� DGG�WKH�RSWLRQ/HJHQG9LVLEOH�SORW�VLQ�[��[� [ FRV�[�� WDQ�[�� [ ������ /HJHQG9LVLEOH�

����

Page 238: Mupad Tutorial

** @kZiab\l

)XQFWLRQV�WKDW�GR�QRW�DOORZ�D�VLPSOH�V\PEROLF�UHSUHVHQWDWLRQ�E\�DQ�H[SUHVVLRQFDQ�DOVR�EH�GHILQHG�E\�D�SURFHGXUH�WKDW�SURGXFHV�D�QXPHULFDO�YDOXH f(x) ZKHQFDOOHG�ZLWK�D�QXPHULFDO�YDOXH x IURP�WKH�SORW�UDQJH� ,Q�WKH�IROORZLQJ�H[DPSOH�ZHFRQVLGHU�WKH�ODUJHVW�HLJHQYDOXH�RI�D�V\PPHWULF 3× 3PDWUL[�WKDW�FRQWDLQV�DSDUDPHWHU x� :H�SORW�WKLV�HLJHQYDOXH�DV�D�IXQFWLRQ�RI x�

I � [ �! PD[�QXPHULF��HLJHQYDOXHV�PDWUL[�>>�[� [� �[ @�

> [� [� [ @�>�[� [� [A�@@����

SORW�I� [ ������

7KH�QDPH x XVHG�LQ�WKH�VSHFLILFDWLRQ�RI�WKH�SORWWLQJ�UDQJH�SURYLGHV�WKH�QDPH�WKDWODEHOV�WKH�KRUL]RQWDO�D[LV�

����

Page 239: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

)XQFWLRQV�FDQ�DOVR�EH�GHILQHG�E\ SLHFHZLVH REMHFWV�

SORW�SLHFHZLVH�>[ � �� � � [@�>� � [ DQG [ � �� ���@�>[ ! �� � � [@��

[ ������

1RWH�WKDW�WKHUH�DUH�JDSV�LQ�WKH�GHILQLWLRQ�RI�WKH�IXQFWLRQ�DERYH� QR�IXQFWLRQ�YDOXHLV�VSHFLILHG�IRU x = 1 DQG x = 2� 7KLV�GRHV�QRW�FDXVH�DQ\�SUREOHP� EHFDXVH SORWVLPSO\�LJQRUHV�DOO�SRLQWV�WKDW�GR�QRW�SURGXFH�UHDO�QXPHULFDO�YDOXHV� 7KXV� LQ�WKHIROORZLQJ�H[DPSOH� WKH�SORW�LV�DXWRPDWLFDOO\�UHVWULFWHG�WR�WKH�UHJLRQV�ZKHUH�WKHIXQFWLRQV�SURGXFH�UHDO�YDOXHV�

����

Page 240: Mupad Tutorial

** @kZiab\l

SORW�VTUW�� � [A��� OQ�[A� � ����[ � ��� [ �� ����

����

Page 241: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

:KHQ�VHYHUDO�IXQFWLRQV�DUH�SORWWHG�LQ�WKH�VDPH�VFHQH� WKH\�DUH�GUDZQ�LQ�GLIIHUHQWFRORUV�WKDW�DUH�FKRVHQ�DXWRPDWLFDOO\� :LWK�WKH &RORUV DWWULEXWH�RQH�PD\�VSHFLI\�DOLVW�RI�5*% FRORUV�WKDW SORW VKDOO�XVH�SORW�[� [A�� [A�� [A�� [A�� [ �����

&RORUV >5*%��5HG� 5*%��2UDQJH� 5*%��<HOORZ�5*%��%OXH/LJKW� 5*%��%OXH@�

/HJHQG9LVLEOH�

����

Page 242: Mupad Tutorial

** @kZiab\l

$QLPDWHG��' SORWV�RI�IXQFWLRQV�DUH�FUHDWHG�E\�SDVVLQJ�IXQFWLRQ�H[SUHVVLRQVGHSHQGLQJ�RQ�D�YDULDEOH��x� VD\��DQG�DQ�DQLPDWLRQ�SDUDPHWHU��a� VD\��DQGVSHFLI\LQJ�D�UDQJH�ERWK�IRU x DQG a� ,Q�WKLV�WH[W� DQLPDWLRQV�DUH�UHSUHVHQWHG�E\�D³ILOP�VWULS�´

SORW�FRV�D [�� [ ���� 3,� D �����

����

Page 243: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

2QFH�WKH�SORW�LV�FUHDWHG� WKH�ILUVW�IUDPH�RI�WKH�SLFWXUH�DSSHDUV�DV�D�VWDWLF�SORW�$IWHU�GRXEOH�FOLFNLQJ�RQ�WKH�SLFWXUH� WKH�JUDSKLFV�WRRO�DOORZV�WR�VWDUW�WKHDQLPDWLRQ�E\�SXVKLQJ�WKH�³6WDUW´�EXWWRQ�

����

Page 244: Mupad Tutorial

** @kZiab\l

$Q�DQLPDWLRQ�FRQVLVWV�RI�VHSDUDWH�IUDPHV� L�H�� LQGLYLGXDO�VWLOO�LPDJHV� 7KH�GHIDXOWQXPEHU�RI�IUDPHV�LV 50� ,I�D�GLIIHUHQW�YDOXH�LV�GHVLUHG� MXVW�SDVV�WKH�DWWULEXWH)UDPHV Q� ZKHUH n LV�WKH�QXPEHU�RI�IUDPHV�WKDW�VKDOO�EH�FUHDWHG�SORW�VLQ�D [�� VLQ�[ � D�� [ ���3,� D ���� 3,�

)UDPHV ���� &RORUV >5*%��%OXH� 5*%��5HG@�

$SDUW�IURP�WKH�FRORU�VSHFLILFDWLRQ�RU�WKH )UDPHV QXPEHU� WKHUH�LV�D�ODUJH�QXPEHURI�IXUWKHU�DWWULEXWHV�WKDW�PD\�EH�SDVVHG�WR SORW� (DFK�DWWULEXWH�LV�SDVVHG�DV�DQHTXDWLRQ $WWULEXWH1DPH $WWULEXWH9DOXH� +HUH� ZH�RQO\�SUHVHQW�VRPH�VHOHFWHGDWWULEXWHV� 6HFWLRQ�³$WWULEXWHV�IRU SORWIXQF�G DQG SORWIXQF�G´�RI�WKH�RQOLQH SORWGRFXPHQWDWLRQ�SURYLGHV�IXUWKHU�WDEOHV�ZLWK�PRUH�DWWULEXWHV�✞

DWWULEXWH�QDPH SRVVLEOH�YDOXHV PHDQLQJ GHIDXOW

+HLJKW � XQLW��FP SK\VLFDO�KHLJKWRI�WKH�SLFWXUH

�� XQLW��PP

:LGWK �� XQLW��FP SK\VLFDO�ZLGWKRI�WKH�SLFWXUH

��� XQLW��PP

)RRWHU VWULQJ IRRWHU�WH[W �� �QR�IRRWHU�+HDGHU VWULQJ KHDGHU�WH[W �� �QR�KHDGHU�7LWOH VWULQJ WLWOH�WH[W �� �QR�WLWOH�7LWOH3RVLWLRQ >UHDO YDOXH�

UHDO YDOXH@FRRUGLQDWHV�RI�WKHORZHU�OHIW�FRUQHU�RIWKH�WLWOH

7DEOH ����� 6RPH�LPSRUWDQW�DWWULEXWHV�IRU�SORWWLQJ��' IXQFWLRQV

�����

Page 245: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

DWWULEXWH�QDPH SRVVLEOH�YDOXHV PHDQLQJ GHIDXOW

*ULG9LVLEOH 758(� )$/6( YLVLELOLW\�RI�³PDMRU´JULG�OLQHV

)$/6(

6XEJULG9LVLEOH 758(� )$/6( YLVLELOLW\�RI�³PLQRU´JULG�OLQHV

)$/6(

$GDSWLYH0HVK LQWHJHU≥ 0 GHSWK�RI�WKH�DGDSWLYHPHVK

$[HV 1RQH� $XWRPDWLF�%R[HG� )UDPH� 2ULJLQ

D[HV�W\SH $XWRPDWLF

$[HV9LVLEOH 758(� )$/6( YLVLELOLW\RI�DOO�D[HV

758(

$[HV7LWOHV >VWULQJ� VWULQJ@ WLWOHV�RI�WKH�D[HV >�[���\�@&RRUGLQDWH7\SH /LQ/LQ� /LQ/RJ�

/RJ/LQ� /RJ/RJOLQHDU�OLQHDU�OLQHDU�ORJDULWKPLF�ORJ�OLQHDU� ORJ�ORJ

/LQ/LQ

&RORUV OLVW�RI�5*% YDOXHV OLQH�FRORUV ILUVW����HQWULHV�RI5*%��&RORU/LVW

)UDPHV LQWHJHU≥ 0 QXPEHU�RI�IUDPHV�RIDQ�DQLPDWLRQ

��

/HJHQG9LVLEOH 758(� )$/6( OHJHQG�RQ�RII 758( LI�SORWWLQJ�PRUHWKDQ�RQH�IXQFWLRQ

/LQH&RORU7\SH 'LFKURPDWLF� )ODW�)XQFWLRQDO�0RQRFKURPH� 5DLQERZ

FRORU�VFKHPH )ODW

0HVK LQWHJHU≥ 2 QXPEHU�RI�VDPSOHSRLQWV

���

6FDOLQJ $XWRPDWLF�&RQVWUDLQHG�8QFRQVWUDLQHG

VFDOLQJ�PRGH 8QFRQVWUDLQHG

7LFNV1XPEHU 1RQH� /RZ� 1RUPDO�+LJK

QXPEHU�RI�ODEHOHGWLFNV

1RUPDO

9LHZLQJ%R[<5DQJH \PLQ��\PD[ UHVWULFWHG�YLHZLQJUDQJH�LQ y GLUHFWLRQ

$XWRPDWLF��$XWRPDWLF

7DEOH ����� 6RPH�LPSRUWDQW�DWWULEXWHV�IRU�SORWWLQJ��' IXQFWLRQV

:H�QRZ�SUHVHQW�VRPH�H[DPSOHV�IRU�WKH�XVH�RI�WKHVH�DWWULEXWHV�

�����

Page 246: Mupad Tutorial

** @kZiab\l

)LUVW� ZH�ZLOO�JLYH�DQ�H[DPSOH�ZKHUH�WKH�QXPEHU�RI�HYDOXDWLRQ�SRLQWV�QHHGVDGMXVWPHQW� 7KH�IROORZLQJ�SORW�H[DPSOH�IHDWXUHV�WKH�QRWRULRXV�IXQFWLRQ VLQ(1/x)WKDW�RVFLOODWHV�ZLOGO\�QHDU�WKH�RULJLQ�

SORW�VLQ���[�� [ ����������

&OHDUO\� WKH�GHIDXOW�RI 121 VDPSOH�SRLQWV�XVHG�E\ SORW GRHV�QRW�VXIILFH�WR�FUHDWH�DVXIILFLHQWO\�UHVROYHG�SORW�

�����

Page 247: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

:H�LQFUHDVH�WKH�QXPEHU�RI�QXPHULFDO�PHVK�SRLQWV�YLD�WKH 0HVK DWWULEXWH�$GGLWLRQDOO\� ZH�LQFUHDVH�WKH�UHVROXWLRQ�GHSWK�RI�WKH�DGDSWLYH�SORWWLQJ�PHFKDQLVPIURP�LWV�GHIDXOW�YDOXH $GDSWLYH0HVK � WR $GDSWLYH0HVK ��SORW�VLQ���[�� [ ���������� 0HVK ����

$GDSWLYH0HVK ��

�����

Page 248: Mupad Tutorial

** @kZiab\l

7KH�IROORZLQJ�FDOO�VSHFLILHV�D�KHDGHU�YLD +HDGHU �7KH�IXQFWLRQ�VLQ�[A����7KH�GLVWDQFH�EHWZHHQ�ODEHOHG�WLFNV�LV�VHW�WR 0.5 DORQJ�WKH x D[LV�DQG�WR 0.2 DORQJ�WKHy D[LV�YLD ;7LFNV'LVWDQFH ��� DQG <7LFNV'LVWDQFH ���� UHVSHFWLYHO\� )RXUDGGLWLRQDO�XQODEHOHG�WLFNV�EHWZHHQ�HDFK�SDLU�RI�ODEHOHG�WLFNV�DUH�VHW�LQ�WKH xGLUHFWLRQ�YLD ;7LFNV%HWZHHQ �� 2QH�DGGLWLRQDO�XQODEHOHG�WLFN�EHWZHHQ�HDFK�SDLURI�ODEHOHG�WLFNV�LQ�WKH y GLUHFWLRQ�LV�UHTXHVWHG�YLD <7LFNV%HWZHHQ �� *ULG�OLQHVDWWDFKHG�WR�WKH�WLFNV�DUH�³VZLWFKHG�RQ´�E\ *ULG9LVLEOH 758( DQG6XEJULG9LVLEOH 758(�SORW�VLQ�[A��� [ �����

+HDGHU �7KH IXQFWLRQ VLQ�[A����;7LFNV'LVWDQFH ���� <7LFNV'LVWDQFH ����;7LFNV%HWZHHQ �� <7LFNV%HWZHHQ ��*ULG9LVLEOH 758(� 6XEJULG9LVLEOH 758(�

�����

Page 249: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

:KHQ�VLQJXODULWLHV�DUH�IRXQG�LQ�WKH�IXQFWLRQ� DQ�DXWRPDWLF�FOLSSLQJ�LV�FDOOHG�WU\LQJWR�UHVWULFW�WKH�YHUWLFDO�YLHZLQJ�UDQJH�LQ�VRPH�ZD\�WR�REWDLQ�D�³UHDVRQDEO\´�VFDOHGSLFWXUH� 7KLV�LV�D�KHXULVWLF�DSSURDFK�WKDW�VRPHWLPHV�QHHGV�D�KHOSLQJ�DGDSWLRQ�³E\KDQG�´ ,Q�WKH�IROORZLQJ�H[DPSOH� WKH�DXWRPDWLFDOO\�FKRVHQ�UDQJH�EHWZHHQ y ≈ −40DQG y ≈ 260 LQ�YHUWLFDO�GLUHFWLRQ�LV�VXLWDEOH�WR�UHSUHVHQW�WKH��WK�RUGHU�SROH�DWx = 1� EXW�LW�GRHV�QRW�SURYLGH�D�JRRG�UHVROXWLRQ�RI�WKH�ILUVW�RUGHU�SROH�DW x = −1�

SORW����[ � ����[ � ��A�� [ ������

7KHUH�LV�QR�JRRG�YLHZLQJ�UDQJH�WKDW�LV�DGHTXDWH�IRU�ERWK�SROHV�EHFDXVH�WKH\�DUH�RIGLIIHUHQW�RUGHU� +RZHYHU� VRPH�FRPSURPLVH�FDQ�EH�IRXQG�

�����

Page 250: Mupad Tutorial

** @kZiab\l

:H�RYHUULGH�WKH�DXWRPDWLF�YLHZLQJ�UDQJH�VXJJHVWHG�E\ SORW DQG�UHTXHVW�D�VSHFLILFYLHZLQJ�UDQJH�LQ�YHUWLFDO�GLUHFWLRQ�YLD 9LHZLQJ%R[<5DQJH�

SORW����[ � ����[ � ��A�� [ ������9LHZLQJ%R[<5DQJH ��������

�����

Page 251: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

7KH�YDOXHV�RI�WKH�IROORZLQJ�IXQFWLRQ�KDYH�D�ORZHU�ERXQG� EXW�QR�XSSHU�ERXQG� :HXVH�WKH�DWWULEXWH 9LHZLQJ%R[<5DQJH $XWRPDWLF���� WR�OHW SORW ILQG�D�ORZHUERXQG�IRU�WKH�YLHZLQJ�ER[�E\�LWVHOI�ZKLOVW�UHTXHVWLQJ�D�VSHFLILF�YDOXH�RI 10 IRU�WKHXSSHU�ERXQG�

SORW�H[S�[� VLQ�3, [� � ���[ � ��A���[ � ��A��[ ������ 9LHZLQJ%R[<5DQJH $XWRPDWLF�����

�����

Page 252: Mupad Tutorial

** @kZiab\l

�' )XQFWLRQ�*UDSKV

:H�FRQVLGHU��' H[DPSOHV� L�H�� SORWV�RI�EL�YDULDWH�IXQFWLRQV f(x, y)� $V�QRWHGDERYH� WR�GLVWLQJXLVK�VWDWLF��' DQG�DQLPDWHG��' SORWV� WKH�IRUPHU�QHHG�WKH�RSWLRQ��'��+HUH�LV�D�SORW�RI�WKH�IXQFWLRQ VLQ(x2 + y2)�

SORW�VLQ�[A� � \A��� [ ������ \ ������ ��'�

�����

Page 253: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

,I�VHYHUDO�IXQFWLRQV�DUH�WR�EH�SORWWHG�LQ�WKH�VDPH�JUDSKLFDO�VFHQH� MXVW�SDVV�DVHTXHQFH�RI�IXQFWLRQ�H[SUHVVLRQV� DOO�IXQFWLRQV�DUH�SORWWHG�RYHU�WKH�VSHFLILHGFRPPRQ�UDQJH�

SORW��[A� � \A����� VLQ�[ � \���[ � \��[ ������ \ ������ ��'�

�����

Page 254: Mupad Tutorial

** @kZiab\l

)XQFWLRQV�WKDW�GR�QRW�DOORZ�D�VLPSOH�V\PEROLF�UHSUHVHQWDWLRQ�E\�DQ�H[SUHVVLRQFDQ�DOVR�EH�GHILQHG�E\�D�SURFHGXUH�WKDW�SURGXFHV�D�QXPHULFDO�YDOXH f(x, y) ZKHQFDOOHG�ZLWK�QXPHULFDO�YDOXHV x� y IURP�WKH�SORW�UDQJH� ,Q�WKH�IROORZLQJ�H[DPSOH�ZHFRQVLGHU�WKH�ODUJHVW�HLJHQYDOXH�RI�D�V\PPHWULF 3× 3PDWUL[�WKDW�FRQWDLQV�WZRSDUDPHWHUV x, y� :H�SORW�WKLV�HLJHQYDOXH�DV�D�IXQFWLRQ�RI x DQG y�

I � �[� \� �! PD[�QXPHULF��HLJHQYDOXHV�PDWUL[�>>�\� [� �[ @�

> [� \� [ @�>�[� [� \A�@@����

SORW���'� I� [ ������ \ ������

7KH�QDPHV x� y XVHG�LQ�WKH�VSHFLILFDWLRQ�RI�WKH�SORWWLQJ�UDQJH�SURYLGH�WKH�ODEHOV�RIWKH�FRUUHVSRQGLQJ�D[HV�

�����

Page 255: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

)XQFWLRQV�FDQ�DOVR�EH�GHILQHG�E\ SLHFHZLVH REMHFWV�

SORW�SLHFHZLVH�>[ � \� \�[@� >[ ! \� �\�[�A�@��[ ����� \ ����� ��'�

1RWH�WKDW�WKHUH�DUH�JDSV�LQ�WKH�GHILQLWLRQ�RI�WKH�IXQFWLRQ�DERYH� QR�IXQFWLRQ�YDOXHLV�VSHFLILHG�IRU x = y� 7KLV�GRHV�QRW�FDXVH�DQ\�SUREOHP� EHFDXVH SORW VLPSO\LJQRUHV�SRLQWV�WKDW�GR�QRW�SURGXFH�UHDO�QXPHULFDO�YDOXHV�LI�LW�ILQGV�VXLWDEOH�YDOXHVLQ�WKH�QHLJKERUKRRG� 7KXV� PLVVLQJ�SRLQWV�RU�VLQJXODULWLHV�GR�QRW�UDLVH�DQ�HUURU�

�����

Page 256: Mupad Tutorial

** @kZiab\l

,I�WKH�IXQFWLRQ�LV�QRW�UHDO�YDOXHG�LQ�UHJLRQV�RI�QRQ�]HUR�PHDVXUH� JDSV�ZLOO�EHYLVLEOH�LQ�WKH�RXWSXW� 7KH�IROORZLQJ�IXQFWLRQ�LV�UHDO�YDOXHG�RQO\�LQ�WKH�GLVFx2 + y2 ≤ 1�

SORW�VTUW�� � [A� � \A��� [ ����� \ ����� ��'�

�����

Page 257: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

:KHQ�VHYHUDO�IXQFWLRQV�DUH�SORWWHG�LQ�WKH�VDPH�VFHQH� WKH\�DUH�GUDZQ�LQ�GLIIHUHQWFRORUV�WKDW�DUH�FKRVHQ�DXWRPDWLFDOO\� :LWK�WKH &RORUV DWWULEXWH�RQH�PD\�VSHFLI\�DOLVW�RI�5*% FRORUV�WKDW SORW VKDOO�XVH�SORW�� � [A� � \A�� � � [A� � \A�� [A� � \A��

[ ������ \ ������ ��'�&RORUV >5*%��5HG� 5*%��*UHHQ� 5*%��%OXH@�

�����

Page 258: Mupad Tutorial

** @kZiab\l

$QLPDWHG��' SORWV�RI�IXQFWLRQV�DUH�FUHDWHG�E\�SDVVLQJ�IXQFWLRQ�H[SUHVVLRQVGHSHQGLQJ�RQ�WZR�YDULDEOHV��x DQG y� VD\��DQG�DQ�DQLPDWLRQ�SDUDPHWHU��a� VD\�DQG�VSHFLI\LQJ�D�UDQJH�IRU x� y� DQG a�

SORW�[AD � \AD� [ ����� \ ����� D ����� ��'�

2QFH�WKH�SORW�LV�FUHDWHG� WKH�ILUVW�IUDPH�RI�WKH�SLFWXUH�DSSHDUV�DV�D�VWDWLF�SORW�$IWHU�GRXEOH�FOLFNLQJ�RQ�WKH�SLFWXUH� WKH�JUDSKLFV�WRRO�DOORZV�WR�VWDUW�WKHDQLPDWLRQ�E\�SXVKLQJ�WKH�³6WDUW´�EXWWRQ�

�����

Page 259: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

$Q�DQLPDWLRQ�FRQVLVWV�RI�VHSDUDWH�IUDPHV� 7KH�GHIDXOW�QXPEHU�RI�IUDPHV�LV 50� ,I�DGLIIHUHQW�YDOXH�LV�GHVLUHG� MXVW�SDVV�WKH�DWWULEXWH )UDPHV Q� ZKHUH n LV�WKH�QXPEHURI�IUDPHV�WKDW�VKDOO�EH�FUHDWHG�

SORW�VLQ�D� VLQ�[� � FRV�D� FRV�\�� [ ���� 3,�\ ���� 3,� D ���� 3,� ��'� )UDPHV ���

$SDUW�IURP�WKH�FRORU�VSHFLILFDWLRQ�RU�WKH )UDPHV QXPEHU� WKHUH�LV�D�ODUJH�QXPEHURI�IXUWKHU�DWWULEXWHV�WKDW�PD\�EH�SDVVHG�WR SORW� (DFK�DWWULEXWH�LV�SDVVHG�DV�DQHTXDWLRQ�WR SORW� +HUH� ZH�RQO\�SUHVHQW�VRPH�VHOHFWHG�DWWULEXWHV� 6HFWLRQ³$WWULEXWHV�IRU SORWIXQF�G DQG SORWIXQF�G´�RI�WKH�RQOLQH SORW GRFXPHQWDWLRQSURYLGHV�IXUWKHU�WDEOHV�ZLWK�PRUH�DWWULEXWHV�✞

DWWULEXWH�QDPH SRVVLEOH�YDOXHV PHDQLQJ GHIDXOW

+HLJKW � XQLW��FP SK\VLFDO�KHLJKW�RI�WKHSLFWXUH

�� XQLW��PP

:LGWK �� XQLW��FP SK\VLFDO�ZLGWK�RI�WKHSLFWXUH

��� XQLW��PP

)RRWHU VWULQJ IRRWHU�WH[W �� �QR�IRRWHU�+HDGHU VWULQJ KHDGHU�WH[W �� �QR�KHDGHU�7LWOH VWULQJ WLWOH�WH[W �� �QR�WLWOH�7LWOH3RVLWLRQ >UHDO YDOXH�

UHDO YDOXH@FRRUGLQDWHV�RI�WKHORZHU�OHIW�FRUQHU�RIWKH�WLWOH

*ULG9LVLEOH 758(� )$/6( YLVLELOLW\�RI�³PDMRU´JULG�OLQHV

)$/6(

7DEOH ����� 6RPH�LPSRUWDQW�DWWULEXWHV�IRU�SORWWLQJ��' IXQFWLRQV

�����

Page 260: Mupad Tutorial

** @kZiab\l

DWWULEXWH�QDPH SRVVLEOH�YDOXHV PHDQLQJ GHIDXOW

6XEJULG9LVLEOH 758(� )$/6( YLVLELOLW\�RI�³PLQRU´JULG�OLQHV

)$/6(

$GDSWLYH0HVK LQWHJHU≥ 0 GHSWK�RI�WKH�DGDSWLYHPHVK

$[HV $XWRPDWLF� %R[HG�)UDPH� 2ULJLQ

D[HV�W\SH %R[HG

$[HV9LVLEOH 758(� )$/6( YLVLELOLW\�RI�DOO�D[HV 758($[HV7LWOHV >VWULQJ� VWULQJ�

VWULQJ@WLWOHV�RI�WKH�D[HV >�[���\���]�@

&RORUV OLVW�RI�5*% FRORUV ILOO�FRORUV)UDPHV LQWHJHU≥ 0 QXPEHU�RI�IUDPHV�RI

WKH�DQLPDWLRQ��

/HJHQG9LVLEOH 758(� )$/6( OHJHQG�RQ�RII 758( LI�SORWWLQJ�PRUHWKDQ�RQH�IXQFWLRQ

)LOO&RORU7\SH 'LFKURPDWLF� )ODW�)XQFWLRQDO�0RQRFKURPH� 5DLQERZ

FRORU�VFKHPH 'LFKURPDWLF

0HVK >LQWHJHU≥ 2� LQWHJHU≥ 2@

QXPEHU�RI�³PDMRU´PHVK�SRLQWV

>��� ��@

6XEPHVK >LQWHJHU≥ 0� LQWHJHU≥ 0@

QXPEHU�RI�³PLQRU´PHVK�SRLQWV

>�� �@

6FDOLQJ $XWRPDWLF�&RQVWUDLQHG�8QFRQVWUDLQHG

VFDOLQJ�PRGH 8QFRQVWUDLQHG

7LFNV1XPEHU 1RQH� /RZ� 1RUPDO�+LJK

QXPEHU�RI�ODEHOHGWLFNV

1RUPDO

9LHZLQJ%R[=5DQJH ]PLQ��]PD[ UHVWULFWHG�YLHZLQJUDQJH�LQ z GLUHFWLRQ

$XWRPDWLF��$XWRPDWLF

7DEOH ����� 6RPH�LPSRUWDQW�DWWULEXWHV�IRU�SORWWLQJ��' IXQFWLRQV

/HW�XV�SUHVHQW�VRPH�H[DPSOH�XVHV�RI�WKHVH�DWWULEXWHV� $V�IRU��' SORWV� ZH¶OO�VWDUWZLWK�WKH�HYDOXDWLRQ�GHQVLW\�

�����

Page 261: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

,Q�WKH�IROORZLQJ�H[DPSOH� WKH�GHIDXOW�PHVK�RI 25× 25 VDPSOH�SRLQWV�XVHG�E\ SORWGRHV�QRW�VXIILFH�WR�FUHDWH�D�VXIILFLHQWO\�UHVROYHG�SORW�

SORW�VLQ�[A� � \A��� [ ������ \ ������ ��'�

�����

Page 262: Mupad Tutorial

** @kZiab\l

:H�LQFUHDVH�WKH�QXPEHU�RI�QXPHULFDO�PHVK�SRLQWV�YLD�WKH 6XEPHVK DWWULEXWH�

SORW�VLQ�[A� � \A��� [ ������ \ ������6XEPHVK >�� �@� ��'�

�����

Page 263: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

7KH�IROORZLQJ�FDOO�VSHFLILHV�D�KHDGHU�YLD +HDGHU �7KH�IXQFWLRQ�VLQ�[���\A����*ULG�OLQHV�DWWDFKHG�WR�WKH�WLFNV�DUH�³VZLWFKHG�RQ´�E\ *ULG9LVLEOH DQG6XEJULG9LVLEOH�SORW�VLQ�[ � \A��� [ �� 3,��� 3,� \ ������ ��'�

+HDGHU �7KH IXQFWLRQ VLQ�[ � \A����*ULG9LVLEOH 758(� 6XEJULG9LVLEOH 758(�

�����

Page 264: Mupad Tutorial

** @kZiab\l

:KHQ�VLQJXODULWLHV�DUH�IRXQG�LQ�WKH�IXQFWLRQ� DQ�DXWRPDWLF�FOLSSLQJ�LV�FDOOHGWU\LQJ�WR�UHVWULFW�WKH�YHUWLFDO�YLHZLQJ�UDQJH�LQ�VRPH�ZD\�WR�REWDLQ�D�³UHDVRQDEO\´VFDOHG�SLFWXUH� 7KLV�LV�D�KHXULVWLF�DSSURDFK�WKDW�VRPHWLPHV�QHHGV�D�KHOSLQJDGDSWLRQ�³E\�KDQG�´ ,Q�WKH�IROORZLQJ�H[DPSOH� WKH�DXWRPDWLFDOO\�FKRVHQ�UDQJHEHWZHHQ z ≈ 0 DQG z ≈ 0.8 LQ�YHUWLFDO�GLUHFWLRQ�LV�VXLWDEOH�WR�UHSUHVHQW�WKH�SROH�DWx = 1� y = 1� EXW�LW�GRHV�QRW�SURYLGH�D�JRRG�UHVROXWLRQ�RI�WKH�SROH�DW x = −1, y = 1�

SORW�����[���A� � �\���A�����[���A� � �\���A��A��[ ������ \ ������ 6XEPHVK >�� �@� ��'�

7KHUH�LV�QR�JRRG�YLHZLQJ�UDQJH�WKDW�LV�DGHTXDWH�IRU�ERWK�SROHV�EHFDXVH�WKH\�DUH�RIGLIIHUHQW�RUGHU�

�����

Page 265: Mupad Tutorial

>Zlr�Iehmmbg`3 @kZial�h_�?ng\mbhgl

:H�RYHUULGH�WKH�DXWRPDWLF�YLHZLQJ�UDQJH�VXJJHVWHG�E\ SORW DQG�UHTXHVW�D�VSHFLILFYLHZLQJ�UDQJH�LQ�WKH�YHUWLFDO�GLUHFWLRQ�YLD 9LHZLQJ%R[=5DQJH�

SORW�����[���A� � �\���A�����[���A� � �\���A��A��[ ������ \ ������ 6XEPHVK >�� �@�9LHZLQJ%R[=5DQJH ������� ��'�

�����

Page 266: Mupad Tutorial

** @kZiab\l

7KH�YDOXHV�RI�WKH�IROORZLQJ�IXQFWLRQ�KDYH�D�ORZHU�ERXQG�EXW�QR�XSSHU�ERXQG� :HXVH�WKH�DWWULEXWH 9LHZLQJ%R[=5DQJH $XWRPDWLF���� WR�OHW SORW ILQG�D�ORZHUERXQG�IRU�WKH�YLHZLQJ�ER[�E\�LWVHOI�ZKLOVW�UHTXHVWLQJ�D�VSHFLILF�YDOXH�RI 20 IRU�WKHXSSHU�ERXQG�

SORW���[A��\A� � H[S��[� VLQ�3, \��[ ������ \ ������ ��'�9LHZLQJ%R[=5DQJH $XWRPDWLF�����

�����

Page 267: Mupad Tutorial

:]oZg\^]�Iehmmbg`3 Ikbg\bie^l�Zg]�?bklm�>qZfie^l

9\nYf[]\�Hdgllaf_2 Hjaf[ahd]k�Yf\�>ajkl�=pYehd]k

,Q�WKH�SUHYLRXV�VHFWLRQ� ZH�LQWURGXFHG�WKH SORW FRPPDQG�IRU�SORWWLQJ�IXQFWLRQV�LQ�' DQG��' ZLWK�VLPSOH�FDOOV�LQ�HDV\�V\QWD[� $OWKRXJK�WKHUH�LV�D�ODUJH�QXPEHU�RIDWWULEXWHV�WR�FRQWURO�WKH�JUDSKLFDO�RXWSXW�DQG�WKHUH�LV�DOVR�D�ODUJH�QXPEHU�RIDGGLWLRQDO�SRVVLELOLWLHV�IRU�SORWWLQJ�SRLQW�OLVWV�HWF�� WKHUH�UHPDLQV�D�VHULRXVUHVWULFWLRQ� WKH�DWWULEXWHV�DUH�XVHG�IRU�DOO�IXQFWLRQV�VLPXOWDQHRXVO\�

,I�DWWULEXWHV DWWU�� HWF��DUH�WR�EH�DSSOLHG�WR�IXQFWLRQV�LQGLYLGXDOO\� RQH�QHHGV�WRLQYRNH�D��VOLJKWO\��PRUH�HODERUDWH�FDOOLQJ�V\QWD[�WR�JHQHUDWH�WKH�SORW�

SORW�SORW��)XQFWLRQ�G�I�� [� D���E�� DWWU��� DWWU��� �����SORW��)XQFWLRQ�G�I�� [� D���E�� DWWU��� DWWU��� ��������

,Q�WKLV�FDOO� HDFK�FDOO�RI SORW��)XQFWLRQ�G FUHDWHV�D�VHSDUDWH�REMHFW�WKDWUHSUHVHQWV�WKH�JUDSK�RI�WKH�IXQFWLRQ�SDVVHG�DV�WKH�ILUVW�DUJXPHQW�RYHU�WKH�SORWWLQJUDQJH�SDVVHG�DV�WKH�VHFRQG�DUJXPHQW� $Q�DUELWUDU\�QXPEHU�RI�SORW�DWWULEXWHV�FDQEH�DVVRFLDWHG�ZLWK�HDFK�IXQFWLRQ�JUDSK� 7KH�REMHFWV�WKHPVHOYHV�DUH�QRW�GLVSOD\HGGLUHFWO\� 7KH SORW FRPPDQG�WULJJHUV�WKH�HYDOXDWLRQ�RI�WKH�IXQFWLRQV�RQ�VRPHVXLWDEOH�QXPHULFDO�PHVK�DQG�FDOOV�WKH�UHQGHUHU�WR�GLVSOD\�WKHVH�QXPHULFDO�GDWD�LQWKH�IRUP�VSHFLILHG�E\�WKH�JLYHQ�DWWULEXWHV�

,Q�IDFW� SORW FDOOHG�RQ�D�SODLQ�H[SUHVVLRQ�GRHV�SUHFLVHO\�WKH�VDPH� LQWHUQDOO\� WKHXWLOLW\�IXQFWLRQ SORW��HDV\ JHQHUDWHV�REMHFWV�RI�W\SH SORW��)XQFWLRQ�G RUSORW��)XQFWLRQ�G� UHVSHFWLYHO\� DQG�WKHVH�DUH�UHQGHUHG�

*HQHUDO�3ULQFLSOHV

,Q�JHQHUDO� JUDSKLFDO�VFHQHV�DUH�FROOHFWLRQV�RI�³JUDSKLFDO�SULPLWLYHV�´ 7KHUH�DUHVLPSOH�SULPLWLYHV�VXFK�DV�SRLQWV� OLQH�VHJPHQWV� SRO\JRQV� UHFWDQJOHV�DQG�ER[HV�FLUFOHV�DQG�VSKHUHV� KLVWRJUDP�SORWV� SLH�FKDUWV�HWF� $Q�H[DPSOH�RI�D�PRUHDGYDQFHG�SULPLWLYH�LV SORW��9HFWRU)LHOG�G WKDW�UHSUHVHQWV�D�FROOHFWLRQ�RI�DUURZVDWWDFKHG�WR�D�UHJXODU�PHVK�YLVXDOL]LQJ�D�YHFWRU�ILHOG�RYHU�D�UHFWDQJXODU�UHJLRQ�LQR2� <HW�PRUH�DGYDQFHG�SULPLWLYHV�DUH�IXQFWLRQ�JUDSKV�DQG�SDUDPHWUL]HG�FXUYHVWKDW�FRPH�HTXLSSHG�ZLWK�VRPH�LQWHUQDO�LQWHOOLJHQFH� H�J�� NQRZLQJ�KRZ�WR�HYDOXDWH

�����

Page 268: Mupad Tutorial

** @kZiab\l

WKHPVHOYHV�QXPHULFDOO\�RQ�DGDSWLYH�PHVKHV�DQG�KRZ�WR�FOLS�WKHPVHOYHV�LQ�FDVH�RIVLQJXODULWLHV� $V�H[DPSOHV�IRU�KLJKO\�DGYDQFHG�SULPLWLYHV�LQ�WKH SORW OLEUDU\� ZHPHQWLRQ SORW��2GH�G� SORW��2GH�G� SORW��6WUHDPOLQHV�G� DQGSORW��,PSOLFLW�G� SORW��,PSOLFLW�G� 7KH�ILUVW�DXWRPDWLFDOO\�VROYH�V\VWHPV�RIRUGLQDU\�GLIIHUHQWLDO�HTXDWLRQV�QXPHULFDOO\�DQG�GLVSOD\�WKH�VROXWLRQV�JUDSKLFDOO\�7KH�ODWWHU�GLVSOD\�WKH�VROXWLRQ�FXUYHV�RU�VXUIDFHV�RI�HTXDWLRQV f(x, y) = 0 RUf(x, y, z) = 0� UHVSHFWLYHO\� E\�VROYLQJ�WKHVH�HTXDWLRQV�QXPHULFDOO\� D�W\SLFDO�WDVNLQ��UHDO��DOJHEUDLF�JHRPHWU\�

$OO�WKHVH�SULPLWLYHV�DUH�MXVW�³REMHFWV´�UHSUHVHQWLQJ�VRPH�JUDSKLFDO�HQWLWLHV� 7KH\DUH�QRW�UHQGHUHG�GLUHFWO\�ZKHQ�WKH\�DUH�FUHDWHG� EXW�MXVW�VHUYH�DV�GDWD�VWUXFWXUHVHQFRGLQJ�WKH�JUDSKLFDO�PHDQLQJ� FROOHFWLQJ�DWWULEXWHV�GHILQLQJ�WKH�SUHVHQWDWLRQVW\OH�DQG�SURYLGLQJ�QXPHULFDO�URXWLQHV�WR�FRQYHUW�WKH�LQSXW�SDUDPHWHUV�WR�VRPHQXPHULFDO�GDWD�WKDW�DUH�WR�EH�VHQW�WR�WKH�UHQGHUHU�

$Q�DUELWUDU\�QXPEHU�RI�VXFK�SULPLWLYHV�FDQ�EH�FROOHFWHG�WR�IRUP�D�JUDSKLFDO�VFHQH�)LQDOO\� D�FDOO�WR SORW SDVVLQJ�D�VHTXHQFH�RI�DOO�SULPLWLYHV�LQ�WKH�VFHQH�LQYRNHV�WKHUHQGHUHU�WR�GUDZ�WKH�SORW� 7KH�IROORZLQJ�H[DPSOH�VKRZV�WKH�JUDSK�RI�WKH�IXQFWLRQf(x) = x · VLQ(x)� $W�WKH�SRLQW (x0, f(x0))� D�JUDSKLFDO�SRLQW�LV�LQVHUWHG�DQG�WKHWDQJHQW�WR�WKH�JUDSK�WKURXJK�WKLV�SRLQW�LV�DGGHG�

I � [ �! [ VLQ�[��[�� ���� G[ � ��J � SORW��)XQFWLRQ�G�I�[�� [ ���� 3,��S � SORW��3RLQW�G�[�� I�[��� 3RLQW6L]H ��� XQLW��PP��W � SORW��/LQH�G�>[� � G[� I�[�� � I�[�� G[@�

>[� � G[� I�[�� � I�[�� G[@��

�����

Page 269: Mupad Tutorial

:]oZg\^]�Iehmmbg`3 Ikbg\bie^l�Zg]�?bklm�>qZfie^l

7KH�SLFWXUH�LV�GUDZQ�E\�FDOOLQJ SORW�

SORW�J� W� S�

(DFK�SULPLWLYH�DFFHSWV�D�YDULHW\�RI�SORW�DWWULEXWHV�WKDW�PD\�EH�SDVVHG�DV�DVHTXHQFH�RI�HTXDWLRQV $WWULEXWH1DPH $WWULEXWH9DOXH WR�WKH�JHQHUDWLQJ�FDOO�0RVW�SURPLQHQWO\� DOPRVW�HYHU\�SULPLWLYH�DOORZV�WR�VHW�WKH�FRORU�H[SOLFLWO\�

J � SORW��)XQFWLRQ�G�I�[�� [ ���� 3,�&RORU 5*%��%OXH��

$OWHUQDWLYHO\� WKH�JHQHUDWHG�REMHFWV�DOORZ�WR�VHW�DWWULEXWHV�YLD�VORW�DVVLJQPHQWV�RIWKH�IRUP REMHFW��$WWULEXWH1DPH� $WWULEXWH9DOXH DV�LQS��&RORU � 5*%��%ODFN�S��3RLQW6L]H � ��� XQLW��PP�W��&RORU � 5*%��5HG�W��/LQH:LGWK � ��� XQLW��PP�

7KH�KHOS�SDJH�RI�HDFK�SULPLWLYH�SURYLGHV�D�OLVW�RI�DOO�DWWULEXWHV�WKH�SULPLWLYH�LVUHDFWLQJ�WR�

�����

Page 270: Mupad Tutorial

** @kZiab\l

&HUWDLQ�DWWULEXWHV� VXFK�DV�D[HV�VW\OH� WKH�YLVLELOLW\�RI�JULG�OLQHV�LQ�WKH�EDFNJURXQGHWF��DUH�DVVRFLDWHG�ZLWK�WKH�ZKROH�VFHQH�UDWKHU�WKDQ�ZLWK�WKH�LQGLYLGXDO�SULPLWLYHV�7KHVH�DWWULEXWHV�PD\�EH�LQFOXGHG�LQ�WKH SORW FDOO�

SORW�J� W� S� *ULG9LVLEOH 758(�

$V�H[SODLQHG�LQ�GHWDLO�LQ�6HFWLRQ�³7KH�)XOO�3LFWXUH´�RQ�SDJH ������ WKH SORWFRPPDQG�DXWRPDWLFDOO\�HPEHGV�WKH�JUDSKLFDO�SULPLWLYHV�LQ�D�FRRUGLQDWH�V\VWHP�ZKLFK�LQ�WXUQ�LV�HPEHGGHG�LQ�D�JUDSKLFDO�VFHQH� ZKLFK�LV�GUDZQ�LQVLGH�D�FDQYDV�7KH�YDULRXV�DWWULEXWHV�DVVRFLDWHG�ZLWK�WKH�JHQHUDO�DSSHDUDQFH�RI�WKH�ZKROH�SLFWXUHDUH�DVVRFLDWHG�ZLWK�WKHVH�³JURXSLQJ�VWUXFWXUHV�´ $ FRQFLVH�OLVW�RI�DOO�VXFKDWWULEXWHV�LV�SURYLGHG�RQ�WKH�KHOS�SDJHV�RI�WKH�FRUUHVSRQGLQJ�SULPLWLYHV�SORW��&DQYDV� SORW��6FHQH�G� SORW��6FHQH�G� SORW��&RRUGLQDWH6\VWHP�G� DQGSORW��&RRUGLQDWH6\VWHP�G� UHVSHFWLYHO\�

7KH�0X3$'� REMHFW�EURZVHU�DOORZV�WR�VHOHFW�HDFK�SULPLWLYH�LQ�WKH�SORW� $IWHUVHOHFWLRQ� WKH�DWWULEXWHV�RI�WKH�SULPLWLYH�FDQ�EH�FKDQJHG�LQWHUDFWLYHO\�LQ�WKHSURSHUW\�LQVSHFWRU��VHH SDJH �������

1H[W� ZH�ZLVK�WR�GHPRQVWUDWH�DQ�DQLPDWLRQ� ,W�LV�UHPDUNDEO\�VLPSOH�WR�JHQHUDWHDQ�DQLPDWHG�SLFWXUH� *RLQJ�EDFN�WR�RXU�SUHYLRXV�H[DPSOH�ZLWK�WKH�WDQJHQW�WR�WKHIXQFWLRQ�JUDSK� ZH�ZDQW�WR�OHW�WKH�SRLQW x0 DW�ZKLFK�WKH�WDQJHQW�LV�DGGHG�PRYHDORQJ�WKH�JUDSK�RI�WKH�IXQFWLRQ� ,Q�0X3$'��\RX�GR�QRW�QHHG�WR�FUHDWH�DQDQLPDWLRQ�IUDPH�E\�IUDPH� ,QVWHDG� HDFK�SULPLWLYH�FDQ�EH�WROG�WR�DQLPDWH�LWVHOI�E\

�����

Page 271: Mupad Tutorial

:]oZg\^]�Iehmmbg`3 Ikbg\bie^l�Zg]�?bklm�>qZfie^l

VLPSO\�GHILQLQJ�LW�ZLWK�D�V\PEROLF�DQLPDWLRQ�SDUDPHWHU�DQG�DGGLQJ�DQ�DQLPDWLRQUDQJH�IRU�WKLV�SDUDPHWHU� 6WDWLF�DQG�DQLPDWHG�REMHFWV�FDQ�EH�PL[HG�DQG�UHQGHUHGWRJHWKHU� 7KH�VWDWLF�IXQFWLRQ�JUDSK�RI f(x) XVHG�LQ�WKH�SUHYLRXV�SORW�LV�UHF\FOHG�IRUWKH�DQLPDWLRQ� 7KH�JUDSKLFDO�SRLQW�DW (x0, f(x0)) DQG�WKH�WDQJHQW�WKURXJK�WKLVSRLQW�VKDOO�EH�DQLPDWHG�XVLQJ�WKH�FRRUGLQDWH [� DV�WKH�DQLPDWLRQ�SDUDPHWHU�'HOHWLQJ�LWV�YDOXH�VHW�DERYH� ZH�FDQ�XVH�WKH�VDPH�GHILQLWLRQV�DV�EHIRUH� QRZ�ZLWK�DV\PEROLF [�� :H�MXVW�KDYH�WR�DGG�WKH�UDQJH�VSHFLILFDWLRQ [� ���� 3, IRU�WKLVSDUDPHWHU�

GHOHWH [��G[ � ��VTUW�� � I�[��A���S � SORW��3RLQW�G�[�� I�[��� [� ���� 3,�

&RORU 5*%��%ODFN�3RLQW6L]H ��� XQLW��PP��

W � SORW��/LQH�G�>[� � G[� I�[�� � I�[�� G[@�>[� � G[� I�[�� � I�[�� G[@�[� ���� 3,� &RORU 5*%��5HG�/LQH:LGWK ��� XQLW��PP��

SORW�J� S� W� *ULG9LVLEOH 758(�

'HWDLOV�RQ�DQLPDWLRQV�DQG�IXUWKHU�H[DPSOHV�DUH�SURYLGHG�VWDUWLQJ�RQ�SDJH ������

�����

Page 272: Mupad Tutorial

** @kZiab\l

:H�VXPPDUL]H�WKH�FRQVWUXFWLRQ�SULQFLSOHV�IRU�JUDSKLFV�ZLWK�0X3$'¶V SORWOLEUDU\�✞

� *UDSKLFDO�VFHQHV�DUH�EXLOW�IURP�JUDSKLFDO�SULPLWLYHV� 7KH�VHFWLRQVWDUWLQJ�RQ�SDJH ����� SURYLGHV�D�VXUYH\�RI�WKH�SULPLWLYHV�WKDW�DUHDYDLODEOH�LQ�WKH SORW OLEUDU\�

� 3ULPLWLYHV � JHQHUDWHG � E\ � WKH SORW OLEUDU\ � DUH � V\P�EROLF � REMHFWV � WKDW � DUH � QRW � UHQGHUHG � GLUHFWO\� 7KH � FDOOSORW�3ULPLWLYH��3ULPLWLYH��«� JHQHUDWHV�WKH�SLFWXUHV�

� *UDSKLFDO � DWWULEXWHV � DUH � VSHFLI LHG � DV � HTXDWLRQV$WWULEXWH1DPH 9DOXH� $WWULEXWHV � IRU � D � JUDSKLFDO � SULPLWLYHPD\�EH�SDVVHG�LQ�WKH�FDOO�WKDW�JHQHUDWHV�WKH�SULPLWLYH� 7KH�KHOSSDJH�RI�HDFK�SULPLWLYH�SURYLGHV�D�FRPSOHWH�OLVW�RI�DOO�DWWULEXWHV�WKHSULPLWLYH�UHDFWV�WR�

� $WWULEXWHV � GHWHUPLQLQJ � WKH � JHQHUDO � DSSHDUDQFH � RI � WKH � SLF�WXUH � PD\ � EH � JLYHQ � LQ � WKH SORW FDOO� 7KH � KHOS � SDJHV � RISORW��&RRUGLQDWH6\VWHP�G� SORW��&RRUGLQDWH6\VWHP�G�SORW��6FHQH�G� SORW��6FHQH�G� DQG SORW��&DQYDV� UHVSHFWLYHO\�SURYLGH�D�FRPSOHWH�OLVW �RI �DOO �DWWULEXWHV�GHWHUPLQLQJ�WKH�JHQHUDODSSHDUDQFH�

� $OPRVW�DOO�DWWULEXWHV�FDQ�EH�FKDQJHG�LQWHUDFWLYHO\�LQ�WKH�YLHZHU�

� 3UHVHQWO\� �' DQG��' SORWV�DUH�VWULFWO\�VHSDUDWHG� 2EMHFWV�RI�GLIIHU�HQW�GLPHQVLRQ�FDQQRW�EH�UHQGHUHG�LQ�WKH�VDPH�SORW�

� $QLPDWLRQV�DUH�QRW�FUHDWHG�IUDPH�E\�IUDPH�EXW�SHU�REMHFW��VHH�DOVR6HFWLRQ ������ $Q�REMHFW�LV�DQLPDWHG�E\�JHQHUDWLQJ�LW�ZLWK�D�V\PEROLFDQLPDWLRQ�SDUDPHWHU�DQG�SURYLGLQJ�D�UDQJH�IRU�WKLV�SDUDPHWHU�LQWKH�JHQHUDWLQJ�FDOO� 7KH�VHFWLRQ�VWDUWLQJ�RQ�SDJH ����� SURYLGHV�IXU�WKHU�GHWDLOV�RQ�DQLPDWLRQV�

3UHVHQWO\� LW�LV�QRW�SRVVLEOH�WR�DGG�REMHFWV�WR�DQ�H[LVWLQJ�SORW� +RZHYHU� XVLQJDQLPDWLRQV� LW�LV�SRVVLEOH�WR�OHW�SULPLWLYHV�DSSHDU�RQH�DIWHU�DQRWKHU�LQ�WKHDQLPDWHG�SLFWXUH�

�����

Page 273: Mupad Tutorial

:]oZg\^]�Iehmmbg`3 Ikbg\bie^l�Zg]�?bklm�>qZfie^l

6RPH�([DPSOHV

([DPSOH��� :H�ZLVK�WR�YLVXDOL]H�WKH�LQWHUSRODWLRQ RI�D�GLVFUHWH�GDWD�VDPSOH�E\FXELF�VSOLQHV� )LUVW� ZH�GHILQH�WKH�GDWD�VDPSOH� FRQVLVWLQJ�RI�D�OLVW�RI�SRLQWV[[x1, y1], [x2, y2], . . . ]� 6XSSRVH�WKH\�DUH�HTXLGLVWDQW�VDPSOH�SRLQWV�IURP�WKH�JUDSKRI�WKH�IXQFWLRQ f(x) = x · e−x · VLQ(5x)�

I � [ �! [ H[S��[� VLQ�� [��GDWD � >>L� I�L�@ � L ���� VWHS ���@�

:H�XVH QXPHULF��FXELF6SOLQH WR�GHILQH�WKH�FXELF�VSOLQH�LQWHUSRODQW�WKURXJKWKHVH�GDWD�

6 � QXPHULF��FXELF6SOLQH�RS�GDWD���

7KH�SORW�VKDOO�FRQVLVW�RI�WKH�IXQFWLRQ f(x) WKDW�SURYLGHV�WKH�GDWD�RI�WKH�VDPSOHSRLQWV�DQG�RI�WKH�VSOLQH�LQWHUSRODQW S(x)� 7KH�JUDSKV�RI f(x) DQG S(x) DUHJHQHUDWHG�YLD SORW��)XQFWLRQ�G� 7KH�GDWD�SRLQWV�DUH�SORWWHG�DV�DSORW��3RLQW/LVW�G�

�����

Page 274: Mupad Tutorial

** @kZiab\l

SORW�SORW��)XQFWLRQ�G�I�[�� [ ����� &RORU 5*%��5HG�/HJHQG7H[W H[SU�WH[W�I�[����

SORW��)XQFWLRQ�G�6�[�� [ ����� &RORU 5*%��%OXH�/HJHQG7H[W �VSOLQH LQWHUSRODQW���

SORW��3RLQW/LVW�G�GDWD� &RORU 5*%��%ODFN��*ULG9LVLEOH 758(� 6XEJULG9LVLEOH 758(�/HJHQG9LVLEOH 758(�

�����

Page 275: Mupad Tutorial

:]oZg\^]�Iehmmbg`3 Ikbg\bie^l�Zg]�?bklm�>qZfie^l

([DPSOH��� $ F\FORLG�LV�WKH�FXUYH�WKDW�\RX�JHW�ZKHQ�IROORZLQJ�D�SRLQW�IL[HG�WR DZKHHO�UROOLQJ�DORQJ�D�VWUDLJKW�OLQH� :H�YLVXDOL]H�WKLV�FRQVWUXFWLRQ�E\�DQ�DQLPDWLRQLQ�ZKLFK�ZH�XVH�WKH x FRRUGLQDWH�RI�WKH�KXE�DV�WKH�DQLPDWLRQ�SDUDPHWHU� 7KHZKHHO�LV�UHDOL]HG�DV�D�FLUFOH� 7KHUH�DUH���SRLQWV�IL[HG�WR�WKH�ZKHHO� D�JUHHQ�SRLQW�RQWKH�ULP� D�UHG�SRLQW�LQVLGH�WKH�ZKHHO�DQG�D�EOXH�SRLQW�RXWVLGH�WKH�ZKHHO�

:KHHO5DGLXV � ��U � >��� :KHHO5DGLXV� ��� :KHHO5DGLXV� ��� :KHHO5DGLXV@�:KHHO&HQWHU � >[� :KHHO5DGLXV@�:KHHO5LP � SORW��&LUFOH�G�:KHHO5DGLXV� :KHHO&HQWHU�

[ ���� 3,�/LQH&RORU 5*%��%ODFN��

:KHHO+XE � SORW��3RLQW�G�:KHHO&HQWHU� [ ���� 3,�3RLQW&RORU 5*%��%ODFN��

:KHHO6SRNH � SORW��/LQH�G�:KHHO&HQWHU�>:KHHO&HQWHU>�@ � PD[�U� VLQ�[��:KHHO&HQWHU>�@ � PD[�U� FRV�[�@�

[ ���� 3,� /LQH&RORU 5*%��%ODFN��FRORU� >5*%��5HG� 5*%��*UHHQ� 5*%��%OXH@�IRU L IURP � WR � GR3RLQW>L@ � SORW��3RLQW�G�>:KHHO&HQWHU>�@ � U>L@ VLQ�[��

:KHHO&HQWHU>�@ � U>L@ FRV�[�@�[ ���� 3,�

3RLQW&RORU FRORU>L@�3RLQW6L]H ��� XQLW��PP��

&\FORLG>L@ � SORW��&XUYH�G�>\ � U>L@ VLQ�\��:KHHO5DGLXV � U>L@ FRV�\�@�

\ ���[� [ ���� 3,�/LQH&RORU FRORU>L@��

HQGBIRU�

�����

Page 276: Mupad Tutorial

** @kZiab\l

SORW�:KHHO5LP� :KHHO+XE� :KHHO6SRNH� 3RLQW>L@ � L �����&\FORLG>L@ � L ����� 6FDOLQJ &RQVWUDLQHG�:LGWK ��� XQLW��PP� +HLJKW �� XQLW��PP�

�����

Page 277: Mupad Tutorial

:]oZg\^]�Iehmmbg`3 Ikbg\bie^l�Zg]�?bklm�>qZfie^l

([DPSOH��� :H�ZLVK�WR�YLVXDOL]H�WKH�VROXWLRQ�RI�WKH�RUGLQDU\�GLIIHUHQWLDOHTXDWLRQ��2'(� y′(x) = −y(x)3 + FRV(x) ZLWK�WKH�LQLWLDO�FRQGLWLRQ y(0) = 0� 7KHVROXWLRQ�VKDOO�EH�GUDZQ�WRJHWKHU�ZLWK�WKH�YHFWRU�ILHOG v⃗(x, y) = (1,−y3 + FRV(x))DVVRFLDWHG�ZLWK�WKLV�2'( �DORQJ�WKH�VROXWLRQ�FXUYH� WKH�YHFWRUV�RI�WKLV�ILHOG�DUHWDQJHQWV�RI�WKH�FXUYH�� :H�XVH QXPHULF��RGHVROYH� WR�JHQHUDWH�WKH�VROXWLRQ�DV�DIXQFWLRQ�SORW� 6LQFH�WKH�QXPHULFDO�LQWHJUDWRU�UHWXUQV�WKH�UHVXOW�DV�D�OLVW�RI�RQHIORDWLQJ�SRLQW�YDOXH� RQH�KDV�WR�SDVV�WKH�VLQJOH�OLVW�HQWU\�DV Y (x)[1] WRSORW��)XQFWLRQ�G� 7KH�YHFWRU�ILHOG�LV�JHQHUDWHG�YLD SORW��9HFWRU)LHOG�G�

I � �[� \� �! �\A� � FRV�[��< � QXPHULF��RGHVROYH��

QXPHULF��RGH�YHFWRUILHOG�^\�[� I�[� \��\��� �`� >\�[�@���

SORW�SORW��9HFWRU)LHOG�G�>�� I�[� \�@� [ ����� \ ������

&RORU 5*%��%OXH� 0HVK >��� ��@��SORW��)XQFWLRQ�G�<�[�>�@� [ ����� &RORU 5*%��5HG�

/LQH:LGWK ��� XQLW��PP��*ULG9LVLEOH 758(� 6XEJULG9LVLEOH 758(� $[HV )UDPH�

�����

Page 278: Mupad Tutorial

** @kZiab\l

([DPSOH��� 7KH�UDGLXV r RI�DQ�REMHFW�ZLWK�URWDWLRQDO�V\PPHWU\�DURXQG�WKHx�D[LV�LV�PHDVXUHG�DW�YDULRXV x SRVLWLRQV�

x ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ����r(x) ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ����

$ VSOLQH�LQWHUSRODWLRQ�LV�XVHG�WR�GHILQH�D�VPRRWK�IXQFWLRQ r(x) IURP�WKHPHDVXUHPHQWV�

VDPSOHSRLQWV � >����� ����@� >����� ����@� >����� ����@� >����� ����@�>����� ����@� >����� ����@� >����� ����@� >����� ����@�>����� ����@� >����� ����@� >����� ����@� >����� ����@�

U � QXPHULF��FXELF6SOLQH�VDPSOHSRLQWV��

:H�UHFRQVWUXFW�WKH�REMHFW�DV�D�VXUIDFH�RI�UHYROXWLRQ�YLD SORW��;5RWDWH� 7KHURWDWLRQ�DQJOH�LV�UHVWULFWHG�WR�D�UDQJH�WKDW�OHDYHV�D�JDS�LQ�WKH�VXUIDFH� 7KH�VSOLQHFXUYH�DQG�WKH�VDPSOH�SRLQWV�DUH�DGGHG�DV�D SORW��&XUYH�G DQG�DSORW��3RLQW/LVW�G� UHVSHFWLYHO\� DQG�GLVSOD\HG�LQ�WKLV�JDS�

�����

Page 279: Mupad Tutorial

:]oZg\^]�Iehmmbg`3 Ikbg\bie^l�Zg]�?bklm�>qZfie^l

SORW�SORW��;5RWDWH�U�[�� [ �����

$QJOH5DQJH ��� 3,����� 3,�&RORU 5*%��0X3$'*ROG��

SORW��&XUYH�G�>[� �� U�[�@� [ �����/LQH:LGWK ��� XQLW��PP�&RORU 5*%��%ODFN��

SORW��3RLQW/LVW�G�>>S>�@� �� S>�@@ � S LQ VDPSOHSRLQWV@�3RLQW6L]H ��� XQLW��PP�&RORU 5*%��5HG��

&DPHUD'LUHFWLRQ >��� ���� ��@�

�����

Page 280: Mupad Tutorial

** @kZiab\l

L`]�>mdd�Ha[lmj]2 ?jYh`a[Yd�Lj]]k

)RU�D�IXOO�XQGHUVWDQGLQJ�RI�WKH�LQWHUDFWLYH�IHDWXUHV�RI�WKH�YLHZHU�WR�EH�GLVFXVVHG�LQWKH�QH[W�VHFWLRQ� ZH�QHHG�WR�NQRZ�WKH�VWUXFWXUH�RI�D�0X3$'� SORW�DV�D�³JUDSKLFDOWUHH�´

7KH�URRW�LV�WKH�³FDQYDV´� WKLV�LV�WKH�GUDZLQJ�DUHD�LQWR�ZKLFK DOO�SDUWV�RI�WKH�SORW�DUHUHQGHUHG� 7KH�W\SH�RI�WKH�FDQYDV�REMHFW�LV SORW��&DQYDV� ,WV�SK\VLFDO�VL]H�PD\�EHVSHFLILHG�YLD�WKH�DWWULEXWHV :LGWK DQG +HLJKW�

,QVLGH�WKH�FDQYDV� RQH�RU�PRUH�³JUDSKLFDO�VFHQHV´�FDQ�EH�GLVSOD\HG� $OO�RI�WKHPPXVW�EH�RI�WKH�VDPH�GLPHQVLRQ� L�H�� REMHFWV�RI�W\SH SORW��6FHQH�G RUSORW��6FHQH�G� UHVSHFWLYHO\� 7KH�IROORZLQJ�FRPPDQG�GLVSOD\V�IRXU�GLIIHUHQW��'VFHQHV� :H�VHW�WKH %RUGHU:LGWK IRU�DOO�REMHFWV�RI�W\SH SORW��6FHQH�G WR�VRPHSRVLWLYH�YDOXH�VR�WKDW�WKH�GUDZLQJ�DUHDV�RI�WKH�VFHQHV�EHFRPH�YLVLEOH�PRUH�FOHDUO\�

6� � SORW��6FHQH�G�SORW��6SKHUH��� >�� �� �@�&RORU 5*%��5HG��

%DFNJURXQG6W\OH /HIW5LJKW��6� � SORW��6FHQH�G�SORW��%R[������� ������ ������

&RORU 5*%��*UHHQ��%DFNJURXQG6W\OH 7RS%RWWRP��

6� � SORW��6FHQH�G�SORW��&RQH��� >�� �� �@� >�� �� �@�&RORU 5*%��%OXH��

%DFNJURXQG6W\OH 3\UDPLG��6� � SORW��6FHQH�G�SORW��&RQH��� >�� �� �@� >�� �� �@�

&RORU 5*%��2UDQJH��%DFNJURXQG6W\OH )ODW�%DFNJURXQG&RORU 5*%��*UD\��

�����

Page 281: Mupad Tutorial

Ma^�?nee�Ib\mnk^3 @kZiab\Ze�Mk^^l

SORW�SORW��&DQYDV�6�� 6�� 6�� 6��:LGWK �� XQLW��PP� +HLJKW �� XQLW��PP�$[HV 1RQH� %RUGHU:LGWK ��� XQLW��PP�SORW��6FHQH�G��%RUGHU:LGWK ��� XQLW��PP��

6HH�6HFWLRQ�³/D\RXW�RI�&DQYDV�DQG�6FHQHV´�RI�WKH�RQOLQH SORW GRFXPHQWDWLRQ�IRUGHWDLOV�RQ�KRZ�WKH�OD\RXW�RI�D�FDQYDV�FRQWDLQLQJ�VHYHUDO�VFHQHV�LV�VHW�

&RRUGLQDWH�V\VWHPV�H[LVW�LQVLGH�D��' VFHQH�RU�D��' VFHQH�DQG�DUH�RI�W\SHSORW��&RRUGLQDWH6\VWHP�G RU SORW��&RRUGLQDWH6\VWHP�G� UHVSHFWLYHO\� 7KHUHPD\�EH�RQH�RU�PRUH�FRRUGLQDWH�V\VWHPV�LQ�D�VFHQH� ,QVLGH�WKH�FRRUGLQDWH�V\VWHPV�DQ�DUELWUDU\�QXPEHU�RI�SULPLWLYHV��RI�WKH�DSSURSULDWH�GLPHQVLRQ��FDQ�EHGLVSOD\HG� 7KXV� ZH�DOZD\V�KDYH�D�FDQYDV� FRQWDLQLQJ�RQH�RU�PRUH�VFHQHV� ZLWKHDFK�VFHQH�FRQWDLQLQJ�RQH�RU�PRUH�FRRUGLQDWH�V\VWHPV� 7KH�JUDSKLFDO�SULPLWLYHV�RU�JURXSV�RI�VXFK�SULPLWLYHV��DUH�FRQWDLQHG�LQ�WKH�FRRUGLQDWH�V\VWHPV�

�����

Page 282: Mupad Tutorial

** @kZiab\l

+HQFH� DQ\�0X3$'�SORW�KDV�WKH�IROORZLQJ�JHQHUDO�VWUXFWXUH�

&DQYDV

6FHQH��

&RRUGLQDWH6\VWHP��

3ULPLWLYH��

3ULPLWLYH��

«

&RRUGLQDWH6\VWHP��

3ULPLWLYH

«

«

6FHQH��

&RRUGLQDWH6\VWHP��

3ULPLWLYH

«

«

«

7KLV�LV�WKH�³JUDSKLFDO�WUHH´�WKDW�LV�GLVSOD\HG�E\�WKH�³REMHFW�EURZVHU´��VHHSDJH �������

6KRUWFXWV� )RU�VLPSOH�SORWV�FRQWDLQLQJ�SULPLWLYHV�LQVLGH�RQH�FRRUGLQDWH�V\VWHPRQO\�WKDW�UHVLGHV�LQ�D�VLQJOH�VFHQH�RI�WKH�FDQYDV� LW�ZRXOG�EH�UDWKHU�FXPEHUVRPH�WRJHQHUDWH�WKH�SLFWXUH�E\�D�FRPPDQG�VXFK�DV

SORW�SORW��&DQYDV�SORW��6FHQH�G�SORW��&RRUGLQDWH6\VWHP�G�3ULPLWLYH�� 3ULPLWLYH�� �������

,Q�IDFW� WKH�FRPPDQG

SORW�3ULPLWLYH�� 3ULPLWLYH�� ����

VXIILFHV� ,W�LV�D�VKRUWFXW�RI�WKH�DERYH�FRPPDQG� ,W�DXWRPDWLFDOO\�JHQHUDWHV�DFRRUGLQDWH�V\VWHP�WKDW�FRQWDLQV�WKH�SULPLWLYHV� HPEHGGHG�LQ�D�VFHQH�ZKLFK�LVDXWRPDWLFDOO\�SODFHG�LQVLGH�D�FDQYDV�REMHFW�

�����

Page 283: Mupad Tutorial

Ma^�?nee�Ib\mnk^3 @kZiab\Ze�Mk^^l

7KXV� WKLV�FRPPDQG�LPSOLFLWO\�FUHDWHV�WKH�JUDSKLFDO�WUHH

&DQYDV

6FHQH

&RRUGLQDWH6\VWHP

3ULPLWLYH��

3ULPLWLYH��

«

WKDW�LV�GLVSOD\HG�LQ�WKH�REMHFW�EURZVHU�

�����

Page 284: Mupad Tutorial

** @kZiab\l

Na]o]j$ :jgok]j$ Yf\�Afkh][lgj2 Afl]jY[lan]�EYfahmdYlagf

$IWHU�D�SORW�FRPPDQG�LV�H[HFXWHG�LQ�D�QRWHERRN� WKH�SORW�ZLOO�W\SLFDOO\�DSSHDUEHORZ�WKH�LQSXW�UHJLRQ� HPEHGGHG�LQ�WKH�QRWHERRN� 7R�³DFWLYDWH´�WKH�SORW� FOLFN�RQLW�ZLWK�WKH�OHIW�PRXVH�EXWWRQ� 7KLV�OHDYHV�WKH�SORW�HPEHGGHG�LQ�WKH�QRWHERRN� EXWWKH�QRWHERRN�PHQXV�DUH�UHSODFHG�E\�WKH�JUDSKLFV�PHQXV�

0DNH�VXUH�WKDW�WKH�LWHP�µ2EMHFW�%URZVHU¶�LV�HQDEOHG�XQGHU�WKH�µ9LHZ¶�PHQX� $SDQH�WLWOHG�µ2EMHFW�%URZVHU¶�LV�YLVLEOH�FRQWDLQLQJ�WZR�VXE�ZLQGRZV� ZKLFK�ZH�ZLOOUHIHU�WR�DV�WKH�µREMHFW�EURZVHU¶�DQG�WKH�µSURSHUW\�LQVSHFWRU�¶

,Q�WKH�REMHFW�EURZVHU� WKH�JUDSKLFDO�WUHH�RI�WKH�SORW�DV�LQWURGXFHG�LQ�WKH�SUHFHGLQJVHFWLRQ�LV�YLVLEOH� ,W�DOORZV�WR�VHOHFW�DQ\�QRGH�RI�WKH�JUDSKLFDO�WUHH�E\�D�PRXVH�FOLFN�

$IWHU�VHOHFWLQJ�DQ�REMHFW�LQ�WKH�REMHFW�EURZVHU� WKH�FRUUHVSRQGLQJ�SDUW�RI�WKH�SORWLV�KLJKOLJKWHG�LQ�VRPH�ZD\�DOORZLQJ�WR�LGHQWLI\�WKH�VHOHFWHG�REMHFW�YLVXDOO\� 7KHSURSHUW\�LQVSHFWRU�WKHQ�GLVSOD\V�WKH�DWWULEXWHV�WKH�VHOHFWHG�REMHFW�UHDFWV�WR� )RUH[DPSOH� IRU�DQ�REMHFW�RI�W\SH )XQFWLRQ�G� WKH�DWWULEXWHV�YLVLEOH�LQ�WKH�SURSHUW\LQVSHFWRU�DUH�FDWHJRUL]HG�DV�µ'HILQLWLRQ�¶ µ$QLPDWLRQ�¶ µ$QQRWDWLRQ�¶ µ&DOFXODWLRQ�¶DQG�µ6W\OH¶�ZLWK�WKH�ODWWHU�VSOLW�LQWR�WKH�VXE�FDWHJRULHV��6W\OH�RI��µ/LQHV�¶ �6W\OH�RI�µ3RLQWV�¶ �6W\OH�RI��µ$V\PSWRWHV�¶ 2SHQLQJ�RQH�RI�WKHVH�FDWHJRULHV� RQH�ILQGV�WKHDWWULEXWHV�DQG�WKHLU�FXUUHQW�YDOXHV�WKDW�DSSO\�WR�WKH�FXUUHQWO\�VHOHFWHG�REMHFW�

�����

Page 285: Mupad Tutorial

Ob^p^k% ;khpl^k% Zg]�Bgli^\mhk3 Bgm^kZ\mbo^�FZgbineZmbhg

$IWHU�VHOHFWLRQ�RI�DQ�DWWULEXWH�ZLWK�WKH�PRXVH� LWV�YDOXH�FDQ�EH�FKDQJHG�

7KHUH�LV�D�VRSKLVWLFDWHG�ZD\�RI�VHWWLQJ�GHIDXOWV��VHH�6HFWLRQ ���� DQG�6HFWLRQ ����IRU�DQ�H[SODQDWLRQ��IRU�WKH�DWWULEXWHV�YLD�WKH�REMHFW�EURZVHU�DQG�WKH�SURSHUW\LQVSHFWRU� 7KH�µ9LHZ¶�PHQX�SURYLGHV�DQ�LWHP�µ+LGH�'HIDXOWV�¶ 'LVDEOLQJ�µ+LGH'HIDXOWV�¶ WKH�REMHFW�EURZVHU�VKRZV�HQWULHV�IRU�GHIDXOWV�

�����

Page 286: Mupad Tutorial

** @kZiab\l

$W�HDFK�QRGH�RI�WKH�JUDSKLFDO�WUHH� GHIDXOW�YDOXHV�IRU�WKH�LQKHULWDEOH�DWWULEXWHV�FDQEH�VHW�YLD�WKH�SURSHUW\�LQVSHFWRU� 7KHVH�GHIDXOWV�DUH�YDOLG�IRU�DOO�SULPLWLYHV�WKDWH[LVW�EHORZ�WKLV�QRGH� XQOHVV�WKHVH�GHIDXOWV�DUH�UH�GHILQHG�DW�VRPH�RWKHU�QRGHIXUWKHU�GRZQ�LQ�WKH�WUHH�KLHUDUFK\�

7KLV�PHFKDQLVP�LV�SDUWLFXODUO\�XVHIXO�ZKHQ�WKHUH�DUH�PDQ\�SULPLWLYHV�RI�WKH�VDPHNLQG�LQ�WKH�SORW� ,PDJLQH�D�SLFWXUH�FRQVLVWLQJ�RI������SRLQWV� ,I�\RX�ZLVK�WRFKDQJH�WKH�FRORU�RI�DOO�SRLQWV�WR�JUHHQ� LW�ZRXOG�EH�TXLWH�FXPEHUVRPH�WR�VHW3RLQW&RORU 5*%��*UHHQ LQ�DOO������SRLQWV� ,QVWHDG� \RX�FDQ�VHW3RLQW&RORU 5*%��*UHHQ LQ�WKH 3RLQW&RORU GHIDXOWV�HQWU\�DW�VRPH�WUHH�QRGH�WKDWFRQWDLQV�DOO�WKH�SRLQWV��H�J�� WKH�FDQYDV�� HLWKHU�GLUHFWO\�LQ�WKH SORW FDOO�RU�YLD�WKHREMHFW�EURZVHU�SURSHUW\�LQVSHFWRU� 6LPLODUO\� LI�WKHUH�DUH������SRLQWV�LQ�RQH�VFHQHDQG�DQRWKHU������SRLQWV�LQ�D�VHFRQG�VFHQH� \RX�FDQ�FKDQJH�WKH�FRORU�RI�DOO�SRLQWVLQ�WKH�ILUVW�VFHQH�E\�DQ�DSSURSULDWH�GHIDXOW�HQWU\�LQ�WKH�ILUVW�VFHQH� ZKLOVW�WKHGHIDXOW�HQWU\�IRU�WKH�VHFRQG�VFHQH�FDQ�EH�VHW�WR�D�GLIIHUHQW�YDOXH� 6HH�SDJH �����IRU�DQ�H[DPSOH�

�����

Page 287: Mupad Tutorial

Ob^p^k% ;khpl^k% Zg]�Bgli^\mhk3 Bgm^kZ\mbo^�FZgbineZmbhg

$ �' SORW�FDQ�EH�URWDWHG�DQG�VKLIWHG�E\�WKH�PRXVH� $OVR�]RRPLQJ�LQ�DQG�RXW�LVSRVVLEOH� ,Q�IDFW� WKHVH�RSHUDWLRQV�DUH�UHDOL]HG�E\�PRYLQJ�WKH�FDPHUD�DURXQG�FORVHU�WR� RU�IDUWKHU�DZD\�IURP�WKH�VFHQH� UHVSHFWLYHO\� 7KHUH�LV�D�FDPHUD�FRQWUROWKDW�PD\�EH�VZLWFKHG�RQ�DQG�RII�YLD�WKH�µ&DPHUD�&RQWURO¶�LWHP�RI�WKH�µ9LHZ¶�PHQXRU�WKH�FDPHUD�LFRQ�LQ�WKH�WRROEDU� ,W�SURYLGHV�WKH�FXUUHQW�YLHZLQJ�SDUDPHWHUV�VXFKDV�FDPHUD�SRVLWLRQ� IRFDO�SRLQW�DQG�WKH�DQJOH�RI�WKH�FDPHUD�OHQV�

7KH�VHFWLRQ�VWDUWLQJ�RQ�SDJH ������ SURYLGHV�PRUH�LQIRUPDWLRQ�RQ�FDPHUDV�

�����

Page 288: Mupad Tutorial

** @kZiab\l

Hjaealan]k

,Q�WKLV�VHFWLRQ� ZH�JLYH�D�EULHI�VXUYH\�RI�WKH�JUDSKLFDO�SULPLWLYHV� JURXSLQJFRQVWUXFWV� WUDQVIRUPDWLRQV�HWF��SURYLGHG�E\�WKH SORW OLEUDU\�

7KH�IROORZLQJ�WDEOH�OLVWV�WKH�µORZ�OHYHO¶�SULPLWLYHV�✞

$UF�G� $UF�G FLUFXODU�DUF�$UURZ�G� $UURZ�G DUURZ�%R[ UHFWDQJXODU�ER[�LQ��'�&LUFOH�G� &LUFOH�G FLUFOH�&RQH FRQH�FRQLFDO�IUXVWXP�LQ��'�&\OLQGHU F\OLQGHU�LQ��'�(OOLSVH�G� (OOLSVH�G HOOLSVH�(OOLSVRLG HOOLSVRLG�LQ��'�/LQH�G� /LQH�G �ILQLWH�RU�LQILQLWH��OLQH�3DUDOOHORJUDP�G� 3DUDOOHORJUDP�G SDUDOOHORJUDP�3ODQH LQILQWH�SODQH�LQ��'�3RLQW�G� 3RLQW�G SRLQW�3RLQW/LVW�G� 3RLQW/LVW�G OLVW�RI�SRLQWV�3RO\JRQ�G� 3RO\JRQ�G OLQH�VHJPHQWV�IRUPLQJ�D�SRO\JRQ�5HFWDQJOH UHFWDQJOH�LQ��'�6SKHUH VSKHUH�LQ��'�6XUIDFH6HW VXUIDFH�LQ��'

�DV�D�FROOHFWLRQ�RI�WULDQJOHV��6XUIDFH67/ LPSRUW�RI��' VWO VXUIDFHV�7H[W�G� 7H[W�G WH[W�REMHFW�

,Q�DGGLWLRQ� WKHUH�DUH�SULPLWLYHV SORW��7HWUDKHGURQ� SORW��+H[DKHGURQ�SORW��2FWDKHGURQ� SORW��'RGHFDKHGURQ� DQG SORW��,FRVDKHGURQ IRU�3ODWR¶VUHJXODU�SRO\KHGUD�DQG SORW��:DWHUPDQ IRU�:DWHUPDQ�SRO\KHGUD�

7KH�IROORZLQJ�WDEOH�OLVWV�WKH�µKLJK�OHYHO¶�SULPLWLYHV�DQG�µVSHFLDO�SXUSRVH¶SULPLWLYHV�

�����

Page 289: Mupad Tutorial

Ikbfbmbo^l

%DUV�G� %DUV�G �VWDWLVWLFDO��GDWD�SORW�%R[SORW �VWDWLVWLFDO��ER[�SORW�&RQIRUPDO SORW�RI�FRQIRUPDO�IXQFWLRQV�LQ��'�&XUYH�G� &XUYH�G SDUDPHWHUL]HG�FXUYH�&\OLQGULFDO VXUIDFH�LQ�F\OLQGULFDO�FRRUGLQDWHV�LQ��'�'HQVLW\ GHQVLW\�SORW�LQ��'�)XQFWLRQ�G� )XQFWLRQ�G IXQFWLRQ�JUDSK�+DWFK KDWFKHG�UHJLRQ�LQ��'�+LVWRJUDP�G �VWDWLVWLFDO��KLVWRJUDP�SORW�LQ��'�,PSOLFLW�G LPSOLFLWO\�GHILQHG�FXUYHV�LQ��'�,PSOLFLW�G LPSOLFLWO\�GHILQHG�VXUIDFHV�LQ��'�,QHTXDOLW\ YLVXDOL]DWLRQ�RI�LQHTXDOLWLHV�LQ��'�,WHUDWLRQ YLVXDOL]DWLRQ�RI�LWHUDWLRQV�LQ��'�/LVWSORW ILQLWH�OLVW�RI�SRLQWV�/V\V /LQGHQPD\HU�V\VWHP�LQ��'�0DWUL[SORW YLVXDOL]DWLRQ�RI�PDWULFHV�LQ��'�2GH�G� 2GH�G JUDSKLFDO�VROXWLRQ�RI�DQ�2'(�3LHFKDUW�G� 3LHFKDUW�G �VWDWLVWLFDO��SLH�FKDUW�3RODU FXUYH�LQ�SRODU�FRRUGLQDWHV�LQ��'�3ULVP SULVP�LQ��'�3\UDPLG S\UDPLGV�DQG�WKHLU�IUXVWUXPV�LQ��'�443ORW �VWDWLVWLFDO��TXDQWLOH�TXDQWLOH�SORWV�5DVWHU UDVWHU�DQG�ELWPDS�SORWV�LQ��'�5RRWORFXV GHSHQGHQF\�RI�UDWLRQDO�URRW�RQ�SDUDPHWHU�6FDWWHUSORW �VWDWLVWLFDO��VFDWWHU�SORW�LQ��'�6HTXHQFH VHTXHQFH��JLYHQ�E\�IRUPXOD��LQ��'�6SDUVH0DWUL[SORW VSDUVLW\�SDWWHUQ�RI�D�PDWUL[� �'�6SKHULFDO VXUIDFH�LQ�VSKHULFDO�FRRUGLQDWHV�LQ��'�6WUHDPOLQHV�G YHFWRU�ILHOG�YLVXDOL]DWLRQ�LQ��'�6XP YLVXDOL]DWLRQ�RI�V\PEROLF�VXPV�LQ��'�6ZHHS VZHHS�VXUIDFH�VSDQQHG�E\�WZR�FXUYHV�LQ��'�7XEH WXEH�SORW�LQ��'�7XUWOH WXUWOH�SORW�LQ��'�9HFWRU)LHOG�G� 9HFWRU)LHOG�G YHFWRU�ILHOG�SORW�6XUIDFH SDUDPHWHUL]HG�VXUIDFH�LQ��'�;5RWDWH VXUIDFH�RI�UHYROXWLRQ�LQ��'�=5RWDWH VXUIDFH�RI�UHYROXWLRQ�LQ��'�

�����

Page 290: Mupad Tutorial

** @kZiab\l

7KH�IROORZLQJ�WDEOH�OLVWV�WKH�YDULRXV�OLJKW�VRXUFHV�DYDLODEOH�WR�LOOXPLQDWH��' SORWV�✞

$PELHQW/LJKW DPELHQW��XQGLUHFWHG��OLJKW�'LVWDQW/LJKW GLUHFWHG�OLJKW��VXQ�OLJKW��3RLQW/LJKW �XQGLUHFWHG��SRLQW�OLJKW�6SRW/LJKW �GLUHFWHG��VSRW�OLJKW�

7KH�IROORZLQJ�WDEOH�OLVWV�YDULRXV�JURXSLQJ�FRQVWUXFWV�✞

&DQYDV GUDZLQJ�DUHD�6FHQH�G FRQWDLQHU�IRU��' FRRUGLQDWH�V\VWHPV�6FHQH�G FRQWDLQHU�IRU��' FRRUGLQDWH�V\VWHPV�&RRUGLQDWH6\VWHP�G FRQWDLQHU�IRU��' SULPLWLYHV�&RRUGLQDWH6\VWHP�G FRQWDLQHU�IRU��' SULPLWLYHV�*URXS�G JURXS�RI�SULPLWLYHV�LQ��'�*URXS�G JURXS�RI�SULPLWLYHV�LQ��'�

3ULPLWLYHV�RU�JURXSV�RI�SULPLWLYHV�FDQ�EH�WUDQVIRUPHG�E\�WKH�IROORZLQJ�REMHFWV�✞

6FDOH�G� 6FDOH�G VFDOLQJ�5HIOHFW�G� 5HIOHFW�G UHIOHFWLRQ�5RWDWH�G� 5RWDWH�G URWDWLRQ�7UDQVODWH�G� 7UDQVODWH�G WUDQVODWLRQ�7UDQVIRUP�G� 7UDQVIRUP�G JHQHUDO�OLQHDU�WUDQVIRUPDWLRQ�

$GGLWLRQDOO\� WKHUH�DUH�✞✝

☎✆

&DPHUD FDPHUD�LQ��'�&OLSSLQJ%R[ FOLSSLQJ�ER[�LQ��'�

�����

Page 291: Mupad Tutorial

:mmkb[nm^l

9lljaZml]k

7KH SORW OLEUDU\�SURYLGHV�PRUH�WKDQ�����DWWULEXWHV�IRU�ILQH�WXQLQJ�RI�WKHJUDSKLFDO�RXWSXW� %HFDXVH�RI�WKLV�ODUJH�QXPEHU� WKH�DWWULEXWHV�DUH�JURXSHG�LQWRYDULRXV�FDWHJRULHV�LQ�WKH�REMHFW�EURZVHU��VHH�SDJH �������DQG�WKH�GRFXPHQWDWLRQ�✞

FDWHJRU\ PHDQLQJ

$QLPDWLRQ SDUDPHWHUV�UHOHYDQW�IRU�DQLPDWLRQV$QQRWDWLRQ IRRWHU� KHDGHU� WLWOHV� DQG�OHJHQGV$[HV D[HV�VW\OH�DQG�D[HV�WLWOHV&DOFXODWLRQ QXPHULFDO�HYDOXDWLRQ'HILQLWLRQ SDUDPHWHUV�WKDW�FKDQJH�WKH�REMHFW�LWVHOI*ULG�/LQHV JULG�OLQHV�LQ�WKH�EDFNJURXQG��UXOLQJV�/D\RXW OD\RXW�SDUDPHWHUV�IRU�FDQYDV�DQG�VFHQHV6W\OH SDUDPHWHUV�WKDW�GR�QRW�FKDQJH�WKH�REMHFWV�EXW�WKHLU

SUHVHQWDWLRQ��YLVLELOLW\� FRORU� OLQH�ZLGWK� SRLQW�VL]H�HWF��$UURZV VW\OH�SDUDPHWHUV�IRU�DUURZV/LQHV VW\OH�SDUDPHWHUV�IRU�OLQH�REMHFWV3RLQWV VW\OH�SDUDPHWHUV�IRU�SRLQW�REMHFWV6XUIDFH VW\OH�SDUDPHWHUV�IRU�VXUIDFH�REMHFWV�LQ��'

DQG�ILOOHG�DUHDV�LQ��'7LFN�0DUNV D[HV�WLFN�PDUNV� VW\OH�DQG�ODEHOV

2Q�WKH�KHOS�SDJH�IRU�HDFK�SULPLWLYH� WKHUH�LV�D�FRPSOHWH�OLVW�RI�DOO�DWWULEXWHV�WKHSULPLWLYH�UHDFWV�WR� &OLFNLQJ�RQ�DQ�DWWULEXWH� \RX�DUH�OHDG�WR�WKH�KHOS�SDJH�IRU�WKLVDWWULEXWH�ZKLFK�SURYLGHV�DOO�WKH�QHFHVVDU\�LQIRUPDWLRQ�FRQFHUQLQJ�LWV�VHPDQWLFVDQG�DGPLVVLEOH�YDOXHV� 7KH�H[DPSOHV�RQ�WKH�KHOS�SDJH�GHPRQVWUDWH�WKH�W\SLFDO�XVHRI�WKH�DWWULEXWH�

'HIDXOW�9DOXHV

0RVW�DWWULEXWHV�KDYH�D�GHIDXOW�YDOXH�WKDW�LV�XVHG�LI�QR�YDOXH�LV�VSHFLILHG�H[SOLFLWO\�$V�DQ�H[DPSOH� ZH�FRQVLGHU�WKH�DWWULEXWH /LQHV9LVLEOH WKDW�LV�XVHG�E\�VHYHUDOSULPLWLYHV�VXFK�DV SORW��%R[� SORW��&LUFOH�G� SORW��&RQH� SORW��&XUYH�G�SORW��5DVWHU HWF� $OWKRXJK�WKH\�DOO�XVH�WKH�VDPH�DWWULEXWH�QDPHG /LQHV9LVLEOH�LWV�GHIDXOW�YDOXH�GLIIHUV�DPRQJ�WKH�GLIIHUHQW�SULPLWLYH�W\SHV� 7KH�VSHFLILF�GHIDXOWV

�����

Page 292: Mupad Tutorial

** @kZiab\l

DUH�DFFHVVLEOH�E\�SDVVLQJ�WKH�VORWV SORW��%R[��/LQHV9LVLEOH�SORW��&LUFOH�G��/LQHV9LVLEOH HWF� WR SORW��JHW'HIDXOW�SORW��JHW'HIDXOW�SORW��%R[��/LQHV9LVLEOH��SORW��JHW'HIDXOW�SORW��&LUFOH�G��/LQHV9LVLEOH��SORW��JHW'HIDXOW�SORW��&RQH��/LQHV9LVLEOH��SORW��JHW'HIDXOW�SORW��5DVWHU��/LQHV9LVLEOH�

758(, 758(, 758(, )$/6(

,I�DQ\�RI�WKH�GHIDXOW�YDOXHV�SURYLGHG�E\�WKH�0X3$'� V\VWHP�GR�QRW�VHHPDSSURSULDWH�IRU�\RXU�DSSOLFDWLRQV� FKDQJH�WKHVH�GHIDXOWV�YLD SORW��VHW'HIDXOW�

SORW��VHW'HIDXOW�SORW��%R[��/LQHV9LVLEOH )$/6(�758(

�7KH�UHWXUQ�YDOXH�LV�WKH�SUHYLRXVO\�YDOLG�GHIDXOW�YDOXH�� 6HYHUDO�GHIDXOWV�FDQ�EHFKDQJHG�VLPXOWDQHRXVO\�

SORW��VHW'HIDXOW�SORW��%R[��/LQHV9LVLEOH )$/6(�SORW��&LUFOH�G��/LQHV9LVLEOH )$/6(�SORW��&LUFOH�G��)LOOHG 758(�

)$/6(, 758(, )$/6(

SORW��JHW'HIDXOW�SORW��%R[��/LQHV9LVLEOH�)$/6(

,QKHULWDQFH�RI�$WWULEXWHV

7KH�VHWWLQJ�RI�GHIDXOW�YDOXHV�IRU�DWWULEXWHV�LV�TXLWH�VRSKLVWLFDWHG� $VVXPH�WKDW�\RXKDYH�WZR�VFHQHV�WKDW�DUH�WR�EH�GLVSOD\HG�LQ�RQH�VLQJOH�FDQYDV� %RWK�VFHQHV�FRQVLVWRI����JUDSKLFDO�SRLQWV� HDFK�

SRLQWV� � SORW��3RLQW�G�L��� 3,� VLQ�L��� 3,�� � L ������SRLQWV� � SORW��3RLQW�G�L��� 3,� FRV�L��� 3,�� � L ������6� � SORW��6FHQH�G�SRLQWV���6� � SORW��6FHQH�G�SRLQWV���

�����

Page 293: Mupad Tutorial

:mmkb[nm^l

SORW�6�� 6��

,I�ZH�ZLVK�WR�FRORU�DOO�SRLQWV�LQ�ERWK�VFHQHV�UHG� ZH�FDQ�VHW�D�GHIDXOW�IRU�WKH�SRLQWFRORU�LQ�WKH SORW FRPPDQG�SORW�6�� 6�� 3RLQW&RORU 5*%��5HG�

,I�ZH�ZLVK�WR�FRORU�DOO�SRLQWV�LQ�WKH�ILUVW�VFHQH�UHG�DQG�DOO�SRLQWV�LQ�WKH�VHFRQGVFHQH�EOXH� ZH�FDQ�JLYH�HDFK�RI�WKH�SRLQWV�WKH�GHVLUHG�FRORU�LQ�WKH�JHQHUDWLQJ�FDOOWR SORW��3RLQW�G� $OWHUQDWLYHO\� ZH�FDQ�VHW�VHSDUDWH�GHIDXOWV�IRU�WKH�SRLQW�FRORULQVLGH�WKH�VFHQHV�

6� � SORW��6FHQH�G�SRLQWV�� 3RLQW&RORU 5*%��5HG��6� � SORW��6FHQH�G�SRLQWV�� 3RLQW&RORU 5*%��%OXH��SORW�6�� 6��

+HUH�LV�WKH�JHQHUDO�UXOH�IRU�VHWWLQJ�GHIDXOWV�LQVLGH�D�JUDSKLFDO�WUHH��VHH�SDJH �������✞

✆:KHQ�DQ�DWWULEXWH�LV�VSHFLILHG�LQ�D�QRGH�RI�WKH�JUDSKLFDO�WUHH� WKH�VSHFLILHG�YDOXHVHUYHV�DV�WKH�GHIDXOW�YDOXH�IRU�WKH�DWWULEXWH�IRU�DOO�SULPLWLYHV�WKDW�H[LVW�LQ�WKHVXEWUHH�VWDUWLQJ�IURP�WKDW�QRGH�

7KH�GHIDXOW�YDOXH�FDQ�EH�RYHUZULWWHQ�E\�DQRWKHU�YDOXH�DW�HDFK�QRGH�IXUWKHU�GRZQLQ�WKH�VXEWUHH��H�J�� ILQDOO\� E\�D�VSHFLILF�YDOXH�LQ�WKH�SULPLWLYHV�� ,Q�WKH�IROORZLQJFDOO� WKH�GHIDXOW�FRORU�µUHG¶�LV�VHW�LQ�WKH�FDQYDV� 7KLV�YDOXH�LV�DFFHSWHG�DQG�XVHG�LQ

�����

Page 294: Mupad Tutorial

** @kZiab\l

WKH�ILUVW�VFHQH� 7KH�VHFRQG�VFHQH�VHWV�WKH�GHIDXOW�FRORU�µEOXH¶�RYHUULGLQJ�WKH�GHIDXOWYDOXH�µUHG¶�VHW�LQ�WKH�FDQYDV� $GGLWLRQDOO\� WKHUH�LV�DQ�H[WUD�SRLQW�ZLWK�D�FRORUH[SOLFLWO\�VHW�WR�µEODFN¶� 7KLV�SRLQW�LJQRUHV�WKH�GHIDXOWV�VHW�IXUWKHU�XS�LQ�WKH�WUHHKLHUDUFK\�

H[WUDSRLQW � SORW��3RLQW�G����� ����3RLQW6L]H � XQLW��PP�3RLQW&RORU 5*%��%ODFN��

6� � SORW��6FHQH�G�SRLQWV�� H[WUDSRLQW��6� � SORW��6FHQH�G�SRLQWV�� H[WUDSRLQW�

3RLQW&RORU 5*%��%OXH��SORW�SORW��&DQYDV�6�� 6�� 3RLQW&RORU 5*%��5HG��

7KH�IROORZLQJ�FDOO�JHQHUDWHV�WKH�VDPH�UHVXOW� 1RWH�WKDW�WKH SORW FRPPDQGDXWRPDWLFDOO\�FUHDWHV�D�FDQYDV�REMHFW�DQG�SDVVHV�WKH�DWWULEXWH3RLQW&RORU 5*%��5HG WR�WKH�FDQYDV�SORW�6�� 6�� 3RLQW&RORU 5*%��5HG�

:H�QRWH�WKDW�WKHUH�DUH�GLIIHUHQW�SULPLWLYHV�WKDW�UHDFW�WR�WKH�VDPH�DWWULEXWH� :HXVHG /LQHV9LVLEOH LQ�WKH�SUHYLRXV�VHFWLRQ�WR�GHPRQVWUDWH�WKLV�IDFW� 2QH�RI�WKHUXOHV�IRU�LQKHULWLQJ�DWWULEXWHV�LQ�D�JUDSKLFDO�WUHH�LV�

,I�DQ�DWWULEXWH�VXFK�DV /LQHV9LVLEOH 758( LV�VSHFLILHG�LQ�VRPH�QRGH�RI�WKHJUDSKLFDO�WUHH� DOO SULPLWLYHV�EHORZ�WKLV�QRGH�WKDW�UHDFW�WR�WKLV�DWWULEXWH�XVH�WKHVSHFLILHG�YDOXH�DV�WKH�GHIDXOW�YDOXH�,I � D � W\SH � VSHFLILF � DWWULEXWH � �XVLQJ � WKH � ³IXOO\ � TXDOLILHG´ � V\QWD[� � VXFK � DVSORW��&LUFOH�G��/LQHV9LVLEOH 758( LV�VSHFLILHG� WKH�QHZ�GHIDXOW�YDOXH�LVYDOLG�RQO\�IRU�SULPLWLYHV�RI�WKDW�VSHFLILF�W\SH�

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�FRQVLGHU�����UDQGRPO\�SODFHG�FLUFOHV�ZLWK�DUHFWDQJOH�LQGLFDWLQJ�WKH�DUHD�LQWR�ZKLFK�DOO�FLUFOH�FHQWHUV�DUH�SODFHG�

UHFWDQJOH � SORW��5HFWDQJOH������ � �����FLUFOHV � SORW��&LUFOH�G������ >IUDQGRP��� IUDQGRP��@�

� L �������

�����

Page 295: Mupad Tutorial

:mmkb[nm^l

SORW�UHFWDQJOH� FLUFOHV� $[HV )UDPH�

:H�ZLVK�WR�WXUQ�WKH�FLUFOHV�LQWR�ILOOHG�FLUFOHV�E\�VHWWLQJ )LOOHG 758( DQG/LQHV9LVLEOH )$/6(�SORW�UHFWDQJOH� FLUFOHV�

)LOOHG 758(� )LOO3DWWHUQ 6ROLG�/LQHV9LVLEOH )$/6(� $[HV )UDPH�

�����

Page 296: Mupad Tutorial

** @kZiab\l

7KLV�LV�QRW�TXLWH�ZKDW�ZH�ZDQWHG� 1RW�RQO\�WKH�FLUFOHV� EXW�DOVR�WKH�UHFWDQJOHUHDFWV�WR�WKH�DWWULEXWHV )LOOHG� )LOO3DWWHUQ� DQG /LQHV9LVLEOH� 7KH�IROORZLQJFRPPDQG�UHVWULFWV�WKHVH�DWWULEXWHV�WR�WKH�FLUFOHV�

SORW�UHFWDQJOH� FLUFOHV�SORW��&LUFOH�G��)LOOHG 758(�SORW��&LUFOH�G��)LOO3DWWHUQ 6ROLG�SORW��&LUFOH�G��/LQHV9LVLEOH )$/6(�$[HV )UDPH�

3ULPLWLYHV�5HTXHVWLQJ�6SHFLDO�6FHQH�$WWULEXWHV� ³+LQWV´

7KH�GHIDXOW�YDOXHV�IRU�WKH�DWWULEXWHV�DUH�FKRVHQ�LQ�VXFK�D�ZD\�WKDW�WKH\�SURGXFHUHDVRQDEOH�SLFWXUHV�LQ�³W\SLFDO´�SORW�FRPPDQGV� )RU�H[DPSOH� WKH�GHIDXOW�D[HVW\SH�LQ��' VFHQHV�LV $[HV %R[HG� EHFDXVH�WKLV�LV�WKH�PRVW�DSSURSULDWH�D[HV�W\SH�LQWKH�PDMRULW\�RI��' SORWV�

SORW��JHW'HIDXOW�SORW��&RRUGLQDWH6\VWHP�G��$[HV�%R[HG

+RZHYHU� WKHUH�DUH�H[FHSWLRQV� (�J�� D�SORW�FRQWDLQLQJ�D��' SLH�FKDUW�VKRXOGSUREDEO\�KDYH�QR�D[HV�DW�DOO� 6LQFH�LW�LV�QRW�GHVLUDEOH�WR�XVH $[HV 1RQH DV�WKHGHIDXOW�VHWWLQJ�IRU�DOO�SORWV� H[FHSWLRQDO�SULPLWLYHV�VXFK�DV SORW��3LHFKDUW�G DUH

�����

Page 297: Mupad Tutorial

:mmkb[nm^l

JLYHQ�D�FKDQFH�WR�RYHUULGH�WKH�GHIDXOW�D[HV�DXWRPDWLFDOO\� ,Q�D�SLH�FKDUW�SORW� QRD[HV�DUH�XVHG�E\�GHIDXOW�

SORW�SORW��3LHFKDUW�G�>��� ��� ��� �� ��� ��@�7LWOHV >� ������ � ������

� ������ � ������ ������ � �����@��

1RWH�WKDW�WKH�DWWULEXWH $[HV WKDW�LV�LQ�FKDUJH�RI�VZLWFKLQJ�D[HV�RQ�DQG�RII�LV�DVFHQH�DWWULEXWH�DQG�KHQFH�FDQQRW�EH�SURFHVVHG�E\�SLH�FKDUW�REMHFWV�GLUHFWO\�+HQFH� D�VHSDUDWH�PHFKDQLVP�IRU�UHTXHVWLQJ�VSHFLDO�VFHQH�DWWULEXWHV�E\�SULPLWLYHVLV�LPSOHPHQWHG� VR�FDOOHG�³KLQWV�´

$ ³KLQW´�LV�DQ�DWWULEXWH�RI�RQH�RI�WKH�VXSHURUGLQDWH�QRGHV�LQ�WKH�JUDSKLFDO�WUHH� L�H��DQ�DWWULEXWH�RI�D�FRRUGLQDWH�V\VWHP� D�VFHQH� RU�WKH�FDQYDV� 7KH�KHOS�SDJHV�RI�WKHSULPLWLYHV�JLYH�LQIRUPDWLRQ�RQ�ZKDW�KLQWV�DUH�VHQW�E\�WKH�SULPLWLYH� ,I�VHYHUDOSULPLWLYHV�VHQG�FRQIOLFWLQJ�KLQWV� WKH�ILUVW�KLQW�LV�XVHG�

+LQWV�DUH�LPSOHPHQWHG�LQWHUQDOO\�DQG�FDQQRW�EH�VZLWFKHG�RII�E\�WKH�XVHU� 1RWH�KRZHYHU� WKDW�WKH�KLQWV�FRQFHSW�RQO\�FRQFHUQV GHIDXOW�YDOXHV IRU�DWWULEXWHV� <RXFDQ�DOZD\V�VSHFLI\�WKH�DWWULEXWH�H[SOLFLWO\�LI�\RX�WKLQN�WKDW�D�GHIDXOW�YDOXH�RU�D�KLQWLV�QRW�DSSURSULDWH�

�����

Page 298: Mupad Tutorial

** @kZiab\l

$V�DQ�H[DPSOH� ZH�H[SOLFLWO\�UHTXHVW $[HV %R[HG LQ�WKH�IROORZLQJ�FDOO�

SORW�SORW��3LHFKDUW�G�>��� ��� ��� �� ��� ��@�7LWOHV >� ������ � ������

� ������ � ������ ������ � �����@��

$[HV %R[HG�

�����

Page 299: Mupad Tutorial

:mmkb[nm^l

7KH�+HOS�3DJHV�RI�$WWULEXWHV

:H�KDYH�D�EULHI�ORRN�DW�D�W\SLFDO�KHOS�SDJH�IRU�D�SORW�DWWULEXWH�WR�H[SODLQ�WKHLQIRUPDWLRQ�SURYLGHG�WKHUH�✞

80HVK� 90HVK� 86XEPHVK� 96XEPHVK ±�QXPEHU�RI�VDPSOHSRLQWV

7KH�DWWULEXWHV 80HVK HWF��GHWHUPLQH�WKH�QXPEHU�RI�VDPSOH�SRLQWV�XVHG�IRU�WKHQXPHULFDO�DSSUR[LPDWLRQ�RI�SDUDPHWHUL]HG�SORW�REMHFWV�VXFK�DV�FXUYHV�DQGVXUIDFHV�→ ([DPSOHV

$WWULEXWH 7\SH 9DOXH86XEPHVK� 96XEPHVK LQKHULWHG QRQ�QHJDWLYH�LQWHJHU80HVK� 90HVK LQKHULWHG SRVLWLYH�LQWHJHU

6HH�$OVR$GDSWLYH0HVK� 0HVK� 6XEPHVK� ;0HVK� <0HVK� =0HVK2EMHFWV�UHDFWLQJ�WR 80HVK� 90HVK� 86XEPHVK� 96XEPHVKSORW��6XUIDFH� SORW��&\OLQGULFDO�SORW��;5RWDWH� SORW��6SKHULFDO�SORW��=5RWDWH

����80HVK� 90HVK����86XEPHVK�96XEPHVK�

SORW��7XEH ����80HVK�����90HVK����86XEPHVK����96XEPHVK�

«

'HWDLO

� 0DQ\�SORW�REMHFWV�KDYH�WR�EH�HYDOXDWHG�QXPHULFDOO\�RQ�D�GLVFUHWH�PHVK�«

([DPSOH�� «

�����

Page 300: Mupad Tutorial

** @kZiab\l

,Q�WKHVH�KHOS�SDJHV�

� WKH�WDEOH�HQWU\�³7\SH´��³LQKHULWHG´��VWDWHV�WKDW�WKHVH�DWWULEXWHV�PD\�QRW�RQO\EH�VSHFLILHG�LQ�WKH�JHQHUDWLQJ�FDOO�RI�JUDSKLFDO�SULPLWLYHV� 7KH\�FDQ�DOVR�EHVSHFLILHG�DW�KLJKHU�QRGHV�RI�D�JUDSKLFDO�WUHH�WR�EH�LQKHULWHG�E\�WKH�SULPLWLYHVLQ�WKH�FRUUHVSRQGLQJ�VXEWUHH��VHH�SDJH �������

� WKH�WDEOH�HQWU\�³9DOXH´�VWDWHV�WKH�W\SH�RI�WKH�QXPEHU Q WKDW�LV�DGPLVVLEOHZKHQ�SDVVLQJ�WKH�DWWULEXWHV 80HVK Q� 90HVK Q HWF�

� WKH�VHFWLRQV�³2EMHFW�W\SHV�UHDFWLQJ�WR 80HVK´�HWF��SURYLGH�FRPSOHWHOLVWLQJV�RI�DOO�SULPLWLYHV�UHDFWLQJ�WR�WKH�DWWULEXWH�V��WRJHWKHU�ZLWK�WKH�VSHFLILFGHIDXOW�YDOXHV�

WKH�³'HWDLOV´�VHFWLRQ�JLYHV�IXUWKHU�LQIRUPDWLRQ�RQ�WKH�VHPDQWLFV�RI�WKHDWWULEXWH�V�� )LQDOO\� WKHUH�DUH�³([DPSOHV´�RI�SORW�FRPPDQGV�XVLQJ�WKHGHVFULEHG�DWWULEXWH�V��

�����

Page 301: Mupad Tutorial

<hehkl

;gdgjk

7KH�PRVW�SURPLQHQW�SORW�DWWULEXWH� DYDLODEOH�IRU�DOPRVW�DOO�SULPLWLYHV� LV &RORU�VHWWLQJ�WKH�µSULPDU\¶�FRORU�RI�DQ�REMHFW� 0X3$'�¶V SORW OLEUDU\�NQRZV���GLIIHUHQWW\SHV�RI�SULPDU\�FRORUV�

� 7KH�DWWULEXWH 3RLQW&RORU UHIHUV�WR�WKH�FRORU�RI�SRLQWV�LQ��' DQG��' �RI�W\SHSORW��3RLQW�G DQG SORW��3RLQW�G� UHVSHFWLYHO\��

� 7KH�DWWULEXWH /LQH&RORU UHIHUV�WR�WKH�FRORU�RI�OLQH�REMHFWV�LQ��' DQG��'��7KLVLQFOXGHV�WKH�FRORU�RI�IXQFWLRQ�JUDSKV�LQ��'��FXUYHV�LQ��' DQG��'��SRO\JRQOLQHV�LQ��' DQG��' HWF� $OVR��' REMHFWV�VXFK�DV�IXQFWLRQ�JUDSKV�LQ��'�SDUDPHWUL]HG�VXUIDFHV�HWF��UHDFW�WR�WKH�DWWULEXWH /LQH&RORU� LW�GHILQHV�WKHFRORU�RI�WKH�FRRUGLQDWH�PHVK�OLQHV�WKDW�DUH�GLVSOD\HG�RQ�WKH�VXUIDFH�

� 7KH�DWWULEXWH )LOO&RORU UHIHUV�WR�WKH�FRORU�RI�FORVHG�DQG�ILOOHG�SRO\JRQV�LQ�' DQG��' DV�ZHOO�DV�KDWFKHG�UHJLRQV�LQ��'��)XUWKHU� LW�VHWV�WKH�VXUIDFH�FRORURI�IXQFWLRQ�JUDSKV�LQ��'��SDUDPHWUL]HG�VXUIDFHV�HWF� 7KLV�LQFOXGHV�VSKHUHV�FRQHV�HWF�

7KH�SULPLWLYHV�DOVR�DFFHSW�WKH�DWWULEXWH &RORU DV�D�VKRUWFXW�IRU�RQH�RI�WKHVH�FRORUV�'HSHQGLQJ�RQ�WKH�SULPLWLYH� HLWKHU 3RLQW&RORU� /LQH&RORU� RU )LOO&RORU LV�VHWZLWK�WKH &RORU DWWULEXWH�

5*% &RORUV

0X3$'� XVHV�WKH�5*% FRORU�PRGHO� L�H�� FRORUV�DUH�VSHFLILHG�E\�OLVWV >U�J�E@ RIUHG� JUHHQ� DQG�EOXH�YDOXHV�EHWZHHQ 0 DQG 1� RU�LQ�+70/ V\QWD[�DV�D�KDVK�PDUNIROORZHG�E\�D�VL[�GLJLW�KH[DGHFLPDO�QXPEHU� DV�LQ �FF����� %ODFN�DQG�ZKLWHFRUUHVSRQG�WR >�����@ ����������DQG >�����@ ��IIIIII�� UHVSHFWLYHO\� 7KHOLEUDU\ 5*% FRQWDLQV�QXPHURXV FRORU�QDPHV�ZLWK�FRUUHVSRQGLQJ�5*% YDOXHV�

5*%��%ODFN� 5*%��:KLWH� 5*%��5HG� 5*%��6N\%OXH[0.0, 0.0, 0.0] , [1.0, 1.0, 1.0] , [1.0, 0.0, 0.0] , [0.0, 0.8, 1.0]

/LVW�DOO�FRORU�QDPHV�DYDLODEOH�LQ�WKH 5*% OLEUDU\�YLD LQIR�5*%�� $OWHUQDWLYHO\� WKHUHLV�WKH�FRPPDQG 5*%��&RORU1DPHV�� UHWXUQLQJ�D�FRPSOHWH�OLVW�RI�QDPHV� :LWK

�����

Page 302: Mupad Tutorial

** @kZiab\l

5*%��&RORU1DPHV�5HG�>&DGPLXP5HG'HHS� &DGPLXP5HG/LJKW� &DUGLQDO5HG�

&DUPLQH5HG� 'DUN5HG� (QJOLVK5HG� ,QGLDQ5HG�

2UDQJH5HG� 3HUPDQHQW5HG9LROHW� 5HG� 9HQHWLDQ5HG�

9LROHW5HG� 9LROHW5HG0HGLXP� 9LROHW5HG3DOH@

RQH�VHDUFKHV�IRU�DOO�FRORU�QDPHV�WKDW�FRQWDLQ�µ5HG�¶ 7R�DFWXDOO\�VHH�WKH�FRORUVIRXQG� XVH 5*%��SORW&RORU3DOHWWH�5*%��SORW&RORU3DOHWWH��UHG��

$IWHU�H[SRUWLQJ�WKH�FRORU�OLEUDU\�YLD XVH�5*%���\RX�FDQ�XVH�WKH�FRORU�QDPHV�LQ�WKHVKRUW�IRUP %ODFN� :KLWH� ,QGLDQ5HG HWF� 1RWH� KRZHYHU� WKDW�WKLV�FRPPDQG�ZLOODVVLJQ�YDOXHV�WR�D�OLWWOH�RYHU 350 LGHQWLILHUV� SUHFOXGLQJ�WKHLU�XVH�LQ�RWKHU�PHDQLQJV�

5*%D�FRORU�YDOXHV�FRQVLVW�RI�OLVWV >U�J�E�D@ FRQWDLQLQJ�D�IRXUWK�HQWU\� WKH³RSDFLW\´�YDOXH a EHWZHHQ 0 DQG 1� �,Q�+70/�OLNH�QRWDWLRQ� WKH�KH[DGHFLPDOVSHFLILFDWLRQ�KDV�HLJKW�FKDUDFWHUV�� )RU a = 0� D�VXUIDFH�SDWFK�SDLQWHG�ZLWK�WKLV5*%D�FRORU�LV�IXOO\�WUDQVSDUHQW��L�H�� LQYLVLEOH�� )RU a = 1� WKH�VXUIDFH�SDWFK�LVRSDTXH� L�H�� LW�KLGHV�SORW�REMHFWV�WKDW�DUH�EHKLQG�LW� )RU 0 < a < 1� WKH�VXUIDFHSDWFK�LV�VHPL�WUDQVSDUHQW� L�H�� SORW�REMHFWV�EHKLQG�LW�VKLQH�WKURXJK�

5*%D�FRORU�YDOXHV�FDQ�EH�FRQVWUXFWHG�HDVLO\�YLD�WKH 5*% OLEUDU\� 2QH�RQO\�KDV�WRDSSHQG�D�IRXUWK�HQWU\�WR�WKH [r, g, b] OLVWV�SURYLGHG�E\�WKH�FRORU�QDPHV� 7KH�HDVLHVW

�����

Page 303: Mupad Tutorial

<hehkl

ZD\�WR�GR�WKLV�LV�WR�DSSHQG�WKH�OLVW [a] WR�WKH�5*% OLVW�YLD�WKH�FRQFDWHQDWLRQRSHUDWRU�µ�¶� :H�FUHDWH�D�VHPL�WUDQVSDUHQW�JUD\�

5*%��*UD\�>���@[0.752907, 0.752907, 0.752907, 0.5]

7KH�IROORZLQJ�FRPPDQG�SORWV�D�KLJKO\�WUDQVSDUHQW�UHG�ER[� FRQWDLQLQJ�DVRPHZKDW�OHVV�WUDQVSDUHQW�JUHHQ�ER[�ZLWK�DQ�RSDTXH�EOXH�ER[�LQVLGH�

SORW�SORW��%R[������� ������ ������)LOO&RORU 5*%��5HG�>���@��

SORW��%R[������� ������ ������)LOO&RORU 5*%��*UHHQ�>���@��

SORW��%R[������� ������ ������)LOO&RORU 5*%��%OXH��

/LQHV9LVLEOH 758(� /LQH&RORU 5*%��%ODFN�6FDOLQJ &RQVWUDLQHG�

+69 &RORUV

%HVLGHV�WKH�5*% PRGHO� WKHUH�DUH�YDULRXV�RWKHU�SRSXODU�FRORU�IRUPDWV�XVHG�LQFRPSXWHU�JUDSKLFV� 2QH�LV�WKH�+69 PRGHO��+XH� 6DWXUDWLRQ� 9DOXH�� 7KH 5*%OLEUDU\�SURYLGHV�WKH�URXWLQHV 5*%��IURP+69 DQG 5*%��WR+69 WR�FRQYHUW�+69

�����

Page 304: Mupad Tutorial

** @kZiab\l

FRORUV�WR�5*% FRORUV�DQG�YLFH�YHUVD�

KVY � 5*%��WR+69�5*%��2UDQJH�[24.0, 1.0, 1.0]

5*%��IURP+69�KVY� 5*%��2UDQJH[1.0, 0.4, 0.0] = [1.0, 0.4, 0.0]

:LWK�WKH 5*%��IURP+69 XWLOLW\� DOO�FRORUV�LQ�D�0X3$' SORW�FDQ�EH�VSHFLILHG�HDVLO\DV�+69 FRORUV� )RU�H[DPSOH� WKH�FRORU�µYLROHW¶�LV�JLYHQ�E\�WKH�+69 YDOXHV>�����������@� ZKHUHDV�µGDUN�JUHHQ¶�LV�JLYHQ�E\�WKH�+69 VSHFLILFDWLRQ>���������@ �����IRU�µJUHHQ�¶ ����IRU�µGDUN¶�� +HQFH� D�VHPL�WUDQVSDUHQW�YLROHWVSKHUH�LQWHUVHFWHG�E\�DQ�RSDTXH�GDUN�JUHHQ�SODQH�PD\�EH�VSHFLILHG�DV�IROORZV�

SORW�SORW��6SKHUH��� >�� �� �@�&RORU 5*%��IURP+69�>���� ���� ���� ���@���

SORW��6XUIDFH�>[� \� ���@� [ ������ \ ������0HVK >�� �@�&RORU 5*%��IURP+69�>���� �� ���@���

�����

Page 305: Mupad Tutorial

:gbfZmbhgl

9faeYlagfk

*HQHUDWLQJ�6LPSOH�$QLPDWLRQV

(DFK�SULPLWLYH�RI�WKH SORW OLEUDU\�NQRZV�KRZ�PDQ\�VSHFLILFDWLRQV�RI�W\SH�³UDQJH´LW�KDV�WR�H[SHFW�

✆:KHQHYHU�D�JUDSKLFDO�SULPLWLYH�UHFHLYHV�D�³VXUSOXV´�UDQJH�VSHFLILFDWLRQ�E\DQ�HTXDWLRQ�VXFK�DV D DPLQ��DPD[� WKH�SDUDPHWHU D LV�LQWHUSUHWHG�DV�DQ�³DQ�LPDWLRQ�SDUDPHWHU´�DVVXPLQJ�YDOXHV�IURP DPLQ WR DPD[�

:KDW�LV�D�³VXUSOXV´�UDQJH" 7KLV�LV�D�VSHFLILFDWLRQ�WKDW�LV�QRW�QHFHVVDU\�IRUJHQHUDWLQJ�D�VWDWLF�REMHFW� )RU�H[DPSOH� D�VWDWLF�XQLYDULDWH�IXQFWLRQ�JUDSK�LQ��'VXFK�DV

SORW��)XQFWLRQ�G�VLQ�[�� [ ���� 3,��

H[SHFWV�RQH�SORW�UDQJH�IRU�WKH x FRRUGLQDWH� ZKHUHDV�D�VWDWLF�EL�YDULDWH�IXQFWLRQJUDSK�LQ��' H[SHFWV�WZR�SORW�UDQJHV�IRU�WKH x DQG y FRRUGLQDWH�

SORW��)XQFWLRQ�G�VLQ�[A� � \A��� [ ����� \ ������

$ VWDWLF�FRQWRXU�SORW�LQ��' H[SHFWV���UDQJHV�IRU�WKH x DQG y FRRUGLQDWH�

SORW��,PSOLFLW�G�[A� � \A� � �� [ ������ \ �������

$ VWDWLF�FRQWRXU�SORW�LQ��' H[SHFWV���UDQJHV�IRU�WKH x� y� DQG z FRRUGLQDWH�

SORW��,PSOLFLW�G�[A� � \A� � ]A� � �� [ ������\ � ����� ] � ������

$Q\�DGGLWLRQDO�HTXDWLRQ�LQYROYLQJ�D�UDQJH�PDNHV�WKH�SULPLWLYH�FUHDWH�DQDQLPDWLRQ� 6LQFH�D�VWDWLF��' IXQFWLRQ�JUDSK�H[SHFWV�RQO\�RQH�UDQJH��IRU�WKHLQGHSHQGHQW�YDULDEOH�� WKH�IROORZLQJ�FDOO�LV�LQWHUSUHWHG�DV�DQ�LQVWUXFWLRQ�WRJHQHUDWH�DQ�DQLPDWHG��' IXQFWLRQ�JUDSK�

SORW��)XQFWLRQ�G�VLQ�[ � D�� [ ���� 3,� D ���� 3,��

7KXV� LW�LV�YHU\�HDV\�LQGHHG�WR�FUHDWH�DQLPDWHG�REMHFWV� MXVW�SDVV�D�³VXUSOXV´�UDQJHHTXDWLRQ D DPLQ��DPD[ WR�WKH�JHQHUDWLQJ�FDOO�RI�WKH�SULPLWLYH� 7KH�QDPH D RI�WKHDQLPDWLRQ�SDUDPHWHU�LV�LUUHOHYDQW� DQ\�V\PEROLF�QDPH�PD\�EH�XVHG� $OO�RWKHU

�����

Page 306: Mupad Tutorial

** @kZiab\l

HQWULHV�DQG�DWWULEXWHV�RI�WKH�SULPLWLYH�WKDW�DUH�V\PEROLF�H[SUHVVLRQV�RI�WKHDQLPDWLRQ�SDUDPHWHU�ZLOO�EH�DQLPDWHG� ,Q�WKH�IROORZLQJ�FDOO� ERWK�WKH�IXQFWLRQH[SUHVVLRQ�DV�ZHOO�DV�WKH x UDQJH�RI�WKH�IXQFWLRQ�JUDSK�GHSHQG�RQ�WKH�DQLPDWLRQSDUDPHWHU� $OVR� WKH�UDQJHV�GHILQLQJ�WKH�ZLGWK�DQG�WKH�KHLJKW�RI�WKH�UHFWDQJOH�DVZHOO�DV�WKH�HQG�SRLQW�RI�WKH�OLQH�GHSHQG�RQ�LW�

SORW�SORW��)XQFWLRQ�G�D VLQ�[�� [ ���D 3,� D ��������SORW��5HFWDQJOH����D 3,� ���D� D �������

/LQH&RORU 5*%��%ODFN��SORW��/LQH�G�>�� �@� >3, D� D@� D ��� ����

/LQH&RORU 5*%��%ODFN��

�����

Page 307: Mupad Tutorial

:gbfZmbhgl

+HUH�LV�DQ�DQLPDWHG�DUF�ZKRVH�UDGLXV�DQG�³DQJOH�UDQJH´�GHSHQG�RQ�WKH�DQLPDWLRQSDUDPHWHU�

SORW�SORW��$UF�G�� � D� >�� �@� $QJOH5DQJH ���D 3,�D ������

+HUH� DQ�DGGLWLRQDO�UDQJH�VSHFLILFDWLRQ�IRU�WKH�DWWULEXWH $QJOH5DQJH LV�XVHG�+RZHYHU� WKH�DWWULEXWH�LV�LGHQWLILHG�E\�LWV�QDPH�DV�DQ�DGPLVVLEOH�DWWULEXWH�IRU�WKHSULPLWLYH�DQG�WKXV�QRW�DVVXPHG�WR�EH�WKH�VSHFLILFDWLRQ�RI�DQ�DQLPDWLRQ�SDUDPHWHU�

✆%HZDUH� 'R�PDNH�VXUH�WKDW�DWWULEXWHV�DUH�VSHFLILHG�E\�WKHLU�FRUUHFW�QDPHV�,I�DQ�LQFRUUHFW�DWWULEXWH�QDPH�LV�XVHG� LW�PD\�EH�PLVWDNHQ�IRU�DQ�DQLPDWLRQSDUDPHWHU�

,Q�WKH�IROORZLQJ�H[DPSOHV� ZH�ZLVK�WR�GHILQH�D�VWDWLF�VHPL�FLUFOH� XVLQJSORW��$UF�G ZLWK $QJOH5DQJH ���3,��+RZHYHU� $QJOH5DQJH LV�VSHOOHGLQFRUUHFWO\� $ SORW�LV�FUHDWHG� ,W�LV�DQ�DQLPDWHG�IXOO�FLUFOH�ZLWK�WKH�DQLPDWLRQSDUDPHWHU $QJHO5DQJH�

SORW�SORW��$UF�G��� >�� �@� $QJHO5DQJH ���3,��

7KH�DQLPDWLRQ�SDUDPHWHU�PD\�EH�DQ\�V\PEROLF�SDUDPHWHU��LGHQWLILHU�RU�LQGH[HGLGHQWLILHU��WKDW�LV�GLIIHUHQW�IURP�WKH�V\PEROV�XVHG�IRU�WKH�PDQGDWRU\�UDQJH

�����

Page 308: Mupad Tutorial

** @kZiab\l

VSHFLILFDWLRQV��VXFK�DV�WKH�QDPHV�RI�WKH�LQGHSHQGHQW�YDULDEOHV�LQ�IXQFWLRQJUDSKV�� 7KH�SDUDPHWHU�PXVW�DOVR�EH�GLIIHUHQW�IURP�DQ\�RI�WKH�SURWHFWHG�QDPHV�RIWKH�SORW�DWWULEXWHV�

✞✝

☎✆

$QLPDWLRQV�DUH�FUHDWHG�REMHFW�E\�REMHFW� 7KH�QDPHV�RI�WKH�DQLPDWLRQ�SDUDP�HWHUV�LQ�GLIIHUHQW�REMHFWV�QHHG�QRW�FRLQFLGH�

,Q�WKH�IROORZLQJ�H[DPSOH� GLIIHUHQW�QDPHV D DQG E DUH�XVHG�IRU�WKH�DQLPDWLRQSDUDPHWHUV�RI�WKH�WZR�IXQFWLRQV�

SORW�SORW��)XQFWLRQ�G�� D [� [ ����� D ������SORW��)XQFWLRQ�G�E [A�� [ ����� E ������

$Q�DQLPDWLRQ�SDUDPHWHU�LV�D�JOREDO�V\PEROLF�QDPH� ,W�FDQ�EH�XVHG�DV�D�JOREDOYDULDEOH�LQ�SURFHGXUHV�GHILQLQJ�WKH�JUDSKLFDO�REMHFW� 7KH�IROORZLQJ�H[DPSOHIHDWXUHV�WKH��' JUDSK�RI�D�EL�YDULDWH�IXQFWLRQ�WKDW�LV�GHILQHG�E\�D�SURFHGXUH�XVLQJWKH�JOREDOO\�GHILQHG�DQLPDWLRQ�SDUDPHWHU� )XUWKHU� D�ILOO�FRORU�IXQFWLRQ P\FRORU LVGHILQHG�WKDW�FKDQJHV�WKH�FRORU�LQ�WKH�FRXUVH�RI�WKH�DQLPDWLRQ� ,W�FRXOG�XVH�WKHDQLPDWLRQ�SDUDPHWHU�DV�D�JOREDO�SDUDPHWHU� MXVW�OLNH�WKH�IXQFWLRQ f GRHV�$OWHUQDWLYHO\� WKH�DQLPDWLRQ�SDUDPHWHU�PD\�EH�GHFODUHG�DV�DQ�DGGLWLRQDO�LQSXWSDUDPHWHU� 5HIHU�WR�WKH�KHOS�SDJH�RI )LOO&RORU)XQFWLRQ WR�ILQG�RXW�KRZ�PDQ\LQSXW�SDUDPHWHUV�WKH�ILOO�FRORU�IXQFWLRQ�H[SHFWV�DQG�ZKLFK�RI�WKH�LQSXW�SDUDPHWHUV

�����

Page 309: Mupad Tutorial

:gbfZmbhgl

UHFHLYHV�WKH�DQLPDWLRQ�SDUDPHWHU� 2QH�ILQGV�WKDW�IRU )XQFWLRQ�G� WKH�ILOO�FRORUIXQFWLRQ�LV�FDOOHG�ZLWK�WKH�FRRUGLQDWHV x� y� z RI�WKH�SRLQWV�RQ�WKH�JUDSK� 7KH�QH[WLQSXW�SDUDPHWHU��WKH��WK�DUJXPHQW�RI P\FRORU��LV�WKH�DQLPDWLRQ�SDUDPHWHU�

I � �[� \� �! � � �[ � D�A� � �\ � D�A��P\FRORU � SURF�[� \� ]� D�

ORFDO W�EHJLQ

W � VTUW��[ � D�A� � �\ � D�A���LI W � ��� WKHQ UHWXUQ�5*%��5HG�HOLI W � ��� WKHQ UHWXUQ�5*%��2UDQJH�HOLI W � ��� WKHQ UHWXUQ�5*%��*UHHQ�HOVH UHWXUQ�5*%��%OXH�HQGBLI�

HQG�SORW�SORW��)XQFWLRQ�G�I� [ ������ \ ������ D ������

)LOO&RORU)XQFWLRQ P\FRORU��

3OD\LQJ�$QLPDWLRQV

:KHQ�DQ�DQLPDWHG�SORW�LV�FUHDWHG�LQ�D�0X3$'� QRWHERRN� WKH�ILUVW�IUDPH�RI�WKHDQLPDWLRQ�DSSHDUV�DV�D�VWDWLF�SLFWXUH�EHORZ�WKH�LQSXW�UHJLRQ� 7R�VWDUW�WKH

�����

Page 310: Mupad Tutorial

** @kZiab\l

DQLPDWLRQ� FOLFN�RQ�WKH�SORW� $Q�LFRQ�IRU�VWRSSLQJ�DQG�UH�VWDUWLQJ�WKH�DQLPDWLRQZLOO�DSSHDU��PDNH�VXUH�WKH�LWHP�µ$QLPDWLRQ�%DU¶�RI�WKH�µ9LHZ¶�PHQX�LV�HQDEOHG��

2QH�FDQ�DOVR�XVH�WKH�VOLGHU�WR�DQLPDWH�WKH�SLFWXUH�³E\�KDQG�´

7KH�1XPEHU�RI�)UDPHV�DQG�WKH�7LPH�5DQJH

%\�GHIDXOW� DQ�DQLPDWLRQ�FRQVLVWV�RI����IUDPHV� 7KH�QXPEHU�RI�IUDPHV�FDQ�EH�VHWWR�EH�DQ\�SRVLWLYH�QXPEHU n E\�VSHFLI\LQJ�WKH�DWWULEXWH )UDPHV Q� 7KLV�DWWULEXWHFDQ�EH�VHW�LQ�WKH�JHQHUDWLQJ�FDOO�RI�WKH�DQLPDWHG�SULPLWLYHV� RU�DW�VRPH�KLJKHUQRGH�RI�WKH�JUDSKLFDO�WUHH� ,Q�WKH�ODWWHU�FDVH� WKLV�DWWULEXWH�LV�LQKHULWHG�E\�DOOSULPLWLYHV�WKDW�H[LVW�EHORZ�WKH�QRGH� :LWK D DPLQ��DPD[� )UDPHV Q� WKH i�WKIUDPH�FRQVLVWV�RI�D�VQDSVKRW�RI�WKH�SULPLWLYH�ZLWK

a = aPLQ +i− 1n− 1

· (aPD[ − aPLQ), i = 1, . . . , n.

,QFUHDVLQJ�WKH�QXPEHU�RI�IUDPHV�GRHV�QRW�PHDQ�WKDW�WKH�DQLPDWLRQ�UXQV�ORQJHU�WKH�UHQGHUHU�GRHV�QRW�ZRUN�ZLWK�D�IL[HG�QXPEHU�RI�IUDPHV�SHU�VHFRQG�EXWSURFHVVHV�DOO�IUDPHV�ZLWKLQ�D�IL[HG�WLPH�LQWHUYDO�

�����

Page 311: Mupad Tutorial

:gbfZmbhgl

,Q�WKH�EDFNJURXQG� WKHUH�LV�D�³UHDO�WLPH�FORFN´�XVHG�WR�V\QFKURQL]H�WKH�DQLPDWLRQRI�GLIIHUHQW�DQLPDWHG�REMHFWV� $Q�DQLPDWLRQ�KDV�D�WLPH�UDQJH�PHDVXUHG�E\�WKLVFORFN� 7KH�WLPH�UDQJH�LV�VHW�E\�WKH�DWWULEXWHV 7LPH%HJLQ W�� 7LPH(QG W� RU�HTXLYDOHQWO\� 7LPH5DQJH W���W�� ZKHUH W�� W� DUH�UHDO�QXPHULFDO�YDOXHVUHSUHVHQWLQJ�SK\VLFDO�WLPHV�LQ�VHFRQGV� 7KHVH�DWWULEXWH�FDQ�EH�VHW�LQ�WKHJHQHUDWLQJ�FDOO�RI�WKH�DQLPDWHG�SULPLWLYHV� RU�DW�VRPH�KLJKHU�QRGH�RI�WKH�JUDSKLFDOWUHH� ,Q�WKH�ODWWHU�FDVH� WKHVH�DWWULEXWHV�DUH�LQKHULWHG�E\�DOO�SULPLWLYHV�WKDW�H[LVWEHORZ�WKH�QRGH�

7KH�DEVROXWH�YDOXH�RI W� LV�LUUHOHYDQW�LI�DOO�DQLPDWHG�REMHFWV�VKDUH�WKH�VDPH�WLPHUDQJH� 2QO\�WKH�WLPH�GLIIHUHQFH W��W�PDWWHUV� ,W�LV��DQ�DSSUR[LPDWLRQ�RI��WKHSK\VLFDO�WLPH�LQ�VHFRQGV�WKDW�WKH�DQLPDWLRQ�ZLOO�ODVW�

7KH�SDUDPHWHU�UDQJH DPLQ��DPD[ LQ�WKH�VSHFLILFDWLRQ�RI�WKH�DQLPDWLRQ�SDUDP�HWHU D DPLQ��DPD[ WRJHWKHU�ZLWK )UDPHV Q GHILQHV�DQ�HTXLGLVWDQW�WLPH�PHVKLQ�WKH�WLPH�LQWHUYDO�VHW�E\ 7LPH%HJLQ W� DQG 7LPH(QG W�� 7KH�IUDPH�ZLWKD DPLQ LV�YLVLEOH�DW�WKH�WLPH W�� WKH�IUDPH�ZLWK D DPD[ LV�YLVLEOH�DW�WKH�WLPHW��

:LWK�WKH�GHIDXOW 7LPH%HJLQ �� WKH�YDOXH�RI�WKH�DWWULEXWH 7LPH(QG JLYHV�WKHSK\VLFDO�WLPH�RI�WKH�DQLPDWLRQ�LQ�VHFRQGV� 7KH�GHIDXOW�YDOXH�LV 7LPH(QG ���L�H�� DQ�DQLPDWLRQ�XVLQJ�WKH�GHIDXOW�YDOXHV�ZLOO�ODVW�DERXW����VHFRQGV� 7KHQXPEHU�RI�IUDPHV�VHW�E\ )UDPHV Q GRHV�QRW�LQIOXHQFH�WKH�WLPH�LQWHUYDO� EXWFKDQJHV�WKH�QXPEHU�RI�IUDPHV�GLVSOD\HG�LQ�WKLV�WLPH�LQWHUYDO�

+HUH�LV�D�VLPSOH�H[DPSOH�

SORW�SORW��3RLQW�G�>D� VLQ�D�@� D ���� 3,�)UDPHV ��� 7LPH5DQJH ������

7KH�SRLQW�ZLOO�EH�DQLPDWHG�IRU�DERXW���SK\VLFDO�VHFRQGV�LQ�ZKLFK�LW�PRYHV�DORQJRQH�SHULRG�RI�WKH�VLQH�JUDSK� (DFK�IUDPH�LV�GLVSOD\HG�IRU�DERXW�����VHFRQGV� $IWHULQFUHDVLQJ�WKH�QXPEHU�RI�IUDPHV�E\�D�IDFWRU�RI��� HDFK�IUDPH�LV�GLVSOD\HG�IRU�DERXW�����VHFRQGV� PDNLQJ�WKH�DQLPDWLRQ�VRPHZKDW�VPRRWKHU�

SORW�SORW��3RLQW�G�>D� VLQ�D�@� D ���� 3,�)UDPHV ���� 7LPH5DQJH ������

�����

Page 312: Mupad Tutorial

** @kZiab\l

1RWH�WKDW�WKH�KXPDQ�H\H�FDQQRW�GLVWLQJXLVK�EHWZHHQ�GLIIHUHQW�IUDPHV�LI�WKH\FKDQJH�ZLWK�D�UDWH�RI�PRUH�WKDQ����IUDPHV�SHU�VHFRQG� 7KXV� WKH�QXPEHU�RIIUDPHV n VHW�IRU�WKH�DQLPDWLRQ�VKRXOG�VDWLVI\

n < 25 · (t1 − t0).

+HQFH� ZLWK�WKH�GHIDXOW�WLPH�UDQJH 7LPH%HJLQ �� 7LPH(QG �� �VHFRQGV�� LW�GRHVQRW�PDNH�VHQVH�WR�VSHFLI\ )UDPHV Q ZLWK n > 250� ,I�D�KLJKHU�IUDPH�QXPEHU�LVUHTXLUHG�WR�REWDLQ�D�VXIILFLHQW�UHVROXWLRQ�RI�WKH�DQLPDWHG�REMHFW� RQH�VKRXOG�DOVRLQFUHDVH�WKH�WLPH�IRU�WKH�DQLPDWLRQ�E\�D�VXIILFLHQWO\�KLJK�YDOXH�RI 7LPH(QG�

:KDW�&DQ�%H�$QLPDWHG"

:H�PD\�UHJDUG�D�JUDSKLFDO�SULPLWLYH�DV�D�FROOHFWLRQ�RI�SORW�DWWULEXWHV� �7KLV�LVDFWXDOO\�ZKDW�WKH\�DUH�LQWHUQDOO\� ,QGHHG� HYHQ�WKH�IXQFWLRQ�H[SUHVVLRQ VLQ�[� LQSORW��)XQFWLRQ�G�VLQ�[��[ ���� 3,� LV�UHDOL]HG�DV�WKH�DWWULEXWH)XQFWLRQ VLQ�[��� 6R� WKH�TXHVWLRQ�LV�

³:KLFK�DWWULEXWHV�FDQ�EH�DQLPDWHG"´

7KH�DQVZHU�LV� ³$OPRVW�DQ\�DWWULEXWH�FDQ�EH�DQLPDWHG�´ ,QVWHDG�RI�OLVWLQJ�WKHDWWULEXWHV�WKDW�DOORZ�DQLPDWLRQ� LW�LV�PXFK�HDVLHU�WR�FKDUDFWHUL]H�WKH�DWWULEXWHVWKDW�FDQQRW�EH�DQLPDWHG�

� 1RQH�RI�WKH�FDQYDV�DWWULEXWHV�FDQ�EH�DQLPDWHG� 7KLV�LQFOXGHV�OD\RXWSDUDPHWHUV�VXFK�DV�WKH�SK\VLFDO�VL]H�RI�WKH�SLFWXUH� 6HH�WKH�KHOS�SDJH�RISORW��&DQYDV IRU�D�FRPSOHWH�OLVW�RI�DOO�FDQYDV�DWWULEXWHV�

� 1RQH�RI�WKH�DWWULEXWHV�RI��' VFHQHV�DQG��' VFHQHV�FDQ�EH�DQLPDWHG� 7KLVLQFOXGHV�OD\RXW�SDUDPHWHUV� EDFNJURXQG�FRORU�DQG�VW\OH� DXWRPDWLF�FDPHUDSRVLWLRQLQJ�LQ��' HWF� 6HH�WKH�KHOS�SDJHV�RI SORW��6FHQH�G DQGSORW��6FHQH�G IRU�D�FRPSOHWH�OLVW�RI�DOO�VFHQH�DWWULEXWHV�1RWH�WKDW�WKHUH�DUH�FDPHUD�REMHFWV�RI�W\SH SORW��&DPHUD WKDW�FDQ�EH�SODFHGLQ�D��' VFHQH� 7KHVH�FDPHUD�REMHFWV�FDQ�EH�DQLPDWHG�DQG�DOORZ�WR�UHDOL]H�D³IOLJKW´�WKURXJK�D��' VFHQH� 6HH�6HFWLRQ ����� IRU�GHWDLOV�

� 1RQH�RI�WKH�DWWULEXWHV�RI��' FRRUGLQDWH�V\VWHPV�DQG��' FRRUGLQDWH�V\VWHPVFDQ�EH�DQLPDWHG� 7KLV�LQFOXGHV�YLHZLQJ�ER[HV� D[HV� D[HV�WLFNV� DQG�JULG

�����

Page 313: Mupad Tutorial

:gbfZmbhgl

OLQHV��UXOLQJV��LQ�WKH�EDFNJURXQG� 6HH�WKH�KHOS�SDJHV�RISORW��&RRUGLQDWH6\VWHP�G DQG SORW��&RRUGLQDWH6\VWHP�G IRU�DFRPSOHWH�OLVW�RI�DOO�DWWULEXWHV�IRU�FRRUGLQDWH�V\VWHPV�

$OWKRXJK�WKH 9LHZLQJ%R[ DWWULEXWH�RI�D�FRRUGLQDWH�V\VWHP�FDQQRW�EHDQLPDWHG� WKH�XVHU�FDQ�VWLOO�DFKLHYH�DQLPDWHG�YLVLELOLW\�HIIHFWV�LQ��' E\FOLSSLQJ�ER[�REMHFWV�RI�W\SH SORW��&OLSSLQJ%R[�

� 1RQH�RI�WKH�DWWULEXWHV�WKDW�DUH�GHFODUHG�DV�³$WWULEXWH�7\SH� LQKHULWHG´RQ�WKHLU�KHOS�SDJH�FDQ�EH�DQLPDWHG� 7KLV�LQFOXGHV�VL]H�VSHFLILFDWLRQV�VXFK�DV3RLQW6L]H� /LQH:LGWK HWF�

� 5*% DQG�5*%D�YDOXHV�FDQQRW�EH�DQLPDWHG� +RZHYHU� LW�LV�SRVVLEOH�WRDQLPDWH�WKH�FRORULQJ�RI�OLQHV�DQG�VXUIDFHV�YLD�XVHU�GHILQHG�SURFHGXUHV� 6HHWKH�KHOS�SDJHV�RI /LQH&RORU)XQFWLRQ DQG )LOO&RORU)XQFWLRQ IRU�GHWDLOV�

� 7KH�WH[WV�RI�DQQRWDWLRQV�VXFK�DV )RRWHU� +HDGHU� 7LWOH� OHJHQG�HQWULHV�HWF�FDQQRW�EH�DQLPDWHG� 7KH�SRVLWLRQ�RI�WLWOHV� KRZHYHU� FDQ�EH�DQLPDWHG�

7KHUH�DUH�VSHFLDO�WH[W�REMHFWV SORW��7H[W�G DQG SORW��7H[W�G WKDW�DOORZ�WRDQLPDWH�WKH�WH[W�DV�ZHOO�DV�WKHLU�SRVLWLRQ�

� )RQWV�FDQQRW�EH�DQLPDWHG�

� $WWULEXWHV�VXFK�DV 'LVFRQWLQXLW\6HDUFK 758( RU )LOO3DWWHUQ 6ROLG WKDWFDQ�DVVXPH�RQO\�ILQLWHO\�PDQ\�YDOXHV�IURP�D�IL[HG�GLVFUHWH�VHW�FDQQRW�EHDQLPDWHG�

1HDUO\�DOO�DWWULEXWHV�QRW�IDOOLQJ�LQWR�RQH�RI�WKHVH�FDWHJRULHV�FDQ�EH�DQLPDWHG� <RXZLOO�ILQG�GHWDLOHG�LQIRUPDWLRQ�RQ�WKLV�LVVXH�RQ�WKH�FRUUHVSRQGLQJ�KHOS�SDJHV�RISULPLWLYHV�DQG�DWWULEXWHV�

$GYDQFHG�$QLPDWLRQV� 7KH�6\QFKURQL]DWLRQ�0RGHO

$V�DOUHDG\�H[SODLQHG�RQ�SDJH ������ WKHUH�LV�D�³UHDO�WLPH�FORFN´�UXQQLQJ�LQ�WKHEDFNJURXQG�WKDW�V\QFKURQL]HV�WKH�DQLPDWLRQ�RI�GLIIHUHQW�DQLPDWHG�REMHFWV�

(DFK�DQLPDWHG�REMHFW�KDV�LWV�RZQ�VHSDUDWH�³UHDO�WLPH�OLIH�VSDQ´�VHW�E\�WKHDWWULEXWHV 7LPH%HJLQ W�� 7LPH(QG W� RU� HTXLYDOHQWO\� 7LPH5DQJH W���W�� 7KHYDOXHV W�� W� UHSUHVHQW�VHFRQGV�PHDVXUHG�E\�WKH�³UHDO�WLPH�FORFN�´

�����

Page 314: Mupad Tutorial

** @kZiab\l

,Q�PRVW�FDVHV� WKHUH�LV�QR�QHHG�WR�ERWKHU�DERXW�VSHFLI\LQJ�WKH�OLIH�VSDQ� ,I7LPH%HJLQ DQG 7LPH(QG DUH�QRW�VSHFLILHG� WKH�GHIDXOW�YDOXHV 7LPH%HJLQ � DQG7LPH(QG �� DUH�XVHG� L�H�� WKH�DQLPDWLRQ�ZLOO�ODVW�DERXW����VHFRQGV� 7KHVH�YDOXHVRQO\�QHHG�WR�EH�PRGLILHG

� LI�D�VKRUWHU�RU�ORQJHU�UHDO�WLPH�SHULRG�IRU�WKH�DQLPDWLRQ�LV�GHVLUHG� RU

� LI�WKH�DQLPDWLRQ�FRQWDLQV�VHYHUDO�DQLPDWHG�REMHFWV� ZKHUH�VRPH�RI�WKHDQLPDWHG�REMHFWV�DUH�WR�UHPDLQ�VWDWLF�ZKLOH�RWKHUV�FKDQJH�

+HUH�LV�DQ�H[DPSOH�IRU�WKH�VHFRQG�VLWXDWLRQ� 7KH�SORW�FRQVLVWV�RI���MXPSLQJ�SRLQWV�)RU�WKH�ILUVW���VHFRQGV� WKH�OHIW�SRLQW�MXPSV�XS�DQG�GRZQ� ZKLOH�WKH�RWKHU�SRLQWVUHPDLQ�DW�WKHLU�LQLWLDO�SRVLWLRQ� 7KHQ� DOO�SRLQWV�VWD\�VWDWLF�IRU���VHFRQG� $IWHU�DWRWDO�RI���VHFRQGV� WKH�PLGGOH�SRLQW�VWDUWV�LWV�DQLPDWLRQ�E\�MXPSLQJ�XS�DQG�GRZQ�ZKLOH�WKH�OHIW�SRLQW�UHPDLQV�VWDWLF�LQ�LWV�ILQDO�SRVLWLRQ�DQG�WKH�ULJKW�SRLQWV�VWD\VVWDWLF�LQ�LWV�LQLWLDO�SRVLWLRQ� $IWHU���VHFRQGV� WKH�ULJKW�SRLQW�EHJLQV�WR�PRYH�DV�ZHOO�7KH�RYHUDOO�WLPH�VSDQ�IRU�WKH�DQLPDWLRQ�LV�WKH�KXOO�RI�WKH�WLPH�UDQJHV�RI�DOODQLPDWHG�REMHFWV� L�H�� ���VHFRQGV�LQ�WKLV�H[DPSOH�

S� � SORW��3RLQW�G���� VLQ�D�� D ���3,�&RORU 5*%��5HG�7LPH%HJLQ �� 7LPH(QG ���

S� � SORW��3RLQW�G��� VLQ�D�� D ���3,�&RORU 5*%��%ODFN�7LPH%HJLQ �� 7LPH(QG ����

S� � SORW��3RLQW�G��� VLQ�D�� D ���3,�&RORU 5*%��%OXH�7LPH%HJLQ �� 7LPH(QG ����

�����

Page 315: Mupad Tutorial

:gbfZmbhgl

SORW�S�� S�� S�� 3RLQW6L]H ��� XQLW��PP�<$[LV9LVLEOH )$/6(�

+HUH� DOO�SRLQWV�XVH�WKH�GHIDXOW�VHWWLQJV 9LVLEOH%HIRUH%HJLQ 758( DQG9LVLEOH$IWHU(QG 758( ZKLFK�PDNH�WKHP�YLVLEOH�DV�VWDWLF�REMHFWV�RXWVLGH�WKHWLPH�UDQJH�RI�WKHLU�DQLPDWLRQ� :H�VHW 9LVLEOH$IWHU(QG )$/6( IRU�WKH�PLGGOHSRLQW� VR�WKDW�LW�GLVDSSHDUV�DIWHU�WKH�HQG�RI�LWV�DQLPDWLRQ� :LWK9LVLEOH%HIRUH%HJLQ )$/6(��WKH�ULJKW�SRLQW�LV�QRW�YLVLEOH�XQWLO�LWV�DQLPDWLRQVWDUWV�

S���9LVLEOH$IWHU(QG � )$/6(�S���9LVLEOH%HIRUH%HJLQ � )$/6(�

�����

Page 316: Mupad Tutorial

** @kZiab\l

SORW�S�� S�� S�� 3RLQW6L]H ��� XQLW��PP�<$[LV9LVLEOH )$/6(�

:H�VXPPDUL]H�WKH�V\QFKURQL]DWLRQ�PRGHO�RI�DQLPDWLRQV�✞

✆7KH�WRWDO�UHDO�WLPH�VSDQ�RI�DQ�DQLPDWHG�SORW�LV�WKH�SK\VLFDO�UHDO�WLPH�JLYHQ�E\WKH�PLQLPXP�RI�WKH 7LPH%HJLQ YDOXHV�RI�DOO�DQLPDWHG�REMHFWV�LQ�WKH�SORW�WR�WKHPD[LPXP�RI�WKH 7LPH(QG YDOXHV�RI�DOO�WKH�DQLPDWHG�REMHFWV�

� :KHQ�D�SORW�FRQWDLQLQJ�DQLPDWHG�REMHFWV�LV�FUHDWHG� WKH�UHDO�WLPH�FORFN�LV�VHWWR�WKH�PLQLPXP�RI�WKH 7LPH%HJLQ YDOXHV�RI�DOO�DQLPDWHG�REMHFWV�LQ�WKH�SORW�7KH�UHDO�WLPH�FORFN�LV�VWDUWHG�ZKHQ�SXVKLQJ�WKH�µSOD\¶�EXWWRQ�IRU�DQLPDWLRQVLQ�WKH�JUDSKLFDO�XVHU�LQWHUIDFH�

� %HIRUH�WKH�UHDO�WLPH�UHDFKHV�WKH 7LPH%HJLQ YDOXH W� RI�DQ�DQLPDWHG�REMHFW�WKLV�REMHFW�LV�VWDWLF�LQ�WKH�VWDWH�FRUUHVSRQGLQJ�WR�WKH�EHJLQ�RI�LWV�DQLPDWLRQ�'HSHQGLQJ�RQ�WKH�DWWULEXWH 9LVLEOH%HIRUH%HJLQ� LW�PD\�EH�YLVLEOH�RULQYLVLEOH�EHIRUH W��

� 'XULQJ�WKH�WLPH�IURP W� WR W�� WKH�REMHFW�FKDQJHV�IURP�LWV�RULJLQDO�WR�LWV�ILQDOVWDWH�

�����

Page 317: Mupad Tutorial

:gbfZmbhgl

� $IWHU�WKH�UHDO�WLPH�UHDFKHV�WKH 7LPH(QG YDOXH W�� WKH�REMHFW�VWD\V�VWDWLF�LQ�WKHVWDWH�FRUUHVSRQGLQJ�WR�WKH�HQG�RI�LWV�DQLPDWLRQ� 'HSHQGLQJ�RQ�WKH�YDOXH�RIWKH�DWWULEXWH 9LVLEOH$IWHU(QG� LW�PD\�VWD\�YLVLEOH�RU�EHFRPH�LQYLVLEOH�DIWHUW��

� 7KH�DQLPDWLRQ�RI�WKH�HQWLUH�SORW�HQGV�ZLWK�WKH�SK\VLFDO�WLPH�JLYHQ�E\�WKHPD[LPXP�RI�WKH 7LPH(QG YDOXHV�RI�DOO�DQLPDWHG�REMHFWV�LQ�WKH�SORW�

)UDPH�E\�)UDPH�$QLPDWLRQV

7KHUH�DUH�VRPH�VSHFLDO�DWWULEXWHV�VXFK�DV 9LVLEOH$IWHU WKDW�DUH�YHU\�XVHIXO�WREXLOG�DQLPDWLRQV�IURP�SXUHO\�VWDWLF�REMHFWV�✞

✆:LWK 9LVLEOH$IWHU W�� DQ�REMHFW�LV�LQYLVLEOH�IURP�WKH�VWDUW�RI�WKH�DQLPDWLRQXQWLO�WLPH W�� 7KHQ�LW�ZLOO�DSSHDU�DQG�UHPDLQ�YLVLEOH�IRU�WKH�UHVW�RI�WKH�DQLPD�WLRQ�

✆:LWK 9LVLEOH%HIRUH W�� DQ�REMHFW�LV�YLVLEOH�IURP�WKH�VWDUW�RI�WKH�DQLPDWLRQXQWLO�WLPH W�� 7KHQ�LW�ZLOO�GLVDSSHDU�DQG�UHPDLQ�LQYLVLEOH�IRU�WKH�UHVW�RI�WKHDQLPDWLRQ�

7KHVH�DWWULEXWHV�VKRXOG�QRW�EH�FRPELQHG�WR�GHILQH�D�³YLVLELOLW\�UDQJH´�IURP W� WRW�� 8VH�WKH�DWWULEXWH 9LVLEOH)URP7R LQVWHDG�✞

✆:LWK 9LVLEOH)URP7R W���W�� DQ�REMHFW�LV�LQYLVLEOH�IURP�WKH�VWDUW�RI�WKH�DQ�LPDWLRQ�XQWLO�WLPH W�� 7KHQ�LW�ZLOO�DSSHDU�DQG�UHPDLQ�YLVLEOH�XQWLO�WLPH W��ZKHQ�LW�ZLOO�GLVDSSHDU�DQG�UHPDLQ�LQYLVLEOH�IRU�WKH�UHVW�RI�WKH�DQLPDWLRQ�

:H�FRQWLQXH�WKH�H[DPSOH�RI�WKH�SUHYLRXV�VHFWLRQ�LQ�ZKLFK�ZH�GHILQHG�WKHIROORZLQJ�DQLPDWHG�SRLQWV�

�����

Page 318: Mupad Tutorial

** @kZiab\l

S� � SORW��3RLQW�G���� VLQ�D�� D ���3,�&RORU 5*%��5HG�7LPH%HJLQ �� 7LPH(QG ���

S� � SORW��3RLQW�G��� VLQ�D�� D ���3,�&RORU 5*%��%ODFN�7LPH%HJLQ �� 7LPH(QG ����

S� � SORW��3RLQW�G��� VLQ�D�� D ���3,�&RORU 5*%��%OXH�7LPH%HJLQ �� 7LPH(QG ����

S���9LVLEOH$IWHU(QG � )$/6(�S���9LVLEOH%HIRUH%HJLQ � )$/6(�

:H�DGG�D�IXUWKHU�SRLQW S� WKDW�LV�QRW�DQLPDWHG� :H�PDNH�LW�LQYLVLEOH�DW�WKH�VWDUW�RIWKH�DQLPDWLRQ�YLD�WKH�DWWULEXWH 9LVLEOH)URP7R� ,W�LV�PDGH�YLVLEOH�DIWHU���VHFRQGVWR�GLVDSSHDU�DJDLQ�DIWHU����VHFRQGV�

S� � SORW��3RLQW�G����� ���� &RORU 5*%��%ODFN�9LVLEOH)URP7R �������

7KH�VWDUW�RI�WKH�DQLPDWLRQ�LV�GHWHUPLQHG�E\ S� ZKLFK�EHDUV�WKH�DWWULEXWH7LPH%HJLQ �� WKH�HQG�RI�WKH�DQLPDWLRQ�LV�GHWHUPLQHG�E\ S� ZKLFK�KDV�VHW7LPH(QG ���SORW�S�� S�� S�� S�� 3RLQW6L]H ��� XQLW��PP�

<$[LV9LVLEOH )$/6(�

�����

Page 319: Mupad Tutorial

:gbfZmbhgl

$OWKRXJK�D�W\SLFDO�0X3$'�DQLPDWLRQ�LV�JHQHUDWHG�REMHFW�E\�REMHFW� HDFK�DQLPDWHGREMHFW�WDNLQJ�FDUH�RI�LWV�RZQ�DQLPDWLRQ� ZH�FDQ�DOVR�XVH�WKH�DWWULEXWHV9LVLEOH$IWHU� 9LVLEOH%HIRUH� 9LVLEOH)URP7R WR�EXLOG�XS�DQ�DQLPDWLRQ�IUDPH�E\IUDPH�✞

³)UDPH�E\�IUDPH�DQLPDWLRQV�´ &KRRVH�D�FROOHFWLRQ�RI��W\SLFDOO\�VWDWLF�JUDSKLFDO�SULPLWLYHV�WKDW�DUH�WR�EH�YLVLEOH�LQ�WKH i�WK�IUDPH�RI�WKH�DQLPDWLRQ�6HW 9LVLEOH)URP7R W>L@��W>L��@ IRU�WKHVH�SULPLWLYHV� ZKHUH W>L@��W>L��@LV�WKH�UHDO�WLPH�OLIH�VSDQ�RI�WKH i�WK�IUDPH��LQ�VHFRQGV�� )LQDOO\� SORW�DOO�IUDPHVLQ�D�VLQJOH SORW FRPPDQG�

+HUH�LV�DQ�H[DPSOH� :H�OHW�WZR�SRLQWV�ZDQGHU�DORQJ�WKH�JUDSKV�RI�WKH�VLQH�DQG�WKHFRVLQH�IXQFWLRQ� UHVSHFWLYHO\� (DFK�IUDPH�LV�WR�FRQVLVW�RI�D�SLFWXUH�RI�WZR�SRLQWV�:H�XVH SORW��*URXS�G WR�GHILQH�WKH�IUDPH� WKH�JURXS�IRUZDUGV�WKH�DWWULEXWH9LVLEOH)URP7R WR�DOO�LWV�HOHPHQWV�IRU L IURP � WR ��� GR

W>L@ � L����HQGBIRU�IRU L IURP � WR ��� GR

[ � L���� 3,�P\IUDPH>L@ � SORW��*URXS�G�

SORW��3RLQW�G�>[� VLQ�[�@� &RORU 5*%��5HG��SORW��3RLQW�G�>[� FRV�[�@� &RORU 5*%��%OXH��9LVLEOH)URP7R W>L@��W>L � �@��

HQGBIRU�

�����

Page 320: Mupad Tutorial

** @kZiab\l

SORW�P\IUDPH>L@ � L ������� 3RLQW6L]H ��� XQLW��PP�

7KLV�³IUDPH�E\�IUDPH´�DQLPDWLRQ�FHUWDLQO\�QHHGV�D�OLWWOH�ELW�PRUH�FRGLQJ�HIIRUWWKDQ�WKH�HTXLYDOHQW�REMHFW�ZLVH�DQLPDWLRQ� ZKHUH�HDFK�RI�WKH�SRLQWV�LV�DQLPDWHG�

�����

Page 321: Mupad Tutorial

:gbfZmbhgl

&RPSDUH�

GHOHWH L�SORW�SORW��3RLQW�G�>L���� 3,� VLQ�L���� 3,�@�

L ������� &RORU 5*%��5HG��SORW��3RLQW�G�>L���� 3,� FRV�L���� 3,�@�

L ������� &RORU 5*%��%OXH��)UDPHV ���� 3RLQW6L]H ��� XQLW��PP�

7KHUH�LV�D�VSHFLDO�NLQG�RI�SORW�ZKHUH�³IUDPH�E\�IUDPH´�DQLPDWLRQV�DUH�YHU\�XVHIXO�1RWH�WKDW�LQ�WKH�SUHVHQW�YHUVLRQ�RI�WKH�JUDSKLFV� QHZ�SORW�REMHFWV�FDQQRW�EH�DGGHGWR�D�VFHQH�WKDW�LV�DOUHDG\�UHQGHUHG� :LWK�WKH�VSHFLDO�³YLVLELOLW\´�DQLPDWLRQV�IRUVWDWLF�REMHFWV� KRZHYHU� RQH�FDQ�HDVLO\�VLPXODWH�D�SORW�WKDW�JUDGXDOO\�EXLOGV�XS� ILOOWKH�IUDPHV�RI�WKH�DQLPDWLRQ�ZLWK�VWDWLF�REMHFWV�WKDW�DUH�YLVLEOH�IRU�D�OLPLWHG�WLPHRQO\� 7KH�YLVLELOLW\�FDQ�EH�FKRVHQ�YHU\�IOH[LEO\�E\�WKH�XVHU� )RU�H[DPSOH� WKH�VWDWLFREMHFWV�FDQ�EH�PDGH�YLVLEOH�RQO\�IRU�RQH�IUDPH��9LVLEOH)URP7R��ZKLFK�DOORZV�WRVLPXODWH�PRYLQJ�REMHFWV�

�����

Page 322: Mupad Tutorial

** @kZiab\l

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�XVH 9LVLEOH$IWHU WR�ILOO�XS�WKH�SORW�JUDGXDOO\� :HGHPRQVWUDWH�WKH�FDXVWLFV�JHQHUDWHG�E\�VXQOLJKW�LQ�D�WHD�FXS� 7KH�ULP�RI�WKH�FXS�UHJDUGHG�DV�D�PLUURU� LV�JLYHQ�E\�WKH�IXQFWLRQ f(x) = −

√1− x2� x ∈ [−1, 1] �D

VHPL�FLUFOH�� 6XQ�UD\V�SDUDOOHO�WR�WKH y�D[LV�DUH�UHIOHFWHG�E\�WKH�ULP� $IWHUUHIOHFWLRQ�DW�WKH�SRLQW (x, f(x)) RI�WKH�ULP� D�UD\�KHDGV�LQWR�WKH�GLUHFWLRQ(−1,−(f ′(x)− 1/f ′(x))/2) LI x LV�SRVLWLYH� ,W�KHDGV�LQWR�WKH�GLUHFWLRQ(1, (f ′(x)− 1/f ′(x))/2) LI x LV�QHJDWLYH� 6ZHHSLQJ�WKURXJK�WKH�PLUURU�IURP�OHIW�WRULJKW� WKH�LQFRPLQJ�UD\V�DV�ZHOO�DV�WKH�UHIOHFWHG�UD\V�DUH�YLVXDOL]HG�DV�OLQHV� ,Q�WKHDQLPDWLRQ� WKH\�EHFRPH�YLVLEOH�DIWHU�WKH�WLPH 5x� ZKHUH x LV�WKH�FRRUGLQDWH�RI�WKHULP�SRLQW�DW�ZKLFK�WKH�UD\�LV�UHIOHFWHG�

�����

Page 323: Mupad Tutorial

:gbfZmbhgl

I � [ �! �VTUW�� � [A���SORW��� 7KH VWDWLF ULP�

SORW��)XQFWLRQ�G�I�[�� [ ������&RORU 5*%��%ODFN��

�� 7KH LQFRPLQJ UD\V�SORW��/LQH�G�>[� �@� >[� I�[�@�

&RORU 5*%��2UDQJH�9LVLEOH$IWHU � [

� � [ LQ >�� � L��� � L �����@��� 7KH UHIOHFWHG UD\V OHDYLQJ WR WKH ULJKW�SORW��/LQH�G�>[� I�[�@�

>�� I�[�����[� �I�[� � ��I�[����@�&RORU 5*%��2UDQJH�9LVLEOH$IWHU � [

� � [ LQ >�� � L��� � L �����@��� 7KH UHIOHFWHG UD\V OHDYLQJ WR WKH OHIW�SORW��/LQH�G�>[� I�[�@�

>��� I�[� � �[��� �I�[� � ��I�[����@�&RORU 5*%��2UDQJH�9LVLEOH$IWHU � [

� � [ LQ >�� � L��� � L ������@�9LHZLQJ%R[ >������ �����@�

�����

Page 324: Mupad Tutorial

** @kZiab\l

([DPSOHV

([DPSOH��� :H�EXLOG�D��' DQLPDWLRQ�WKDW�GLVSOD\V�D�IXQFWLRQ f(x) WRJHWKHU�ZLWKWKH�LQWHJUDO F (x) =

∫ x0 f(y)Gy� 7KH�DUHD�EHWZHHQ�WKH�JUDSK�RI f DQG�WKH x�D[LV�LV

GLVSOD\HG�DV�DQ�DQLPDWHG�KDWFK�REMHFW� 7KH�FXUUHQW�YDOXH�RI F (x) LV�GLVSOD\HG�E\DQ�DQLPDWHG�WH[W�

',*,76 � ���� WKH IXQFWLRQ� I�[��I � [ �! FRV�[A����� WKH DQWL�GHULYDWLYH� )�[��) � [ �! QXPHULF��LQW�I�\�� \ ���[���� WKH JUDSK RI I�[��J � SORW��)XQFWLRQ�G�I�[�� [ �����

&RORU 5*%��%OXH���� WKH JUDSK RI )�[��* � SORW��)XQFWLRQ�G�)�[�� [ �����

&RORU 5*%��%ODFN���� D SRLQW PRYLQJ DORQJ WKH JUDSK RI )�[��S � SORW��3RLQW�G�>D� )�D�@� D �����

&RORU 5*%��%ODFN���� KDWFKHG UHJLRQ EHWZHHQ WKH RULJLQ DQG�� WKH PRYLQJ SRLQW S�K � SORW��+DWFK�J� �� ���D� D �����

&RORU 5*%��5HG���� WKH ULJKW ERUGHU OLQH RI WKH KDWFKHG UHJLRQ�O � SORW��/LQH�G�>D� �@� >D� I�D�@� D �����

&RORU 5*%��5HG���� D GDVKHG YHUWLFDO OLQH IURP I�[� WR )�[��/� � SORW��/LQH�G�>D� I�D�@� >D� )�D�@�

D ����� &RORU 5*%��%ODFN�/LQH6W\OH 'DVKHG��

�� D GDVKHG KRUL]RQWDO OLQH IURP WKH \ D[LV WR )�[��/� � SORW��/LQH�G�>����� )�D�@� >D� )�D�@�

D ����� &RORU 5*%��%ODFN�/LQH6W\OH 'DVKHG��

�� WKH FXUUHQW YDOXH RI ) DW WKH PRYLQJ SRLQW S�W � SORW��7H[W�G�D �! )�D�� >����� )�D�@� D �����

+RUL]RQWDO$OLJQPHQW 5LJKW��

�����

Page 325: Mupad Tutorial

:gbfZmbhgl

SORW�J� *� S� K� O� /�� /�� W�<7LFNV1XPEHU 1RQH� <7LFNV$W >��� �@��

GHOHWH ',*,76�

([DPSOH��� :H�EXLOG�WZR��' DQLPDWLRQV� 7KH�ILUVW�RQH�VWDUWV�ZLWK�D�UHFWDQJXODUVWULS�WKDW�LV�GHIRUPHG�WR�DQ�DQQXOXV�LQ�WKH x� y SODQH�

F � D �! ��� �� � ��VLQ�3,�� D���P\FRORU � �X� Y� [� \� ]� �!

>�X � ��������� �� ���� � X�����@�UHFWDQJOH�DQQXOXV � SORW��6XUIDFH�

>F�D� � �X�F�D�� FRV�3, Y�� �X�F�D�� VLQ�3, Y�� �@�X ��������� Y �D��D� D ����A������)LOO&RORU)XQFWLRQ P\FRORU� 0HVK >�� ��@�)UDPHV ����

�����

Page 326: Mupad Tutorial

** @kZiab\l

SORW�UHFWDQJOH�DQQXOXV� $[HV 1RQH�&DPHUD'LUHFWLRQ >���� ��� �@�

7KH�VHFRQG�DQLPDWLRQ�WZLVWV�WKH�DQQXOXV�WR�EHFRPH�D�0RHELXV�VWULS�

DQQXOXV�PRHELXV � SORW��6XUIDFH�>��X � �� FRV�D Y 3,��� � �� FRV�3, Y����X � �� FRV�D Y 3,��� � �� VLQ�3, Y���X � �� VLQ�D Y 3,���@�

X ��������� Y ������ D ����� 0HVK >�� ��@�)LOO&RORU)XQFWLRQ P\FRORU� )UDPHV ����

�����

Page 327: Mupad Tutorial

:gbfZmbhgl

SORW�DQQXOXV�PRHELXV� $[HV 1RQH�&DPHUD'LUHFWLRQ >���� ��� �@�

1RWH�WKDW�WKH�ILQDO�IUDPH�RI�WKH�ILUVW�DQLPDWLRQ�FRLQFLGHV�ZLWK�WKH�ILUVW�IUDPH�RI�WKHVHFRQG�DQLPDWLRQ� 7R�MRLQ�WKH�WZR�VHSDUDWH�DQLPDWLRQV� ZH�FDQ�VHW�DSSURSULDWHYLVLELOLW\ UDQJHV�DQG�SORW�WKHP�WRJHWKHU� $IWHU���VHFRQGV� WKH�ILUVW�DQLPDWLRQREMHFW�YDQLVKHV�DQG�WKH�VHFRQG�WDNHV�RYHU�

UHFWDQJOH�DQQXOXV��9LVLEOH)URP7R � �����DQQXOXV�PRHELXV��9LVLEOH)URP7R � �����

�����

Page 328: Mupad Tutorial

** @kZiab\l

SORW�UHFWDQJOH�DQQXOXV� DQQXOXV�PRHELXV�$[HV 1RQH� &DPHUD'LUHFWLRQ >���� ��� �@�

�����

Page 329: Mupad Tutorial

@khnil�h_�Ikbfbmbo^l

?jgmhk�g^�Hjaealan]k

$Q�DUELWUDU\�QXPEHU�RI�JUDSKLFDO�SULPLWLYHV�LQ��' RU��' FDQ�EH�FROOHFWHG�LQJURXSV�RI�W\SH SORW��*URXS�G RU SORW��*URXS�G� UHVSHFWLYHO\� 7KLV�LV�XVHIXO�IRUKDQGLQJ�GRZQ�DWWULEXWH�YDOXHV�WR�DOO�HOHPHQWV�LQ�D�JURXS�

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�YLVXDOL]H�UDQGRP�JHQHUDWRUV�ZLWK�GLIIHUHQWGLVWULEXWLRQV�E\�XVLQJ�WKHP�WR�SRVLWLRQ�UDQGRP�SRLQWV�

U� � VWDWV��QRUPDO5DQGRP��� ���JURXS� � SORW��*URXS�G�SORW��3RLQW�G�U���� U����

� L ��������U� � VWDWV��XQLIRUP5DQGRP���� ���JURXS� � SORW��*URXS�G�SORW��3RLQW�G�U���� U����

� L ��������SORW�JURXS�� JURXS�� $[HV )UDPH�

:H�FDQQRW�GLVWLQJXLVK�EHWZHHQ�WKH�WZR�NLQGV�RI�SRLQWV� 'XH�WR�WKH�JURXSLQJ� LW�LVYHU\�HDV\�WR�FKDQJH�WKHLU�FRORU� VL]H� DQG�VW\OH�E\�VHWWLQJ�WKH�DSSURSULDWH�DWWULEXWHVLQ�WKH�JURXSV� 1RZ� WKH�WZR�NLQGV�RI�SRLQWV�FDQ�EH�GLVWLQJXLVKHG�HDVLO\�

�����

Page 330: Mupad Tutorial

** @kZiab\l

JURXS���3RLQW&RORU � 5*%��5HG�JURXS���3RLQW6W\OH � ;&URVVHV�JURXS���3RLQW&RORU � 5*%��%OXH�JURXS���3RLQW6L]H � ��� XQLW��PP�SORW�JURXS�� JURXS�� $[HV )UDPH�

�����

Page 331: Mupad Tutorial

MkZgl_hkfZmbhgl

LjYfk^gjeYlagfk

$IILQH�OLQHDU�WUDQVIRUPDWLRQV x⃗→ A · x⃗+ b⃗ ZLWK�D�YHFWRU b⃗ DQG�D�PDWUL[ A FDQ�EHDSSOLHG�WR�JUDSKLFDO�REMHFWV�YLD�WUDQVIRUPDWLRQ�REMHFWV� 7KHUH�DUH�VSHFLDOWUDQVIRUPDWLRQV�VXFK�DV�WUDQVODWLRQV� VFDOLQJ� DQG�URWDWLRQV�DV�ZHOO�DV�JHQHUDODIILQH�OLQHDU�WUDQVIRUPDWLRQV�

� SORW��7UDQVODWH�G�>E��E�@�3ULPLWLYH��3ULPLWLYH������ DSSOLHV�WKHWUDQVODWLRQ x⃗→ x⃗+ b⃗ E\�WKH�YHFWRU b⃗ = (b1, b2)T WR�DOO�SRLQWV�RI��' SULPLWLYHV�

� SORW��7UDQVODWH�G�>E��E��E�@�3ULPLWLYH������ DSSOLHV�WKH�WUDQVODWLRQx⃗→ x⃗+ b⃗ E\�WKH�YHFWRU b = (b1, b2, b3)T WR�DOO�SRLQWV�RI��' SULPLWLYHV�

� SORW��5HIOHFW�G�>[��\�@�>[��\�@�3ULPLWLYH������ UHIOHFWV�DOO��'SULPLWLYHV�DERXW�WKH�OLQH�WKURXJK�WKH�SRLQWV >[��\�@ DQG >[��\�@�

� SORW��5HIOHFW�G�>[�\�]@�>Q[�Q\�Q]@�3ULPLWLYH������ UHIOHFWV�DOO��'SULPLWLYHV�DERXW�WKH�SODQH�WKURXJK�WKH�SRLQW >[�\�]@ ZLWK�WKH�QRUPDO >Q[�Q\�Q]@�

� SORW��5RWDWH�G�DQJOH�>F��F�@�3ULPLWLYH������ URWDWHV�DOO�SRLQWV�RI�' SULPLWLYHV�FRXQWHU�FORFNZLVH�E\�WKH�JLYHQ�DQJOH�DERXW�WKH�SLYRW�SRLQW(c1, c2)T �

� SORW��5RWDWH�G�DQJOH�>F��F��F�@�>G��G��G�@�3ULPLWLYH������URWDWHV�DOO�SRLQWV�RI��' SULPLWLYHV�E\�WKH�JLYHQ�DQJOH�DURXQG�WKH�URWDWLRQD[LV�VSHFLILHG�E\�WKH�SLYRW�SRLQW (c1, c2, c3)T DQG�WKH�GLUHFWLRQ (d1, d2, d3)T �

� SORW��6FDOH�G�>V��V�@�3ULPLWLYH������ DSSOLHV�WKH�GLDJRQDO�VFDOLQJPDWUL[ GLDJ(s1, s2) WR�DOO�SRLQWV�RI��' SULPLWLYHV�

� SORW��6FDOH�G�>V��V��V�@�3ULPLWLYH������ DSSOLHV�WKH�GLDJRQDOVFDOLQJ�PDWUL[ GLDJ(s1, s2, s3) WR�DOO�SRLQWV�RI��' SULPLWLYHV�

� SORW��7UDQVIRUP�G�>E��E�@�$�3ULPLWLYH������ DSSOLHV�WKH�JHQHUDODIILQH�OLQHDU�WUDQVIRUPDWLRQ x⃗→ A · x⃗+ b⃗ ZLWK�D 2× 2PDWUL[ A DQG�D�YHFWRUb⃗ = (b1, b2)T WR�DOO�SRLQWV�RI��' SULPLWLYHV�

� SORW��7UDQVIRUP�G�>E��E��E�@�$�3ULPLWLYH������ DSSOLHV�WKH�JHQHUDODIILQH�OLQHDU�WUDQVIRUPDWLRQ x⃗→ A · x⃗+ b⃗ ZLWK�D 3× 3PDWUL[ A DQG�D�YHFWRUb⃗ = (b1, b2, b3)T WR�DOO�SRLQWV�RI��' SULPLWLYHV�

�����

Page 332: Mupad Tutorial

** @kZiab\l

7KH�HOOLSVHV SORW��(OOLSVH�G SURYLGHG�E\�WKH SORW OLEUDU\�KDYH�D[HV�SDUDOOHO�WRWKH�FRRUGLQDWH�D[HV� :H�XVH�D�URWDWLRQ�WR�FUHDWH�DQ�HOOLSVH�ZLWK�D�GLIIHUHQWRULHQWDWLRQ�

FHQWHU � >�� �@�HOOLSVH � SORW��(OOLSVH�G��� �� FHQWHU��SORW�SORW��5RWDWH�G�3,��� FHQWHU� HOOLSVH��

7UDQVIRUP�REMHFWV�FDQ�EH�DQLPDWHG� :H�EXLOG�D�JURXS�FRQVLVWLQJ�RI�WKH�HOOLSVH�DQGLWV�V\PPHWU\�D[HV� $Q�DQLPDWHG�URWDWLRQ�LV�DSSOLHG�WR�WKH�JURXS�

J � SORW��*URXS�G�HOOLSVH�SORW��/LQH�G�>FHQWHU>�@ � �� FHQWHU>�@@�

>FHQWHU>�@ � �� FHQWHU>�@@��SORW��/LQH�G�>FHQWHU>�@� FHQWHU>�@ � �@�

>FHQWHU>�@� FHQWHU>�@ � �@���

�����

Page 333: Mupad Tutorial

MkZgl_hkfZmbhgl

SORW�SORW��5RWDWH�G�D� FHQWHU� D ���� 3,� J��

�����

Page 334: Mupad Tutorial

** @kZiab\l

D]_]f\k

7KH�DQQRWDWLRQV�RI�D�0X3$'� SORW�PD\�LQFOXGH�D�OHJHQG� $ OHJHQG�LV�D�VPDOO�WDEOHWKDW�UHODWHV�WKH�FRORU�RI�DQ�REMHFW�ZLWK�VRPH�WH[W�H[SODLQLQJ�WKH�REMHFW�

I � � [ VLQ�� [��J � � [A� FRV�[��K � VLQ�� [��[�SORWIXQF�G�I� J� K� [ ���3,����

%\�GHIDXOW� OHJHQGV�DUH�SURYLGHG�RQO\�E\ SORWIXQF�G DQG SORWIXQF�G� 7KHVHURXWLQHV�GHILQH�WKH�OHJHQG�WH[WV�DV�WKH�H[SUHVVLRQV�ZLWK�ZKLFK�WKH�IXQFWLRQV�DUHSDVVHG�WR SORWIXQF�G RU SORWIXQF�G� UHVSHFWLYHO\�

������

Page 335: Mupad Tutorial

E^`^g]l

$ FRUUHVSRQGLQJ�SORW�FRPPDQG�XVLQJ�SULPLWLYHV�RI�WKH SORW OLEUDU\�GRHV�QRWJHQHUDWH�WKH�OHJHQG�DXWRPDWLFDOO\�

SORW�SORW��)XQFWLRQ�G�I� [ ���3,��� &RORU 5*%��5HG��SORW��)XQFWLRQ�G�J� [ ���3,��� &RORU 5*%��*UHHQ��SORW��)XQFWLRQ�G�K� [ ���3,��� &RORU 5*%��%OXH��

������

Page 336: Mupad Tutorial

** @kZiab\l

+RZHYHU� OHJHQGV�FDQ�EH�UHTXHVWHG�H[SOLFLWO\�

SORW�SORW��)XQFWLRQ�G�I� [ ���3,��� &RORU 5*%��5HG�/HJHQG �)XQFWLRQ �� ��H[SU�WH[W�I���

SORW��)XQFWLRQ�G�J� [ ���3,��� &RORU 5*%��*UHHQ�/HJHQG �)XQFWLRQ �� ��H[SU�WH[W�J���

SORW��)XQFWLRQ�G�K� [ ���3,��� &RORU 5*%��%OXH�/HJHQG �)XQFWLRQ �� ��H[SU�WH[W�K���

(DFK�JUDSKLFDO�SULPLWLYH�DFFHSWV�WKH�DWWULEXWH /HJHQG� 3DVVLQJ�WKLV�DWWULEXWH�WR�DQREMHFW�WULJJHUV�VHYHUDO�DFWLRQV�

� 7KH�REMHFW�DWWULEXWH /HJHQG7H[W LV�VHW�WR�WKH�JLYHQ�VWULQJ�

� 7KH�REMHFW�DWWULEXWH /HJHQG(QWU\ LV�VHW�WR 758(�

� $ KLQW�LV�VHQW�WR�WKH�VFHQH�FRQWDLQLQJ�WKH�REMHFW�DGYLVLQJ�LW�WR�XVH�WKH�VFHQHDWWULEXWH /HJHQG9LVLEOH 758(�

7KH�DWWULEXWHV /HJHQG7H[W DQG /HJHQG(QWU\ DUH�YLVLEOH�LQ�WKH�³REMHFW�LQVSHFWRU´�RIWKH�LQWHUDFWLYH�YLHZHU��SDJH �������DQG�FDQ�EH�PDQLSXODWHG�LQWHUDFWLYHO\�IRU�HDFKVLQJOH�SULPLWLYH�DIWHU�VHOHFWLRQ�LQ�WKH�³REMHFW�EURZVHU�´ 7KH�DWWULEXWH/HJHQG9LVLEOH LV�DVVRFLDWHG�ZLWK�WKH�VFHQH�REMHFW�DFFHVVLEOH�YLD�WKH�³REMHFWEURZVHU�´

������

Page 337: Mupad Tutorial

E^`^g]l

$W�PRVW����HQWULHV�FDQ�EH�GLVSOD\HG�LQ�D�OHJHQG� ,I�PRUH�HQWULHV�DUH�VSHFLILHG�LQ�DSORW�FRPPDQG� WKH�VXUSOXV�HQWULHV�DUH�LJQRUHG� )XUWKHU� WKH�OHJHQG�PD\�QRW�FRYHUPRUH�WKDQ�����RI�WKH�KHLJKW�RI�WKH�GUDZLQJ�DUHD�RI�D�VFHQH� 2QO\�WKRVH�OHJHQGHQWULHV�ILWWLQJ�LQWR�WKLV�VSDFH�DUH�GLVSOD\HG� UHPDLQLQJ�HQWULHV�DUH�LJQRUHG�

,I�WKH�DWWULEXWH /HJHQG(QWU\ 758( LV�VHW�IRU�D�SULPLWLYH� LWV�OHJHQG�HQWU\�LVGHWHUPLQHG�DV�IROORZV�

� ,I�WKH�DWWULEXWH /HJHQG7H[W LV�VSHFLILHG� LWV�YDOXH�LV�XVHG�IRU�WKH�OHJHQG�WH[W�

� ,I�QR /HJHQG7H[W LV�VSHFLILHG� EXW�WKH 1DPH DWWULEXWH�LV�VHW� WKH�QDPH�LV�XVHGIRU�WKH�OHJHQG�WH[W�

� ,I�QR 1DPH DWWULEXWH�LV�VSHFLILHG�HLWKHU� WKH�W\SH�RI�WKH�REMHFW�VXFK�DV)XQFWLRQ�G� &XUYH�G HWF��LV�XVHG�IRU�WKH�OHJHQG�WH[W�

+HUH�DUH�DOO�DWWULEXWHV�UHOHYDQW�IRU�OHJHQGV�✞

DWWULEXWH�QDPH SRVVLEOH�YDOXHV PHDQLQJ GHIDXOW

/HJHQG VWULQJ VHWV /HJHQG7H[W WRWKH�JLYHQ�VWULQJ�/HJHQG(QWU\ WR758(��DQG/HJHQG9LVLEOH WR758(�

/HJHQG(QWU\ 758(� )$/6( DGG�WKLV�REMHFW�WR�WKHOHJHQG"

758( IRU�IXQFWLRQJUDSKV� FXUYHV� DQGVXUIDFHV� )$/6(RWKHUZLVH

/HJHQG7H[W VWULQJ OHJHQG�WH[W/HJHQG9LVLEOH 758(� )$/6( OHJHQG�RQ�RII 758( IRU

SORWIXQF�G��GSORWWLQJ�PRUH�WKDQRQH�IXQFWLRQ� )$/6(RWKHUZLVH

/HJHQG3ODFHPHQW 7RS� %RWWRP YHUWLFDO�SODFHPHQW %RWWRP/HJHQG$OLJQPHQW /HIW� &HQWHU� 5LJKW KRUL]RQWDO�DOLJQPHQW &HQWHU/HJHQG)RQW VHH�SDJH ������ IRQW�IRU�WKH�OHJHQG

WH[W6DQV�6HULI��

7DEOH ������ $WWULEXWHV�IRU�WKH�OHJHQG

������

Page 338: Mupad Tutorial

** @kZiab\l

>gflk

7KH�SORW�DWWULEXWHV�IRU�VSHFLI\LQJ�IRQWV�DUH $[HV7LWOH)RQW� )RRWHU)RQW�+HDGHU)RQW� /HJHQG)RQW� 7H[W)RQW� 7LFNV/DEHO)RQW� DQG 7LWOH)RQW� (DFK�VXFKIRQW�LV�VSHFLILHG�E\�D�0X3$'� OLVW�FRQWDLQLQJ�DQ\�RI�WKH�IROORZLQJ�

� $ VWULQJ�GHQRWLQJ�WKH�IRQW�IDPLO\� WKH�DYDLODEOH�IRQW�IDPLOLHV�GHSHQG�RQ�WKHIRQWV�WKDW�DUH�LQVWDOOHG�RQ�\RXU�PDFKLQH� 7\SLFDO�IRQW�IDPLOLHV�DYDLODEOH�RQ:LQGRZV�V\VWHPV�DUH �7LPHV�1HZ�5RPDQ� �RI�W\SH �VHULI��� �$ULDO� �RIW\SH �VDQV�VHULI��� RU �&RXULHU�1HZ� �RI�W\SH �PRQRVSDFH���7R�ILQG�RXW�ZKLFK�IRQWV�DUH�DYDLODEOH�RQ�\RXU�PDFKLQH� RSHQ�WKH�PHQXµ)RUPDW�¶ VXEPHQX�µ)RQW¶�LQ�\RXU�0X3$'�QRWHERRN� 7KH�ILUVW�FROXPQ�LQ�WKHIRQW�GLDORJ�SURYLGHV�WKH�QDPHV�RI�WKH�IRQW�IDPLOLHV�WKDW�\RX�PD\�VSHFLI\�

7KH�PRVW�SRUWDEOH�ZD\�RI�GHILQLQJ�IRQWV�LV�WR�XVH�RQH�WKH�WKUHH�JHQHULFIDPLO\�QDPHV �VHULI�� �VDQV�VHULI�� RU �PRQRVSDFH�� 7KH�V\VWHP�ZLOODXWRPDWLFDOO\�FKRRVH�RQH�RI�WKH�DYDLODEOH�IRQW�IDPLOLHV�RI�WKH�VSHFLILHG�W\SHIRU�\RX�

� $ SRVLWLYH�LQWHJHU�VSHFLI\LQJ�WKH�VL]H�RI�WKH�IRQW�LQ�SRLQWV�

� :KHQ�WKH�SDUDPHWHU %ROG LV�VSHFLILHG� WKH�IRQW�LV�EROG�

� :KHQ�WKH�SDUDPHWHU ,WDOLF LV�VSHFLILHG� WKH�IRQW�LV�LWDOLF�

� $ FRORU�VSHFLILFDWLRQ�

� $Q�DOLJQPHQW�VSHFLILFDWLRQ� LW�PXVW�EH�RQH�RI�WKH�IODJV /HIW� &HQWHU� RU5LJKW�

������

Page 339: Mupad Tutorial

?hgml

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�VSHFLI\�WKH�IRQW�IRU�WKH�FDQYDV�KHDGHU�DQG�IRRWHU�

SORW�SORW��)XQFWLRQ�G�VLQ�[�� [ ���� 3,��+HDGHU �7KH VLQH IXQFWLRQ��+HDGHU)RQW >�PRQRVSDFH�� ��� %ROG@�)RRWHU �7KH VLQH IXQFWLRQ��)RRWHU)RQW >�7LPHV 1HZ 5RPDQ�� ��� ,WDOLF@�

������

Page 340: Mupad Tutorial

** @kZiab\l

$OO�IRQW�SDUDPHWHUV�DUH�RSWLRQDO� VRPH�GHIDXOW�YDOXHV�DUH�FKRVHQ�IRU�HQWULHV�WKDWDUH�QRW�VSHFLILHG� )RU�H[DPSOH� LI�\RX�GR�QRW�FDUH�DERXW�WKH�IRRWHU�IRQW�IDPLO\�IRU\RXU�SORW� EXW�\RX�LQVLVW�RQ�D�VSHFLILF�IRQW�VL]H� \RX�PD\�VSHFLI\�D��� SW�IRQW�DVIROORZV�

SORW�SORW��)XQFWLRQ�G�VLQ�[�� [ ���� 3,��)RRWHU �7KH VLQH IXQFWLRQ�� )RRWHU)RQW >��@�

������

Page 341: Mupad Tutorial

LZobg`�Zg]�>qihkmbg`�Ib\mnk^l

KYnaf_�Yf\�=phgjlaf_�Ha[lmj]k

,QWHUDFWLYH�6DYLQJ�DQG�([SRUWLQJ

7KH�0X3$'� NHUQHO�XVHV�DQ [PO IRUPDW�WR�FRPPXQLFDWH�ZLWK�WKH�UHQGHUHU�8VXDOO\� D SORW FRPPDQG�VHQGV�D�VWUHDP�RI [PO GDWD�GLUHFWO\�WR�WKH�YLHZHU�ZKLFKUHQGHUV�WKH�SLFWXUH�

$IWHU�FOLFNLQJ�RQ�WKH�SLFWXUH� WKH�QRWHERRN�SURYLGHV�D�PHQX�LWHP�µ)LOH�([SRUW*UDSKLFV«¶�WKDW�RSHQV�D�GLDORJ�DOORZLQJ�WR�VDYH�WKH�SLFWXUH�LQ�D�YDULHW\�RIJUDSKLFDO�IRUPDWV�

� 7KH�LPDJH�PD\�DOVR�EH�VWRUHG�LQ�YDULRXV�VWDQGDUG�ELWPDS�IRUPDWV�VXFK�DVSQJ� MSJ HWF�

� *UDSKLFV�FDQ�EH�H[SRUWHG�LQ HSV �HQFDSVXODWHG�SRVWVFULSW��IRUPDW� 6LQFH HSVGRHV�QRW�VXSSRUW�WUDQVSDUHQF\� KRZHYHU� JUDSKLFV�PDNLQJ�XVH�RIWUDQVSDUHQF\�ZLOO�GHJUDGH�LQ�TXDOLW\�LQ�WKLV�VWHS� 7KLV�LQFOXGHV�PRVW��'SORWV�

� 0X3$'¶V�QDWLYH�IRUPDW�LV�LQGLFDWHG�E\�WKH�ILOH�H[WHQVLRQ�µ[Y]¶�IRU�WKHFRPSUHVVHG�DQG�µ[YF¶�IRU�WKH�XQFRPSUHVVHG�YHUVLRQ�

2QH�FDQ�XVH�0X3$' WR�RSHQ�VXFK�ILOHV�DQG�GLVSOD\�DQG�PDQLSXODWH�WKH�SORWVFRQWDLQHG�WKHUHLQ�

� )XUWKHU� WKHUH�LV MY[ H[SRUW��-DYD9LHZ��

� 0X3$' DQLPDWLRQV�FDQ�EH�VDYHG�DV�DQLPDWHG JLI ILOHV�

� 2Q�:LQGRZV� DQG�0DFLQWRVK� V\VWHPV� 0X3$' DQLPDWLRQV�FDQ�DOVR�EHVDYHG�DV DYL ILOHV�

� ,Q�D�0DF�26� V\VWHP� 0X3$' DQLPDWLRQV�FDQ�IXUWKHUPRUH�EH�VDYHG�DV4XLFNWLPH� PRYLH�ILOHV�

������

Page 342: Mupad Tutorial

** @kZiab\l

%DWFK�0RGH

0X3$' SORWV�FDQ�DOVR�EH�VDYHG�LQ�³EDWFK�PRGH´�E\�VSHFLI\LQJ�WKH�DWWULEXWH2XWSXW)LOH ILOHQDPH LQ�D SORW FDOO�SORW�3ULPLWLYH�� 3ULPLWLYH�� ����

2XWSXW)LOH �P\SLFWXUH�[Y]��

+HUH� WKH�H[WHQVLRQ [Y] RI�WKH�ILOH�QDPH�LQGLFDWHV�WKDW�0X3$'¶V [PO GDWD�DUH�WR�EHZULWWHQ�LQ�FRPSUHVVHG�IRUPDW� $OWHUQDWLYHO\� WKH�H[WHQVLRQ [YFPD\�EH�XVHG�WRZULWH�WKH [PO GDWD�ZLWKRXW�FRPSUHVVLRQ�RI�WKH�ILOH��WKH�UHVXOWLQJ DVFLL ILOH�FDQ�EHUHDG�ZLWK�DQ\�WH[W�HGLWRU�� )LOHV�LQ�ERWK�IRUPDWV�FDQ�EH�RSHQHG�E\�0X3$' WRJHQHUDWH�WKH�SORW�HQFRGHG�E\�WKH [PO GDWD�

,I�WKH�0X3$' HQYLURQPHQW�YDULDEOH :5,7(3$7+ GRHV�QRW�KDYH�D�YDOXH� WKHSUHYLRXV�FDOO�FUHDWHV�WKH�ILOH�LQ�WKH�GLUHFWRU\�ZKHUH�0X3$'�LV�LQVWDOOHG� $QDEVROXWH�SDWKQDPH�FDQ�EH�VSHFLILHG�WR�SODFH�WKH�ILOH�DQ\ZKHUH�HOVH�

SORW�3ULPLWLYH�� 3ULPLWLYH�� ����2XWSXW)LOH �&�??'RFXPHQWV??P\SLFWXUH�[Y]��

$OWHUQDWLYHO\� WKH�HQYLURQPHQW�YDULDEOH :5,7(3$7+ FDQ�EH�VHW�

:5,7(3$7+ � �&�??'RFXPHQWV��SORW�3ULPLWLYH�� 3ULPLWLYH�� ����

2XWSXW)LOH �P\SLFWXUH�[Y]��

1RZ� WKH�SORW�GDWD�DUH�VWRUHG�LQ�WKH�ILOH�µ&�\'RFXPHQWV\P\SLFWXUH�[Y]¶�

$SDUW�IURP�VDYLQJ�ILOHV�DV [PO GDWD� 0X3$'�SLFWXUHV�FDQ�DOVR�EH�VDYHG�LQ�D�YDULHW\RI�VWDQGDUG�JUDSKLFDO�IRUPDWV�VXFK�DV MSJ� HSV� VYJ� EPS HWF� ,Q�EDWFK�PRGH� WKHH[SRUW�LV�WULJJHUHG�E\�WKH 2XWSXW)LOH DWWULEXWH�LQ�WKH�VDPH�ZD\�DV�IRU�VDYLQJ�LQ[PO IRUPDW� -XVW�XVH�DQ�DSSURSULDWH�H[WHQVLRQ�RI�WKH�ILOHQDPH�LQGLFDWLQJ�WKHIRUPDW� 7KH�IROORZLQJ�FRPPDQGV�VDYH�WKH�SORW�LQ�IRXU�GLIIHUHQW�ILOHV�LQ MSJ� HSV�VYJ� DQG EPS IRUPDW� UHVSHFWLYHO\�

SORW�3ULPLWLYH�� ���� 2XWSXW)LOH �P\SLFWXUH�MSJ���SORW�3ULPLWLYH�� ���� 2XWSXW)LOH �P\SLFWXUH�HSV���SORW�3ULPLWLYH�� ���� 2XWSXW)LOH �P\SLFWXUH�VYJ���SORW�3ULPLWLYH�� ���� 2XWSXW)LOH �P\SLFWXUH�EPS���

������

Page 343: Mupad Tutorial

LZobg`�Zg]�>qihkmbg`�Ib\mnk^l

$Q�DQLPDWHG�0X3$'�SORW�FDQ�EH�H[SRUWHG�WR DYL IRUPDW�

SORW�SORW��)XQFWLRQ�G�H[S�D [�� [ ����� D ������2XWSXW)LOH �P\DQLPDWLRQ�DYL��

,I�QR�ILOH�H[WHQVLRQ�LV�VSHFLILHG�E\�WKH�ILOH�QDPH� WKH�GHIDXOW�H[WHQVLRQ [YF LV�XVHG�L�H�� XQFRPSUHVVHG [PO GDWD�DUH�ZULWWHQ�

,I�D�QRWHERRN�LV�VDYHG�WR�D�ILOH� LWV�ORFDWLRQ�LQ�WKH�ILOH�V\VWHP�LV�DYDLODEOH�LQVLGH�WKHQRWHERRN�DV�WKH�YDOXH�RI�WKH�HQYLURQPHQW�YDULDEOH 127(%22.3$7+��,I�\RX�ZLVK�WRVDYH�\RXU�SORW�LQ�WKH�VDPH�IROGHU�DV�WKH�QRWHERRN� \RX�PD\�FDOO

SORW�3ULPLWLYH�� 3ULPLWLYH�� ����2XWSXW)LOH 127(%22.3$7+��P\SLFWXUH�[Y]��

,Q�DGGLWLRQ�WR 2XWSXW)LOH� WKHUH�LV�WKH�DWWULEXWH 2XWSXW2SWLRQV WR�VSHFLI\SDUDPHWHUV�IRU�VRPH�RI�WKH�H[SRUW�IRUPDWV� 7KH�KHOS�SDJH�RI�WKLV�DWWULEXWHSURYLGHV�GHWDLOHG�LQIRUPDWLRQ�

������

Page 344: Mupad Tutorial

** @kZiab\l

Aehgjlaf_�Ha[lmj]k

0X3$'� GRHV�QRW�SURYLGH�IRU�PDQ\�WRROV�WR�LPSRUW�VWDQGDUG�JUDSKLFDO YHFWRUIRUPDWV� \HW� 3UHVHQWO\� WKH�RQO\�VXSSRUWHG�YHFWRU�W\SH�LV�WKH VWO IRUPDW� SRSXODU�LQVWHUHR�OLWKRJUDSK\� ZKLFK�HQFRGHV��' VXUIDFHV� ,W�FDQ�EH�LPSRUWHG�YLD�WKH�URXWLQHSORW��6XUIDFH67/�

,Q�FRQWUDVW�WR�JUDSKLFDO YHFWRU IRUPDWV� WKHUH�DUH�PDQ\�VWDQGDUG ELWPDS IRUPDWVVXFK�DV EPS� JLI� MSJ� SSP HWF��WKDW�FDQ�EH�LPSRUWHG� 2QH�FDQ�UHDG�VXFK�D�ILOH�YLDLPSRUW��UHDGELWPDS� WKXV�FUHDWLQJ�D�0X3$' DUUD\�RI�5*% FRORU�YDOXHV�RU�DQHTXLYDOHQW�WKUHH�GLPHQVLRQDO�KDUGZDUH�IORDW�DUUD\�WKDW�FDQ�EH�PDQLSXODWHG�DWZLOO� ,Q�SDUWLFXODU� LW�FDQ�EH�IHG�LQWR�WKH�URXWLQH SORW��5DVWHU ZKLFK�FUHDWHV�DQREMHFW�WKDW�FDQ�EH�XVHG�LQ�DQ\��' 0X3$'�SORW� 1RWH� KRZHYHU� WKDW�WKH�LPSRUW�RIELWPDS�GDWD�FRQVXPHV�D�ORW�RI�PHPRU\� L�H�� RQO\�UHDVRQDEO\�VPDOO�ELWPDSV��XS�WRD�IHZ�KXQGUHG�SL[HOV�LQ�HDFK�GLUHFWLRQ��VKRXOG�EH�SURFHVVHG� 0HPRU\FRQVXPSWLRQ�LV�PXFK�VPDOOHU�ZLWK�KDUGZDUH�IORDWLQJ�SRLQW�DUUD\V� ZKLFK�LV�ZK\LPSRUW��UHDVRQDEO\ XVHV�WKHP�E\�GHIDXOW�

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�SORW�WKH�SUREDELOLW\�GHQVLW\�IXQFWLRQ�DQG�WKHFXPXODWLYH�GHQVLW\�IXQFWLRQ�RI�WKH�VWDQGDUG�QRUPDO��³*DXVVLDQ´��GLVWULEXWLRQ�3D\LQJ�WULEXWH�WR�&DUO�)ULHGULFK�*DXVV� ZH�ZLVK�WR�GLVSOD\����������PLQH�KLVSLFWXUH�LQ�WKLV�SORW� $VVXPH�WKDW�ZH�KDYH�KLV�SLFWXUH�DV�D SQJ ELWPDS�ILOH³*DXVV�SQJ�´ :H�LPSRUW�WKH�ILOH�YLD LPSRUW��UHDGELWPDS WKDW�SURYLGHV�XV�ZLWKWKH�ZLGWK�DQG�KHLJKW�LQ�SL[HOV�DQG�WKH�FRORU�GDWD� �KLV�SLFWXUH�LQ�WKLVSORW� $VVXPH�WKDW�ZH�KDYH�KLV�SLFWXUH�DV�D SQJ ELWPDS�ILOH�³*DXVV�SQJ�´ :HLPSRUW�WKH�ILOH�YLD LPSRUW��UHDGELWPDS WKDW�SURYLGHV�XV�ZLWK�WKH�ZLGWK�DQGKHLJKW�LQ�SL[HOV�DQG�WKH�FRORU�GDWD� !!!!!!!��U�����

>ZLGWK� KHLJKW� JDXVV@ � LPSRUW��UHDGELWPDS��*DXVV�SQJ���

1RWH�WKH�FRORQ�DW�WKH�HQG�RI�WKH�DERYH�FRPPDQG� :LWKRXW�LW� 0X3$'�ZLOO�SULQWWKH�FRORU�LQIRUPDWLRQ�RQ�WKH�VFUHHQ� DQG�IRUPDWWLQJ�VXFK�D�KXJH�RXWSXW�WDNHV�WLPH�

:H�KDYH�WR�XVH 6FDOLQJ &RQVWUDLQHG WR�SUHVHUYH�WKH�DVSHFW�UDWLR�RI�WKH�LPDJH�8QIRUWXQDWHO\� WKLV�VHWWLQJ�LV�QRW�DSSURSULDWH�IRU�WKH�IXQFWLRQ�SORWV� 6R�ZH�XVH�WZRGLIIHUHQW�VFHQHV�WKDW�DUH�SRVLWLRQHG�YLD /D\RXW 5HODWLYH� 6HH�6HFWLRQ�µ/D\RXW�RI&DQYDV�DQG�6FHQHV¶�RI�WKH�RQOLQH SORW GRFXPHQWDWLRQ�IRU�GHWDLOV�RQ�KRZ�WKHOD\RXW�RI�D�FDQYDV�FRQWDLQLQJ�VHYHUDO�VFHQHV�LV�VHW�

������

Page 345: Mupad Tutorial

Bfihkmbg`�Ib\mnk^l

SGI � VWDWV��QRUPDO3')��� ���FGI � VWDWV��QRUPDO&')��� ���SORW�SORW��6FHQH�G�SORW��)XQFWLRQ�G�SGI�[�� [ �������

SORW��)XQFWLRQ�G�FGI�[�� [ �������:LGWK �� +HLJKW ���

SORW��6FHQH�G�SORW��5DVWHU�JDXVV��6FDOLQJ &RQVWUDLQHG�:LGWK ���� +HLJKW ����%RWWRP ����� /HIW ����%RUGHU:LGWK ��� XQLW��PP�)RRWHU �&�)� *DXVV��)RRWHU)RQW >�@��

/D\RXW 5HODWLYH�

������

Page 346: Mupad Tutorial

** @kZiab\l

;Ye]jYk�af�+<

7KH�0X3$'� �' JUDSKLFV�PRGHO�LQFOXGHV�DQ�REVHUYHU�DW�D�VSHFLILF�SRVLWLRQ�SRLQWLQJ�D�FDPHUD�ZLWK�D�OHQV�RI�D�VSHFLILF�RSHQLQJ�DQJOH�WR�VRPH�VSHFLILF�IRFDOSRLQW� 7KH�VSHFLILF�SDUDPHWHUV�³SRVLWLRQ�´ ³DQJOH�´ DQG�³IRFDO�SRLQW´�GHWHUPLQH�WKHSLFWXUH�WKDW�WKH�FDPHUD�ZLOO�WDNH�

:KHQ�D��' SLFWXUH�LV�FUHDWHG�D�FDPHUD�ZLWK�DQ�DSSURSULDWH�GHIDXOW�OHQV�LVSRVLWLRQHG�DXWRPDWLFDOO\� ,WV�IRFDO�SRLQW�LV�FKRVHQ�DV�WKH�FHQWHU�RI�WKH�JUDSKLFDOVFHQH� 7KH�LQWHUDFWLYH�YLHZHU�DOORZV�WR�URWDWH�WKH�VFHQH�ZKLFK� LQ�IDFW� LVLPSOHPHQWHG�LQWHUQDOO\�DV�D�FKDQJH�RI�WKH�FDPHUD�SRVLWLRQ� $OVR�LQWHUDFWLYH]RRPLQJ�LQ�DQG�]RRPLQJ�RXW�LV�UHDOL]HG�E\�PRYLQJ�WKH�FDPHUD�FORVHU�WR�RU�IDUWKHUDZD\�IURP�WKH�VFHQH�

$SDUW�IURP�LQWHUDFWLYH�FDPHUD�PRWLRQV� WKH�SHUVSHFWLYH�RI�D��' SLFWXUH�FDQ�DOVR�EHVHW�LQ�WKH�FDOOV�JHQHUDWLQJ�WKH�SORW� 2QH�ZD\�LV�WR�VSHFLI\�WKH�GLUHFWLRQ�IURP�ZKLFKWKH�FDPHUD�LV�SRLQWLQJ�WRZDUGV�WKH�VFHQH� 7KLV�LV�GRQH�YLD�WKH�DWWULEXWH&DPHUD'LUHFWLRQ�SORW�SORW��)XQFWLRQ�G�VLQ�[ � \A��� [ ������ \ �������

&DPHUD'LUHFWLRQ >���� ��� ��@�

������

Page 347: Mupad Tutorial

<Zf^kZl�bg�,=

SORW�SORW��)XQFWLRQ�G�VLQ�[ � \A��� [ ������ \ �������&DPHUD'LUHFWLRQ >��� ���� ��@�

,Q�WKHVH�FDOOV� &DPHUD'LUHFWLRQ GRHV�QRW�IXOO\�VSHFLI\�WKH�SRVLWLRQ�RI�WKH�FDPHUD�7KLV�DWWULEXWH�MXVW�UHTXHVWV�WKH�FDPHUD�WR�EH�SODFHG�DW�VRPH�ODUJH�GLVWDQFH�IURPWKH�VFHQH�DORQJ�WKH�UD\�LQ�WKH�GLUHFWLRQ�JLYHQ�E\�WKH�DWWULEXWH� 7KH�DFWXDO�GLVWDQFHIURP�WKH�VFHQH�LV�GHWHUPLQHG�DXWRPDWLFDOO\�WR�OHW�WKH�JUDSKLFDO�VFHQH�ILOO�WKHSLFWXUH�RSWLPDOO\�

)RU�D�IXOO�VSHFLILFDWLRQ�RI�WKH�SHUVSHFWLYH� WKHUH�DUH�FDPHUD�REMHFWV�RI�W\SHSORW��&DPHUD WKDW�DOORZ�WR�VSHFLI\�WKH�SRVLWLRQ�RI�WKH�FDPHUD� LWV�IRFDO�SRLQW�DQGWKH�RSHQLQJ�DQJOH�RI�LWV�OHQV�

SRVLWLRQ � >��� ���� �@�IRFDOSRLQW � >�� �� �@�DQJOH � 3,����FDPHUD � SORW��&DPHUD�SRVLWLRQ� IRFDOSRLQW� DQJOH��

7KLV�FDPHUD�FDQ�EH�SDVVHG�OLNH�DQ\�JUDSKLFDO�REMHFW�WR�WKH SORW FRPPDQGJHQHUDWLQJ�WKH�VFHQH�

������

Page 348: Mupad Tutorial

** @kZiab\l

2QFH�D�FDPHUD�REMHFW�LV�VSHFLILHG�LQ�D�JUDSKLFDO�VFHQH� LW�GHWHUPLQHV�WKH�YLHZ� 1R³DXWRPDWLF�FDPHUD´�LV�XVHG�DQ\�PRUH�

SORW�SORW��)XQFWLRQ�G�VLQ�[ � \A��� [ ������ \ �������FDPHUD�

������

Page 349: Mupad Tutorial

<Zf^kZl�bg�,=

&DPHUD�REMHFWV�FDQ�EH�DQLPDWHG�

FDPHUD � SORW��&DPHUD�>� FRV�D�� � VLQ�D�� � � FRV�� D�@�>�� �� �@� 3,��� D ���� 3,�)UDPHV �����

,QVHUWLQJ�WKH�DQLPDWHG�FDPHUD�LQ�D�JUDSKLFDO�VFHQH� ZH�REWDLQ�DQ�DQLPDWHG�SORWVLPXODWLQJ�D�³IOLJKW�DURXQG�WKH�VFHQH�´

SORW�SORW��)XQFWLRQ�G�VLQ�[ � \A��� [ ������ \ �������FDPHUD�

,Q�IDFW� VHYHUDO�FDPHUDV�FDQ�EH�LQVWDOOHG�VLPXOWDQHRXVO\�LQ�D�VFHQH� 3HU�GHIDXOW�WKH�ILUVW�FDPHUD�SURGXFHV�WKH�YLHZ�UHQGHUHG� $IWHU�FOLFNLQJ�RQ�DQRWKHU�FDPHUD�LQWKH�REMHFW�EURZVHU�RI�WKH�YLHZHU��SDJH ������� WKH�VHOHFWHG�FDPHUD�WDNHV�RYHU�DQGWKH�QHZ�YLHZ�LV�VKRZQ�

������

Page 350: Mupad Tutorial

** @kZiab\l

1H[W� ZH�KDYH�D�ORRN�DW�D�PRUH�DSSHDOLQJ�H[DPSOH� WKH�VR�FDOOHG�³/RUHQ]DWWUDFWRU�´ 7KH�/RUHQ]�2'( LV�WKH�V\VWHP

GGt

⎝xyz

⎠ =

⎝p · (y − x)

−x · z + r · x− yx · y − b · z

ZLWK�IL[HG�SDUDPHWHUV p� r� b� $V�D�G\QDPLFDO�V\VWHP�IRU Y = [x, y, z]� ZH�KDYH�WRVROYH�WKH�2'( GY /Gt = f(t, Y ) ZLWK�WKH�IROORZLQJ�YHFWRU�ILHOG�

I � SURF�W� <�ORFDO [� \� ]�EHJLQ

>[� \� ]@ � <�>S �\ � [�� �[ ] � U [ � \� [ \ � E ]@

HQGBSURF�

&RQVLGHU�WKH�IROORZLQJ�SDUDPHWHUV�DQG�WKH�IROORZLQJ�LQLWLDO�FRQGLWLRQ <��

S � ��� U � ��� E � �� <� � >�� �� �@�

7KH�URXWLQH SORW��2GH�G VHUYHV�IRU�JHQHUDWLQJ�D�JUDSKLFDO��' VROXWLRQ�RI�DG\QDPLFDO�V\VWHP� ,W�VROYHV�WKH�2'( QXPHULFDOO\�DQG�JHQHUDWHV�JUDSKLFDO�GDWDIURP�WKH�QXPHULFDO�PHVK� 7KH�SORW�GDWD�DUH�VSHFLILHG�E\�WKH�XVHU�YLD�³JHQHUDWRUV´�SURFHGXUHV��WKDW�PDS�D�VROXWLRQ�SRLQW (t, Y ) WR�D�SRLQW (x, y, z) LQ��'�

7KH�IROORZLQJ�JHQHUDWRU *[\] SURGXFHV�D��' SKDVH�SORW�RI�WKH�VROXWLRQ� 7KHJHQHUDWRU *\] SURMHFWV�WKH�VROXWLRQ�FXUYH�WR�WKH (y, z) SODQH�ZLWK x = −20� WKHJHQHUDWRU *[] SURMHFWV�WKH�VROXWLRQ�FXUYH�WR�WKH (x, z) SODQH�ZLWK y = −20� WKHJHQHUDWRU *[\ SURMHFWV�WKH�VROXWLRQ�FXUYH�WR�WKH (x, y) SODQH�ZLWK z = 0�

*[\] � �W� <� �! <�*\] � �W� <� �! >���� <>�@� <>�@@�*[] � �W� <� �! ><>�@� ���� <>�@@�*[\ � �W� <� �! ><>�@� <>�@� �@�

:LWK�WKHVH�JHQHUDWRUV� ZH�FUHDWH�D��' SORW�REMHFW�FRQVLVWLQJ�RI�WKH�SKDVH�FXUYHDQG�LWV�SURMHFWLRQV�

REMHFW � SORW��2GH�G�I� >� ����� VWHS ����@� <��>*[\]� 6W\OH 6SOLQHV� &RORU 5*%��5HG@�>*\]� 6W\OH 6SOLQHV� &RORU 5*%��/LJKW*UD\@�>*[]� 6W\OH 6SOLQHV� &RORU 5*%��/LJKW*UD\@�>*[\� 6W\OH 6SOLQHV� &RORU 5*%��/LJKW*UD\@��

������

Page 351: Mupad Tutorial

<Zf^kZl�bg�,=

:H�GHILQH�DQ�DQLPDWHG�FDPHUD�PRYLQJ�DURXQG�WKH�VFHQH�

FDPHUD � SORW��&DPHUD�>�� � ��� FRV�D�� � � ��� VLQ�D�� ���@�>��� �� ��@� 3,��� D ���� 3,� )UDPHV �����

7KH�IROORZLQJ SORW FDOO�WDNHV�DERXW�KDOI�D�PLQXWH�RQ�D��*+]�FRPSXWHU�

SORW�REMHFW� FDPHUD� $[HV %R[HG� 7LFNV1XPEHU /RZ�

1H[W� ZH�ZLVK�WR�IO\ DORQJ�WKH�/RUHQ]�DWWUDFWRU� :H�FDQQRW�XVH SORW��2GH�G�EHFDXVH�ZH�QHHG�DFFHVV�WR�WKH�QXPHULFDO�GDWD�RI�WKH�DWWUDFWRU�WR�EXLOG�D�VXLWDEOHDQLPDWHG�FDPHUD�REMHFW� :H�XVH�WKH�QXPHULFDO�2'( VROYHU QXPHULF��RGHVROYH�DQG�FRPSXWH�D�OLVW�RI�QXPHULFDO�VDPSOH�SRLQWV�RQ�WKH�/RUHQ]�DWWUDFWRU� 7KLV�WDNHVDERXW�WZHQW\�VHFRQGV�RQ�D��*+]�FRPSXWHU�

< � QXPHULF��RGHVROYH��I� �� <�� 5HPHPEHU/DVW��WLPHPHVK � >� ������ VWHS ����@�< � ><�W� � W LQ WLPHPHVK@�

������

Page 352: Mupad Tutorial

** @kZiab\l

6LPLODU�WR�WKH�SLFWXUH�DERYH� ZH�GHILQH�D�ER[�DURXQG�WKH�DWWUDFWRU�ZLWK�WKHSURMHFWLRQV�RI�WKH�VROXWLRQ�FXUYH�

ER[ � >���� ��� ���� ��� �� ��@�<\] � PDS�<� SW �! >ER[>�@� SW>�@� SW>�@@��<[\ � PDS�<� SW �! >SW>�@� SW>�@� ER[>�@@��<[] � PDS�<� SW �! >SW>�@� ER[>�@� SW>�@@��

:H�FUHDWH�DQ�DQLPDWHG�FDPHUD�XVLQJ�DQ�DQLPDWLRQ�SDUDPHWHU D WKDW�FRUUHVSRQGVWR�WKH�LQGH[�RI�WKH�OLVW�RI�QXPHULFDO�VDPSOH�SRLQWV� 7KH�IROORZLQJ�SURFHGXUHUHWXUQV�WKH i�WK�FRRUGLQDWH��i = 1, 2, 3��RI�WKH a�WK�SRLQW�LQ�WKH�OLVW�RI�VDPSOH�SRLQWV�VHH�&KDSWHU ��� HVSHFLDOO\�WKH�VHFWLRQ�VWDUWLQJ�RQ�SDJH ���� IRU�GHWDLOV��

3RLQW � SURF�D� L�EHJLQ

LI GRPW\SH�IORDW�D�� �! '20B)/2$7 WKHQSURFQDPH�DUJV����

HOVH <>URXQG�D�@>L@�HQGBLI�

HQGBSURF�

,Q�WKH a�WK�IUDPH�RI�WKH�DQLPDWLRQ� WKH�FDPHUD�LV�SRVLWLRQHG�DW�WKH a�WK�VDPSOHSRLQW�RI�WKH�/RUHQ]�DWWUDFWRU� SRLQWLQJ�WRZDUG�WKH�QH[W�VDPSOH�SRLQW� 6HWWLQJ7LPH5DQJH ���Q���� WKH�FDPHUD�YLVLWV�DERXW����SRLQWV�SHU�VHFRQG�

������

Page 353: Mupad Tutorial

<Zf^kZl�bg�,=

Q � QRSV�WLPHPHVK� � ��SORW�SORW��6FHQH�G�

SORW��&DPHUD�>3RLQW�D� L� � L ����@�>3RLQW�D � �� L� � L ����@�3,��� D ���Q� )UDPHV Q�7LPH5DQJH ���Q�����

SORW��3RO\JRQ�G�<� /LQH&RORU 5*%��5HG�3RLQWV9LVLEOH 758(��

SORW��3RO\JRQ�G�<[\� /LQH&RORU 5*%��'LP*UD\��SORW��3RO\JRQ�G�<[]� /LQH&RORU 5*%��'LP*UD\��SORW��3RO\JRQ�G�<\]� /LQH&RORU 5*%��'LP*UD\��SORW��%R[�ER[>�@��ER[>�@� ER[>�@��ER[>�@�

ER[>�@��ER[>�@� )LOOHG 758(�/LQH&RORU 5*%��%ODFN�)LOO&RORU 5*%��*UD\�>���@��

%DFNJURXQG6W\OH )ODW��

������

Page 354: Mupad Tutorial

** @kZiab\l

KljYf_]�=^^][lk�af�+<7�9[[]d]jYl]\�Gh]f?D� daZjYjq7

7KH�UHQGHULQJ�HQJLQH�IRU��' SORWV�XVHV�WKH�2SHQ*/� OLEUDU\� 2SHQ*/ OLEUDU\�LV�DZLGHO\�XVHG�JUDSKLFV�VWDQGDUG�DQG��DOPRVW��DQ\�FRPSXWHU�KDV�DSSURSULDWH�GULYHUVLQVWDOOHG�LQ�LWV�V\VWHP�

%\�GHIDXOW� 0X3$'� XQGHU�:LQGRZV� XVHV�D�VRIWZDUH�2SHQ*/ GULYHU�SURYLGHG�E\WKH�:LQGRZV�RSHUDWLQJ�V\VWHP� 'HSHQGLQJ�RQ�WKH�JUDSKLFV�FDUG�RI�\RXU�PDFKLQH�\RX�PD\�DOVR�KDYH�IXUWKHU�2SHQ*/ GULYHUV�RQ�\RXU�V\VWHP� PD\EH�XVLQJ�KDUGZDUHVXSSRUW�WR�DFFHOHUDWH�\RXU�2SHQ*/ OLEUDU\�

7KH�0X3$'��' JUDSKLFV�ZDV�ZULWWHQ�DQG�WHVWHG�XVLQJ�FHUWDLQ�VWDQGDUG�2SHQ*/GULYHUV� 7KH�QXPHURXV�GULYHUV�DYDLODEOH�RQ�WKH�PDUNHW�KDYH�GLIIHUHQW�UHQGHULQJTXDOLW\�DQG�GLIIHU�VOLJKWO\� 7KLV�PD\� LQ�UDUH�FDVHV� OHDG�WR�VRPH�XQH[SHFWHGJUDSKLFDO�HIIHFWV�RQ�\RXU�PDFKLQH�

$IWHU�GRXEOH�FOLFNLQJ�RQ�D��' SORW� WKH�µ+HOS¶�PHQX�FRQWDLQV�DQ�HQWU\�µ2SHQ*/,QIR�«¶�SURYLGLQJ�WKH�LQIRUPDWLRQ�ZKLFK�2SHQ*/ GULYHU�\RX�DUH�FXUUHQWO\�XVLQJ�<RX�DOVR�JHW�WKH�LQIRUPDWLRQ�KRZ�PDQ\�OLJKW�VRXUFHV�DQG�KRZ�PDQ\�FOLSSLQJSODQHV�WKLV�GULYHU�VXSSRUWV�

7KH�HQWU\�µ&RQILJXUH�«¶�RI�WKH�µ9LHZ¶�PHQX�RSHQV�D�GLDORJ� ZKLFK�FRQWDLQV�DVHFWLRQ�µ8VHU�,QWHUIDFH�¶ 7KLV�VHFWLRQ�FRQWDLQV�WZR�FKHFN�ER[HV�ODEHOHG�µ$FFHOHUDWH2SHQ*/¶ DQG�µ'LVDEOH�2SHQ*/�¶�ZLWK�WKH�ILUVW�RQH�VHOHFWLQJ�EHWZHHQ�WKH�GLYHUVRSWLPL]HG�IRU�WKH�KDUGZDUH�LQVWDOOHG��ZKLFK� DV�RXWOLQHG�DERYH� PD\�FDXVHSUREOHPV�GXH�WR�EDG�GULYHUV�� 7KH�VHFRQG�FKHFNER[�VKRXOG�RQO\�EH�XVHG�LQ�WKHH[WUHPHO\�XQOLNHO\�HYHQW�WKDW��' JUDSKLFV�GR�QRW�ZRUN�DW�DOO� RQ�VRPH�V\VWHPV�RSHQLQJ�DQ\�2SHQ*/ SURJUDP�IUHH]HV�WKH�FRPSXWHU� &KHFNLQJ�WKLV�ER[�GLVDEOHV0X3$' �' JUDSKLFV�

7KXV� LI�\RX�HQFRXQWHU�VWUDQJH�JUDSKLFDO�HIIHFWV�LQ��'��ZH�UHFRPPHQG�WR�XVH�WKLVFRQILJXUDWLRQ�GLDORJ�WR�WRJJOH�KDUGZDUH�DFFHOHUDWLRQ�

([HUFLVH������ )LQG�DOWHUQDWLYH�GLVSOD\�VW\OHV�IRU�WKH�SORWV�LQ�WKLV�FKDSWHU�(VSHFLDOO\� PRGLI\�WKH�H[DPSOH�IURP�SDJH ����� WR�XVH SORW��2GH�G DQG�RUSORW��6WUHDPOLQHV�G�

������

Page 355: Mupad Tutorial

)*

,QSXW�DQG�2XWSXW

Gmlhml�g^�=phj]kkagfk

0X3$'� GRHV�QRW�GLVSOD\�DOO�FRPSXWHG�UHVXOWV�RQ�WKH�VFUHHQ� 7\SLFDO�H[DPSOHVDUH�WKH�FRPPDQGV�ZLWKLQ IRU ORRSV��&KDSWHU ����RU�SURFHGXUHV��&KDSWHU ���� RQO\WKH�ILQDO�UHVXOW��L�H�� WKH�UHVXOW�RI�WKH�ODVW�FRPPDQG��LV�SULQWHG�DQG�WKH�RXWSXW�RILQWHUPHGLDWH�UHVXOWV�LV�VXSSUHVVHG� 1HYHUWKHOHVV� \RX�FDQ�OHW�0X3$' SULQWLQWHUPHGLDWH�UHVXOWV�RU�FKDQJH�WKH�RXWSXW�IRUPDW�

3ULQWLQJ�([SUHVVLRQV�RQ�WKH�6FUHHQ

7KH�IXQFWLRQ SULQW RXWSXWV�0X3$' REMHFWV�RQ�WKH�VFUHHQ�

IRU L IURP � WR � GRSULQW��7KH �� L� �WK SULPH LV �� LWKSULPH�L��

HQGBIRU�7KH �� �� �WK SULPH LV �� �

�7KH �� �� �WK SULPH LV �� ��

:H�UHFDOO�WKDW LWKSULPH�L� FRPSXWHV�WKH i�WK�SULPH� 0X3$' HQFORVHV�WKH�WH[W�LQGRXEOH�TXRWHV� 8VH�WKH�RSWLRQ 8QTXRWHG WR�VXSSUHVV�WKLV�

Page 356: Mupad Tutorial

*+ Bginm�Zg]�Hnminm

IRU L IURP � WR � GRSULQW�8QTXRWHG�

�7KH �� L� �WK SULPH LV �� LWKSULPH�L��HQGBIRU

7KH � �� WK SULPH LV � �

7KH � �� WK SULPH LV � ��

1RWH�WKDW�WKH�RSWLRQ 8QTXRWHG EUHDNV�W\SHVHWWLQJ�

SULQW�[A��� SULQW�8QTXRWHG� [A��x2

�[

)XUWKHUPRUH� \RX�FDQ�HOLPLQDWH�WKH�FRPPDV�IURP�WKH�RXWSXW�E\�PHDQV�RI�WKHWRROV�IRU�PDQLSXODWLQJ�VWULQJV�WKDW�DUH�SUHVHQWHG�LQ�6HFWLRQ �����

IRU L IURP � WR � GRSULQW�8QTXRWHG�

�7KH � � H[SU�WH[W�L� � �WK SULPH LV � �H[SU�WH[W�LWKSULPH�L�� � ����

HQGBIRU7KH �WK SULPH LV ��

7KH �WK SULPH LV ���

+HUH� WKH�IXQFWLRQ H[SU�WH[W LV�XVHG�WR�FRQYHUW�WKH�YDOXH�RI L DQG�WKH�SULPHUHWXUQHG�E\ LWKSULPH�L� WR�VWULQJV� 7KHQ� WKH�FRQFDWHQDWLRQ�RSHUDWRU � FRPELQHVWKHP�ZLWK�WKH�RWKHU�VWULQJV�WR�D�VLQJOH�VWULQJ� 1RWH�WKDW�FRQYHUWLQJ�DQ�H[SUHVVLRQWR�D�VWULQJ�EUHDNV�W\SHVHWWLQJ�DV�ZHOO�DV�WKH�³SUHWW\�SULQW´�IRUPDW�H[SODLQHG�EHORZ�

SULQW�8QTXRWHG� H[SU�WH[W�[A���[A�

$OWHUQDWLYHO\�\RX�FDQ�XVH�WKH�IXQFWLRQ ISULQW� ZKLFK�ZULWHV�GDWD�WR�D�ILOH�RU�RQ�WKHVFUHHQ� ,Q�FRQWUDVW�WR SULQW� LW�GRHV�QRW�RXWSXW�LWV�DUJXPHQWV�DV�LQGLYLGXDOH[SUHVVLRQV� ,QVWHDG� ISULQW FRPELQHV�WKHP�WR�D�VLQJOH�VWULQJ��LI�\RX�XVH�WKHRSWLRQ 8QTXRWHG��

D � RQH� E � VWULQJ�

����

Page 357: Mupad Tutorial

Hnminm�h_�>qik^llbhgl

ISULQW�8QTXRWHG� �� �7KLV LV �� D� � �� E�7KLV LV RQH VWULQJ

7KH�VHFRQG�DUJXPHQW � WHOOV ISULQW WR�GLUHFW�LWV�RXWSXW�WR�WKH�VFUHHQ�

0RGLI\LQJ�WKH�2XWSXW�)RUPDW

8VXDOO\� 0X3$'�ZLOO�IRUPDW�RXWSXWV�LQ�D�ZD\�VLPLODU�WR�KRZ�PDWKHPDWLFDOIRUPXODV�DUH�XVXDOO\�ZULWWHQ�LQ�ERRNV�RU�RQ�D�EODFNERDUG��³W\SHVHW�H[SUHVVLRQV´��LQWHJUDO�VLJQV�DUH�GLVSOD\HG�DV

∫� V\PEROLF�VXPV�DSSHDU�DV

∑HWF� 7KLV�LV�WKH

IRUPDW�XVHG�LQ�PRVW�³UHVXOW�UHJLRQV´�RI�WKLV�ERRN� 7KHUH�DUH�WZR�RWKHU�RXWSXWIRUPDWV�DYDLODEOH� 7KH�IROORZLQJ�WH[W�UHIHUV�WR�WKHVH�$6&,,�EDVHG�RXWSXW�RQO\�ZKLFK�DUH�XVHG�LI�WKH�W\SHVHWWLQJ�LV�VZLWFKHG�RII�YLD�WKH�FRUUHVSRQGLQJ�PHQX�RI�WKHQRWHERRN�LQWHUIDFH�RU�ZKHQ SULQW ZLWK�WKH�RSWLRQ 3ODLQ LV�XVHG�

:LWKRXW�W\SHVHWWLQJ� 0X3$' XVXDOO\�SULQWV�H[SUHVVLRQV�LQ�D�WZR�GLPHQVLRQDOIRUP�ZLWK�VLPSOH��DVFLL��FKDUDFWHUV�

GLII�VLQ�[��FRV�[�� [��

VLQ�[�������� � �

�FRV�[�

7KLV�IRUPDW�LV�NQRZQ�DV SUHWW\�SULQW� ,W�UHVHPEOHV�WKH�XVXDO�PDWKHPDWLFDOQRWDWLRQ� 7KHUHIRUH� LW�LV�RIWHQ�HDVLHU�WR�UHDG�WKDQ�D�VLQJOH�OLQH�RXWSXW� +RZHYHU�0X3$' RQO\�XVHV�SUHWW\�SULQW�IRU�RXWSXW�DQG�LW�LV�QRW�D�YDOLG�LQSXW�IRUPDW� LQ�DJUDSKLFDO�XVHU�LQWHUIDFH�\RX�FDQQRW�FRS\�VRPH�RXWSXW�WH[W�ZLWK�WKH�PRXVH�DQGSDVWH�LW�DV�LQSXW�VRPHZKHUH�HOVH� �7KLV�LV�SRVVLEOH�ZLWK�W\SHVHW�IRUPXODV� DOWKRXJKQRW�IRU�SDUWV�RI�D�ODUJHU�IRUPXOD��

7KH�HQYLURQPHQW�YDULDEOH 35(77<35,17 FRQWUROV�WKH�RXWSXW�IRUPDW� 7KH�GHIDXOWYDOXH�RI�WKH�YDULDEOH�LV 758(��L�H�� WKH�SUHWW\�SULQW�IRUPDW�LV�XVHG�IRU�WKH�RXWSXW� ,I\RX�VHW�WKLV�YDULDEOH�WR )$/6(��\RX�REWDLQ�D�RQH�GLPHQVLRQDO�RXWSXW�IRUP�ZKLFKFDQ�RIWHQ�DOVR�EH�XVHG�DV�LQSXW�

35(77<35,17 � )$/6(� GLII�VLQ�[��FRV�[�� [���FRV�[�A� VLQ�[�A� � �

����

Page 358: Mupad Tutorial

*+ Bginm�Zg]�Hnminm

,I�DQ�RXWSXW�ZRXOG�H[FHHG�WKH�OLQH�ZLGWK� WKH�V\VWHP�DXWRPDWLFDOO\�EUHDNV�WKHOLQHV�

35(77<35,17 � 758(� WD\ORU�VLQ�[�� [ �� ���� � � � ��

[ [ [ [ [[ � �� � ��� � ���� � ������ � �������� �

� ��� ���� ������ ��������

�� ��[ [ ��

���������� � ������������� � 2�[ ����������� �������������

<RX�FDQ�VHW�WKH�HQYLURQPHQW�YDULDEOH 7(;7:,'7+ WR�WKH�GHVLUHG�OLQH�ZLGWK� ,WVGHIDXOW�YDOXH�LV 75 �FKDUDFWHUV�� DQG�\RX�FDQ�DVVLJQ�DQ\�LQWHJHU�EHWZHHQ 10 DQG231 − 1 WR�LW� )RU�H[DPSOH� LI�\RX�FRPSXWH (OQ OQx)′′� WKHQ�\RX�REWDLQ�WKH�IROORZLQJRXWSXW�

GLII�OQ�OQ�[��� [� [�� �

� �������� � ���������� � �

[ OQ�[� [ OQ�[�

,I�\RX�UHGXFH�WKH�YDOXH�RI 7(;7:,'7+��WKH�V\VWHP�EUHDNV�WKH�RXWSXW�DFURVV�WZROLQHV�

7(;7:,'7+ � ��� GLII�OQ�OQ�[���[�[��

� �������� ��

[ OQ�[�

����������� �

[ OQ�[�

7KH�GHIDXOW�YDOXH�LV�UHVWRUHG�E\�GHOHWLQJ 7(;7:,'7+�

GHOHWH 7(;7:,'7+�

<RX�FDQ�DOVR�FRQWURO�WKH�RXWSXW�E\�XVHU�GHILQHG�SUHIHUHQFHV� 7KLV�LV�GLVFXVVHG�LQWKH�VHFWLRQ�VWDUWLQJ�RQ�SDJH �����

����

Page 359: Mupad Tutorial

K^Z]bg`�Zg]�Pkbmbg`�?be^l

J]Y\af_�Yf\�Ojalaf_�>ad]k

<RX�FDQ�VDYH�WKH�YDOXHV�RI�LGHQWLILHUV�RU�D�FRPSOHWH�0X3$'� VHVVLRQ�WR�D�ILOH�DQGUHDG�WKH�ILOH�ODWHU�LQWR�DQRWKHU�0X3$' VHVVLRQ�

7KH�)XQFWLRQV ZULWH DQG UHDG

7KH�IXQFWLRQ ZULWH VWRUHV�WKH�YDOXHV�RI�LGHQWLILHUV�LQ�D�ILOH� VR�WKDW�\RX�FDQ�UHXVHWKH�FRPSXWHG�UHVXOWV�LQ�DQRWKHU�0X3$' VHVVLRQ� ,Q�WKH�IROORZLQJ�H[DPSOH� ZHVDYH�WKH�YDOXHV�RI�WKH�LGHQWLILHUV D DQG E WR�WKH�ILOH DE�PE�D � ���� E � GLII�VLQ�FRV�[��� [��ZULWH��DE�PE�� D� E�

<RX�SDVV�WKH�ILOH�QDPH�DV�D�VWULQJ��6HFWLRQ ������HQFORVHG�LQ�GRXEOH�TXRWHV �� 7KHV\VWHP�WKHQ�FUHDWHV�D�ILOH�ZLWK�WKLV�QDPH��ZLWKRXW ��� ,I�\RX�UHDG�WKLV�ILOH�LQWRDQRWKHU�0X3$' VHVVLRQ�YLD�WKH�IXQFWLRQ UHDG� \RX�FDQ�DFFHVV�WKH�YDOXHV�RI�WKHLGHQWLILHUV D DQG E ZLWKRXW�UHFRPSXWLQJ�WKHP�

UHVHW���UHDG��DE�PE��� D� E

23, − FRV(FRV(x)) VLQ(x)

,I�\RX�XVH�WKH�IXQFWLRQ ZULWH DV�LQ�WKH�DERYH�H[DPSOH� LW�FUHDWHV�D�ILOH�LQ�WKH0X3$' ELQDU\�IRUPDW� %\�FRQYHQWLRQ� D�ILOH�LQ�WKLV�IRUPDW�VKRXOG�KDYH�WKH�ILOHQDPH�H[WHQVLRQ�³�PE´� <RX�FDQ�FDOO�WKH�IXQFWLRQ ZULWH ZLWK�WKH�RSWLRQ 7H[W� 7KLVJHQHUDWHV�D�ILOH�LQ�D�UHDGDEOH�WH[W�IRUPDW��

D � ���� E � H[S�[����ZULWH�7H[W� �DE�PX�� D� E�

�8VXDOO\�WKH�ILOH�QDPH�H[WHQVLRQ�IRU�0X3$' WH[W�ILOHV�VKRXOG�EH�³�PX´� 7KH�0X3$' HGLWRU�DOORZVHGLWLQJ�VXFK�ILOHV�LQ�D�FRPIRUWDEOH�ZD\�

����

Page 360: Mupad Tutorial

*+ Bginm�Zg]�Hnminm

7KH�ILOH DE�PX QRZ�FRQWDLQV�WKH�IROORZLQJ�WZR�V\QWDFWLFDOO\�FRUUHFW�0X3$'FRPPDQGV�

D � ����E � KROG�H[S��KROG�BSOXV��KROG�[�� ����

<RX�FDQ�XVH�WKH�IXQFWLRQ UHDG WR�UHDG�WKLV�ILOH�D � �� E � �� UHDG��DE�PX��� D� E

23, Hx+1

7KH�WH[W�IRUPDW�ILOHV�JHQHUDWHG�E\ ZULWH FRQWDLQ�YDOLG�0X3$' FRPPDQGV� 2IFRXUVH� \RX�FDQ�XVH�DQ\�HGLWRU�WR�JHQHUDWH�VXFK�D�WH[W�ILOH�³E\�KDQG´�DQG�UHDG�LWLQWR�D�0X3$' VHVVLRQ� XVLQJ�WKH�FRPPDQG UHDG RU�WKH�HQWU\�µ5HDG�&RPPDQGV «¶IURP�WKH�µ1RWHERRN¶�PHQX� ,Q�IDFW�WKLV�LV�D�QDWXUDO�ZD\�WR�SURFHHG�ZKHQ�\RXGHYHORS�PRUH�FRPSOH[�0X3$' SURFHGXUHV�

6DYLQJ�D�0X3$'� 6HVVLRQ

,I�\RX�FDOO�WKH�IXQFWLRQ ZULWH ZLWKRXW�VXSSO\LQJ�DQ\�LGHQWLILHUV�DV�DUJXPHQWV� WKHV\VWHP�ZULWHV�WKH�YDOXHV�RI DOO LGHQWLILHUV�KDYLQJ�D�YDOXH�WR�D�ILOH� H[FHSW�IRU�WKRVHGHILQHG�E\�WKH�0X3$'V\VWHP�LWVHOI� 7KXV� LW�LV�SRVVLEOH�WR�UHVWRUH�WKH�VWDWH�RI�WKHFXUUHQW�VHVVLRQ�YLD UHDG DW�D�ODWHU�WLPH�UHVXOW� � ���� UHVXOW� � ���� ���ZULWH��UHVXOWV�PE��

5HDGLQJ�'DWD�IURP�D�7H[W�)LOH

2IWHQ�\RX�ZDQW�WR�XVH�GDWD�LQ�0X3$' WKDW�DUH�JHQHUDWHG�E\�RWKHU�VRIWZDUH��IRUH[DPSOH� \RX�PLJKW�ZDQW�WR�UHDG�LQ�VWDWLVWLFDO�YDOXHV�IRU�IXUWKHU�SURFHVVLQJ�� RUDFFHVV�DOO�ILOHV�LQ�VRPH�GLUHFWRU\�DXWRPDWLFDOO\� 7KLV�LV�SRVVLEOH�ZLWK�WKH�KHOS�RIWKH�IXQFWLRQ LPSRUW��UHDGGDWD IURP�WKH�OLEUDU\ LPSRUW� 7KLV�IXQFWLRQ�FRQYHUWVWKH�FRQWHQWV�RI�D�ILOH�WR�D�QHVWHG�0X3$' OLVW� <RX�PD\�UHJDUG�WKH�ILOH�DV�D�³PDWUL[´ZLWK�OLQH�EUHDNV�LQGLFDWLQJ�WKH�EHJLQQLQJ�RI�D�QHZ�URZ� 1RWH�WKDW�WKH�URZV�PD\

����

Page 361: Mupad Tutorial

K^Z]bg`�Zg]�Pkbmbg`�?be^l

KDYH�GLIIHUHQW�OHQJWK� <RX�PD\�SDVV�DQ�DUELWUDU\�FKDUDFWHU�WR LPSRUW��UHDGGDWDDV�D�FROXPQ�VHSDUDWRU�

6XSSRVH�\RX�KDYH�D�ILOH QXPHULFDO'DWD ZLWK�WKH�IROORZLQJ 4 OLQHV�

� ��� ������ ���

�� ������ �� ���

%\�GHIDXOW� EODQN�FKDUDFWHUV�DUH�DVVXPHG�DV�FROXPQ�VHSDUDWRUV� 6R�\RX�FDQ�UHDGWKLV�ILOH�LQWR�D�0X3$' VHVVLRQ�DV�IROORZV�

GDWD � LPSRUW��UHDGGDWD��QXPHULFDO'DWD���GDWD>�@� GDWD>�@� GDWD>�@� GDWD>�@

[1, 1.2, 12]

[2.34, 234]

[34, 345.6]

[4, 44, 444]

7KH�KHOS�SDJH�IRU LPSRUW��UHDGGDWD SURYLGHV�IXUWKHU�LQIRUPDWLRQ� $OVR�VHHLPSRUW��FVY IRU�DQRWKHU�LPSRUWDQW�GDWD�H[FKDQJH�IRUPDW�

����

Page 362: Mupad Tutorial
Page 363: Mupad Tutorial

)+

8WLOLWLHV

,Q�WKLV�FKDSWHU�ZH�SUHVHQW�VRPH�XVHIXO�IXQFWLRQV� 'XH�WR�VSDFH�OLPLWDWLRQV� ZH�GRQRW�H[SODLQ�WKHLU�FRPSOHWH�IXQFWLRQDOLW\�DQG�UHIHU�WR�WKH�KHOS�SDJHV�IRU�PRUHGHWDLOHG�LQIRUPDWLRQ�

Page 364: Mupad Tutorial

*, Nmbebmb^l

Mk]j%<]Úf]\�Hj]^]j]f[]k

<RX�FDQ�FXVWRPL]H�0X3$'�¶V�EHKDYLRU�E\�XVLQJ SUHIHUHQFHV� 7KH�IROORZLQJFRPPDQG�OLVWV�DOO�SUHIHUHQFHV�

3UHI��3UHI��DOLDV � 758(3UHI��DXWR([SDQVLRQ/LPLW� ����3UHI��DXWR3ORW � )$/6(3UHI��FDOO%DFN � 1,/3UHI��FDOO2Q([LW � 1,/3UHI��GEJ$XWR'LVSOD\ � 758(3UHI��GEJ$XWR/LVW � 758(3UHI��IORDW)RUPDW � �J�3UHI��LJQRUH1R'HEXJ � )$/6(3UHI��NHHS2UGHU � 'RPDLQV2QO\3UHI��NHUQHO � >�� �� �@3UHI��PD[0HP � �3UHI��PD[7LPH � �3UHI��RXWSXW � 1,/3UHI��SRVW,QSXW � 1,/3UHI��SRVW2XWSXW � 1,/3UHI��UHSRUW � �3UHI��WUDLOLQJ=HURHV � )$/6(3UHI��W\SH&KHFN � ,QWHUDFWLYH3UHI��XQORDGDEOH0RGXOHV � )$/6(3UHI��XVHU2SWLRQV � ��3UHI��YHUERVH5HDG � �3UHI��ZDUQ'HDG3URF(QY � )$/6(

:H�UHIHU�WR�WKH�KHOS�SDJH "3UHI IRU�D�FRPSOHWH�GHVFULSWLRQ�RI�DOO�SUHIHUHQFHV� 2QO\D�IHZ�RSWLRQV�DUH�GLVFXVVHG�EHORZ�

<RX�FDQ�XVH�WKH UHSRUW SUHIHUHQFH WR�UHTXHVW�UHJXODU�LQIRUPDWLRQ�RQ�0X3$'¶VDOORFDWHG�PHPRU\� WKH�PHPRU\�UHDOO\�XVHG� DQG�WKH�HODSVHG�FRPSXWLQJ�WLPH�9DOLG�DUJXPHQWV�IRU UHSRUW DUH�LQWHJHUV�EHWZHHQ 0 DQG 9� 7KH�GHIDXOW�YDOXH 0PHDQV�WKDW�QR�LQIRUPDWLRQ�LV�GLVSOD\HG� ,I�\RX�FKRRVH�WKH�YDOXH 9� \RXSHUPDQHQWO\�REWDLQ�LQIRUPDWLRQ�DERXW�0X3$'¶V�FXUUHQW�VWDWH� <RX�DOVR�JHW�WKLVLQIRUPDWLRQ�LQ�WKH�VWDWXV�EDU�RI�WKH�QRWHERRN� EXW�WKH�RXWSXW�RI 3UHI��UHSRUW ZLOOVWD\�LQ�WKH�RXWSXW� DOORZLQJ�\RX�WR�UHYLHZ�WKH�GHYHORSPHQW�RYHU�WLPH�

����

Page 365: Mupad Tutorial

Nl^k&=^Ûg^]�Ik^_^k^g\^l

3UHI��UHSRUW���� LQW�VLQ�[�A��[� [ �����>XVHG ����N� UHVHUYHG ����N� VHFRQGV �@>XVHG ����N� UHVHUYHG ����N� VHFRQGV �@>XVHG ����N� UHVHUYHG ����N� VHFRQGV �@

γ

2− &L (2)

2+OQ (2)

2

7KH�SUHIHUHQFH IORDW)RUPDW FRQWUROV�WKH�RXWSXW�RI�IORDWLQJ�SRLQW�QXPEHUV� )RUH[DPSOH� LI�\RX�VXSSO\�WKH�DUJXPHQW �H�� IORDWLQJ�SRLQW�QXPEHUV�DUH�SULQWHG�ZLWKPDQWLVVD�DQG�H[SRQHQW��H�J�� �����H�� = 1.234 · 10−7�� ,I�\RX�XVH�WKH�DUJXPHQW�I�� 0X3$' SULQWV�WKHP�LQ�IL[HG�SRLQW�UHSUHVHQWDWLRQ�

3UHI��IORDW)RUPDW��H��� IORDW�H[S������1.928749848 · 10−22

3UHI��IORDW)RUPDW��I��� IORDW�H[S������0.0000000000000000000001928749848

<RX�FDQ�XVH�SUHIHUHQFHV�WR�FRQWURO�WKH�VFUHHQ�RXWSXW�LQ�RWKHU�ZD\V�DV�ZHOO� )RUH[DPSOH� DIWHU�FDOOLQJ 3UHI��RXWSXW�)���0X3$' SDVVHV�HYHU\�UHVXOW�FRPSXWHG�E\WKH�NHUQHO�WR�WKH�IXQFWLRQ ) EHIRUH�SULQWLQJ�LW�RQ�WKH�VFUHHQ� 7KH�VFUHHQ�RXWSXW�LVWKHQ�WKH�UHVXOW�RI�WKH�IXQFWLRQ ) LQVWHDG�RI�WKH�UHVXOW�RULJLQDOO\�FRPSXWHG�E\�WKHNHUQHO� ,Q�WKH�IROORZLQJ�H[DPSOH� ZH�XVH�WKLV�WR�FRPSXWH�DQG�RXWSXW�WKHQRUPDOL]DWLRQ�RI�HYHU\�UHTXHVWHG�H[SUHVVLRQ� :H�GHILQH�D�SURFHGXUH )�&KDSWHU ����DQG�SDVV�LW�WR 3UHI��RXWSXW�

3UHI��RXWSXW�[ �! �[� QRUPDO�[����� � [��[ � �� � ��[

x

x+ 1− 2x

+ 1, −−2x2 + x+ 2x2 + x

7KH�OLEUDU\ JHQHUDWH FRQWDLQV�IXQFWLRQV�IRU�FRQYHUWLQJ�0X3$' H[SUHVVLRQV�WR�WKHLQSXW�IRUPDW�RI�RWKHU�SURJUDPPLQJ�ODQJXDJHV��&��)RUWUDQ� RU 7(;�� ,Q�WKHIROORZLQJ�H[DPSOH� WKH�0X3$' RXWSXW�LV�FRQYHUWHG�WR�D�VWULQJ� <RX�PLJKW�WKHQZULWH�WKLV�VWULQJ�WR�D�WH[W�ILOH�IRU�IXUWKHU�SURFHVVLQJ�ZLWK 7(;�

3UHI��RXWSXW�JHQHUDWH��7H;�� GLII�I�[��[�´\IUDF^\SDUWLDO`^\SDUWLDO�[`�I\�\OHIW�[\ULJKW�´

����

Page 366: Mupad Tutorial

*, Nmbebmb^l

7KH�IROORZLQJ�FRPPDQG�UHVHWV�WKH�RXWSXW�URXWLQH�WR�LWV�RULJLQDO�VWDWH�

3UHI��RXWSXW�1,/��

6RPH�XVHUV�ZDQW�WR�REWDLQ�LQIRUPDWLRQ�RQ�FHUWDLQ�FKDUDFWHULVWLFV�RI�DOOFRPSXWDWLRQV� VXFK�DV�FRPSXWLQJ�WLPHV� 7KLV�FDQ�EH�DFKLHYHG�ZLWK�WKH�IXQFWLRQV3UHI��SRVW,QSXW DQG 3UHI��SRVW2XWSXW� %RWK�WDNH�0X3$' SURFHGXUHV�DVDUJXPHQWV� ZKLFK�DUH�WKHQ�FDOOHG�DIWHU�HDFK�LQSXW�RU�RXWSXW� UHVSHFWLYHO\� ,Q�WKHIROORZLQJ�H[DPSOH� ZH�XVH�D�SURFHGXUH�WKDW�DVVLJQV�WKH�V\VWHP�WLPH�UHWXUQHG�E\WLPH�� WR�WKH�JOREDO�LGHQWLILHU 7LPH� 7KLV VWDUWV�D�³WLPHU´�EHIRUH�HDFK�FRPSXWDWLRQ�

3UHI��SRVW,QSXW�SURF�� EHJLQ 7LPH � WLPH�� HQGBSURF��

:H�GHILQH�D�SURFHGXUH P\,QIRUPDWLRQ ZKLFK²DPRQJ�RWKHU�WKLQJV²XVHV�WKLVWLPHU�WR�GHWHUPLQH�WKH�WLPH�WDNHQ�E\�WKH�FRPSXWDWLRQ� ,W HPSOR\V H[SU�WH[W�6HFWLRQ ������WR�FRQYHUW�WKH�QXPHULFDO�WLPH�YDOXH�WR�D�VWULQJ�DQG�FRQFDWHQDWHV�LWZLWK�VRPH�RWKHU�VWULQJV� 0RUHRYHU� WKH�SURFHGXUH�XVHV GRPW\SH WR�ILQG�WKHGRPDLQ�W\SH�RI�WKH�REMHFW�DQG�FRQYHUWV�LW�WR�D�VWULQJ�DV�ZHOO� )LQDOO\�LWFRQFDWHQDWHV�WKH�WLPH�LQIRUPDWLRQ� VRPH�EODQNV � �� DQG�WKH�W\SH�LQIRUPDWLRQ�YLDBFRQFDW� 7KH�IXQFWLRQ OHQJWK LV�XVHG�WR�GHWHUPLQH�WKH�SUHFLVH�QXPEHU�RI�EODQNVVR�WKDW�WKH�GRPDLQ�W\SH�DSSHDUV�IOXVKHG�ULJKW�RQ�WKH�VFUHHQ�

P\,QIRUPDWLRQ � SURF�� EHJLQ�GRPDLQ W\SH� �� H[SU�WH[W�GRPW\SH�DUJV������WLPH� �� H[SU�WH[W�WLPH�� � 7LPH�� � PVHF��BFRQFDW����

� � � 7(;7:,'7+���OHQJWK�����OHQJWK��������

HQGBSURF�

:H�SDVV�WKLV�SURFHGXUH�DV�DUJXPHQW�WR 3UHI��SRVW2XWSXW�

3UHI��SRVW2XWSXW�P\,QIRUPDWLRQ��

$IWHU�HDFK�FRPSXWHG�UHVXOW� WKH�V\VWHP�QRZ�SULQWV�WKH�VWULQJ�JHQHUDWHG�E\�WKHSURFHGXUH P\,QIRUPDWLRQ RQ�WKH�VFUHHQ�

IDFWRU�[A� � ��(x− 1) ·

(x+ x2 + 1

)

WLPH� �� PVHF GRPDLQ W\SH� )DFWRUHG

����

Page 367: Mupad Tutorial

Nl^k&=^Ûg^]�Ik^_^k^g\^l

<RX�FDQ�UHVHW�D�SUHIHUHQFH�WR�LWV�GHIDXOW�YDOXH�E\�VSHFLI\LQJ 1,/ DV�DUJXPHQW� )RUH[DPSOH� WKH�FRPPDQG 3UHI��UHSRUW�1,/� UHVHWV�WKH�YDOXH�RI 3UHI��UHSRUW WR0� 6LPLODUO\� 3UHI�1,/� UHVHWV DOO SUHIHUHQFHV�WR�WKHLU�GHIDXOW�YDOXHV�

([HUFLVH������ 7KH�0X3$' IXQFWLRQ E\WHV UHWXUQV�WKH�DPRXQW�RI�ORJLFDO�DQGSK\VLFDO�PHPRU\�XVHG�E\�WKH�FXUUHQW�0X3$' VHVVLRQ� /HW�WKLV�LQIRUPDWLRQ�DSSHDURQ�WKH�VFUHHQ�DIWHU�HDFK�RXWSXW� XVLQJ 3UHI��SRVW2XWSXW�

����

Page 368: Mupad Tutorial

*, Nmbebmb^l

L`]�@aklgjq�E][`Yfake

(YHU\�LQSXW�WR�0X3$'� \LHOGV�D�UHVXOW�DIWHU�HYDOXDWLRQ�E\�WKH�V\VWHP� 7KHFRPSXWHG�REMHFWV�DUH�VWRUHG�LQWHUQDOO\�LQ�D KLVWRU\�WDEOH� 1RWH�WKDW�WKH�UHVXOW�RIHYHU\�VWDWHPHQW�LV�VWRUHG� HYHQ�LI�LW�LV�QRW�SULQWHG�RQ�WKH�VFUHHQ� <RX�FDQ�XVH�LWODWHU�E\�PHDQV�RI�WKH�IXQFWLRQ ODVW� 7KH�FRPPDQG ODVW��� UHWXUQV�WKH�SUHYLRXVUHVXOW� ODVW��� WKH�ODVW�EXW�RQH� DQG�VR�RQ� ,QVWHDG�RI ODVW�L�� \RX�PD\�XVH�WKHVKRUWHU�QRWDWLRQ �L� 0RUHRYHU� � LV�VKRUW�IRU �� RU ODVW���� 7KXV�WKH�LQSXW

I � GLII�OQ�OQ�[��� [�� LQW�I� [�

FDQ�EH�SDVVHG�WR�WKH�V\VWHP�LQ�WKH�IROORZLQJ�HTXLYDOHQW�IRUP�

GLII�OQ�OQ�[��� [�� LQW��� [�

7KLV�HQDEOHV�\RX�WR�DFFHVV�LQWHUPHGLDWH�UHVXOWV�WKDW�KDYH�QRW�EHHQ�DVVLJQHG�WR�DQLGHQWLILHU� ,W�LV�UHPDUNDEOH�WKDW�WKH�XVH�RI ODVWPD\�VSHHG�XS�FHUWDLQ�LQWHUDFWLYHHYDOXDWLRQV�ZKHQ�FRPSDUHG�WR�WKH�XVH�RI�LGHQWLILHUV�WR�VWRUH�LQWHUPHGLDWH�UHVXOWV�,Q�WKH�IROORZLQJ�H[DPSOH� ZH�ILUVW�WU\�WR�FRPSXWH�D�GHILQLWH�LQWHJUDO�V\PEROLFDOO\�$IWHU�UHFRJQL]LQJ�WKDW�0X3$' GRHV�QRW�FRPSXWH�D�V\PEROLF�YDOXH� ZH�DVN�IRU�DIORDWLQJ�SRLQW�DSSUR[LPDWLRQ�

I � LQW�VLQ�[� H[S�[A���[A� FRV�H[S�[��� [ �����∫ 1

0x2 FRV(Hx) + Hx

3VLQ(x) Gx

VWDUWLQJ7LPH � WLPH���IORDW�I���WLPH�� � VWDUWLQJ7LPH� PVHF

0.5356260737

750PVHF

7KH�IXQFWLRQ WLPH UHWXUQV�WKH�WRWDO�FRPSXWLQJ�WLPH��LQ�PLOOLVHFRQGV��XVHG�E\�WKHV\VWHP�VLQFH�WKH�EHJLQQLQJ�RI�WKH�VHVVLRQ� 7KXV�WKH�SULQWHG�GLIIHUHQFH�LV�WKH�WLPHIRU�FRPSXWLQJ�WKH�IORDWLQJ�SRLQW�DSSUR[LPDWLRQ�

����

Page 369: Mupad Tutorial

Ma^�Ablmhkr�F^\aZgblf

,Q�WKLV�H[DPSOH� ZH�FDQ�UHGXFH�WKH�FRPSXWLQJ�WLPH�GUDPDWLFDOO\�E\�HPSOR\LQJODVW�I � LQW�VLQ�[� H[S�[A���[A� FRV�H[S�[��� [ �����∫ 1

0x2 FRV(Hx) + Hx

3VLQ(x) Gx

VWDUWLQJ7LPH � WLPH���IORDW������WLPH�� � VWDUWLQJ7LPH� PVHF

0.5356260737

40PVHF

,Q�WKLV�FDVH� WKH�UHDVRQ�IRU�WKH�JDLQ�LQ�VSHHG�LV�WKDW�0X3$' GRHV QRW�UH�HYDOXDWHWKH�REMHFWV�WKDW ODVW�L�� �L� RU � UHIHU�WR�� 7KXV�FDOOV�WR ODVW IRUP�DQ�H[FHSWLRQ�WRWKH�XVXDO�FRPSOHWH�HYDOXDWLRQ�DW�LQWHUDFWLYH�OHYHO��6HFWLRQ �����

GHOHWH [� VLQ�[�� [ � �� ��VLQ(x)

<RX�FDQ�HQIRUFH�FRPSOHWH�HYDOXDWLRQ�E\�XVLQJ HYDO�

GHOHWH [� VLQ�[�� [ � �� HYDO����0

3OHDVH�QRWH�WKDW�WKH�YDOXH�RI ODVW�L�PD\�GLIIHU�IURP�WKH i�WK�EXW�ODVW YLVLEOHRXWSXW�LI�\RX�KDYH�VXSSUHVVHG�WKH�VFUHHQ�RXWSXW�RI�VRPH�LQWHUPHGLDWH�UHVXOWV�E\WHUPLQDWLQJ�WKH�FRUUHVSRQGLQJ�FRPPDQGV�ZLWK�D�FRORQ� $OVR�QRWH�WKDW�WKH�YDOXHRI�WKH�H[SUHVVLRQ ODVW�L� FKDQJHV�SHUPDQHQWO\�GXULQJ�D�FRPSXWDWLRQ�

�� ODVW��� � �� ODVW��� � �2

3

�1RWH�WKDW�WKLV�JDLQ�LQ�VSHHG�LV�RQO\�DFKLHYHG�ZKHQ�ZRUNLQJ�LQWHUDFWLYHO\� VLQFH�LGHQWLILHUV�DUH�HYDO�XDWHG�ZLWK�OHYHO 1 ZLWKLQ�SURFHGXUHV�DQ\ZD\��6HFWLRQ �������

����

Page 370: Mupad Tutorial

*, Nmbebmb^l

7KH�HQYLURQPHQW�YDULDEOH +,6725< GHWHUPLQHV�WKH�QXPEHU RI�UHVXOWV�WKDW�0X3$'VWRUHV�LQ�D�VHVVLRQ�DQG�WKDW�FDQ�EH�DFFHVVHG�YLD ODVW�+,6725<

20

7KLV�GHIDXOW�PHDQV�WKDW�0X3$' VWRUHV�WKH�SUHYLRXV 20 H[SUHVVLRQV� 2I�FRXUVH�\RXFDQ�FKDQJH�WKLV�GHIDXOW�E\�DVVLJQLQJ�D�GLIIHUHQW�YDOXH�WR +,6725<��7KLV�PD\�EHDSSURSULDWH�ZKHQ�0X3$' KDV�WR�KDQGOH�KXJH�REMHFWV��VXFK�DV�YHU\�ODUJHPDWULFHV��WKDW�ILOO�XS�D�VLJQLILFDQW�SDUW�RI�WKH�PDLQ�PHPRU\�RI�\RXU�FRPSXWHU�&RSLHV�RI�WKHVH�REMHFWV�DUH�VWRUHG�LQ�WKH�KLVWRU\�WDEOH� UHTXLULQJ�DGGLWLRQDO�VWRUDJHVSDFH� ,Q�WKLV�FDVH� \RX�ZRXOG�UHGXFH�WKH�PHPRU\�ORDG�E\�FKRRVLQJ�VPDOO�YDOXHV�LQ+,6725<��1RWH�WKDW +,6725< RQO\�\LHOGV�WKH�YDOXH�RI�WKH�LQWHUDFWLYH�³KLVWRU\GHSWK�´ ,QVLGH�D�SURFHGXUH� ODVW RQO\�DFFHSWV�WKH�DUJXPHQWV �� � DQG ��

:H�VWURQJO\�UHFRPPHQG�WR�XVH ODVW RQO\�LQWHUDFWLYHO\� 7KH�XVH�RI ODVW ZLWKLQSURFHGXUHV�LV�FRQVLGHUHG�EDG�SURJUDPPLQJ�VW\OH�DQG�VKRXOG�EH�DYRLGHG�

����

Page 371: Mupad Tutorial

Bg_hkfZmbhg�hg�FnI:=� :e`hkbmafl

Af^gjeYlagf�gf�EmH9<� 9d_gjal`ek

6RPH�0X3$'� V\VWHP�IXQFWLRQV�PD\�SURYLGH�UXQWLPH�LQIRUPDWLRQ� 7KH�IROORZLQJFRPPDQG�PDNHV�DOO�VXFK�SURFHGXUHV�SURGXFH�DGGLWLRQDO�LQIRUPDWLRQ�RQ�WKHVFUHHQ�

VHWXVHULQIR�$Q\� ���

$V�DQ�H[DPSOH� ZH�LQYHUW�WKH�IROORZLQJ�PDWUL[��6HFWLRQ ������RYHU�WKH�ULQJ�RILQWHJHUV�PRGXOR 11�

0 � 'RP��0DWUL[�'RP��,QWHJHU0RG������$ � 0�>>�� �� �@� >�� �� �@� >�� �� �@@��$A����,QIR� XVLQJ *DXVVLDQ HOLPLQDWLRQ �/5 GHFRPSRVLWLRQ�⎛

⎜⎝1PRG 11 0PRG 11 6PRG 113PRG 11 4PRG 11 8PRG 119PRG 11 1PRG 11 0PRG 11

⎟⎠

<RX�REWDLQ�PRUH�GHWDLOHG�LQIRUPDWLRQ�E\�LQFUHDVLQJ�WKH�VHFRQG�DUJXPHQW�RIVHWXVHULQIR �WKH�³LQIRUPDWLRQ�OHYHO´��VHWXVHULQIR�$Q\� ��� $A����,QIR� XVLQJ *DXVVLDQ HOLPLQDWLRQ �/5 GHFRPSRVLWLRQ�,QIR� VHDUFKLQJ IRU SLYRW HOHPHQW LQ FROXPQ �,QIR� FKRRVLQJ SLYRW HOHPHQW � PRG �� �URZ ��,QIR� VHDUFKLQJ IRU SLYRW HOHPHQW LQ FROXPQ �,QIR� FKRRVLQJ SLYRW HOHPHQW � PRG �� �URZ ��,QIR� VHDUFKLQJ IRU SLYRW HOHPHQW LQ FROXPQ �,QIR� FKRRVLQJ SLYRW HOHPHQW � PRG �� �URZ ��⎛

⎜⎝1PRG 11 0PRG 11 6PRG 113PRG 11 4PRG 11 8PRG 119PRG 11 1PRG 11 0PRG 11

⎟⎠

����

Page 372: Mupad Tutorial

*, Nmbebmb^l

,I�\RX�HQWHU

VHWXVHULQIR�$Q\� ���

WKH�V\VWHP�VWRSV�SULQWLQJ�DGGLWLRQDO�LQIRUPDWLRQ�

$A����⎛

⎜⎝1PRG 11 0PRG 11 6PRG 113PRG 11 4PRG 11 8PRG 119PRG 11 1PRG 11 0PRG 11

⎟⎠

7KH�ILUVW�DUJXPHQW�RI VHWXVHULQIRPD\�EH�DQ�DUELWUDU\�SURFHGXUH�QDPH�RU�OLEUDU\QDPH� 7KHQ�WKH�FRUUHVSRQGLQJ�SURFHGXUH�V��SURYLGH�DGGLWLRQDO�LQIRUPDWLRQ�3URJUDPPHUV�RI�WKH�V\VWHP�IXQFWLRQV�KDYH�EXLOW�RXWSXW�FRPPDQGV�LQWR�WKH�FRGHYLD XVHULQIR� 7KHVH�FRPPDQGV�DUH�DFWLYDWHG�E\ VHWXVHULQIR� <RX�FDQ�XVH�WKLV�LQ\RXU�RZQ�SURFHGXUHV�DV�ZHOO��"XVHULQIR��

�����

Page 373: Mupad Tutorial

K^lmZkmbg`�Z�FnI:=� L^llbhg

J]klYjlaf_�Y�EmH9<� K]kkagf

7KH�FRPPDQG UHVHW�� UHVHWV�D�0X3$'� VHVVLRQ�WR�LWV�LQLWLDO�VWDWH� $IWHUZDUGV�DOO�LGHQWLILHUV�WKDW�\RX�GHILQHG�SUHYLRXVO\�KDYH�QR�YDOXH�DQG�DOO�HQYLURQPHQWYDULDEOHV�DUH�UHVHW�WR�WKHLU�GHIDXOW�YDOXHV�

D � KHOOR� ',*,76 � ���� UHVHW��� D� ',*,76a, 10

�����

Page 374: Mupad Tutorial

*, Nmbebmb^l

=p][mlaf_�;geeYf\k�g^�l`]�Gh]jYlaf_�Kqkl]e

<RX�FDQ�XVH�WKH�IXQFWLRQ V\VWHP �RU�WKH�H[FODPDWLRQ�V\PERO � IRU�VKRUW��WRH[HFXWH�D�FRPPDQG�RI�WKH�RSHUDWLQJ�V\VWHP� 2Q�81,;� SODWIRUPV� WKH�IROORZLQJFRPPDQG�OLVWV�WKH�FRQWHQWV�RI�WKH�FXUUHQW�GLUHFWRU\�

�OVELQ &RQWULEXWLRQV GRF IRQWV OLE PPJ

<RX�FDQ�QHLWKHU�XVH�WKH�RXWSXW�RI�VXFK�D�FRPPDQG�IRU�IXUWKHU�FRPSXWDWLRQ�QRUVDYH�LW�WR�D�ILOH�� V\VWHP UHWXUQV�WKH�HUURU�VWDWXV�RI�WKH�RSHUDWLQJ�V\VWHP�WR�WKH0X3$'� VHVVLRQ� :KHQ�FDOOHG�ZLWK�WKH � V\QWD[� WKH�RXWSXW�RI�WKLV�YDOXH�LVVXSSUHVVHG�

�,I�WKLV�LV�GHVLUHG� \RX�FDQ�XVH�DQRWKHU�FRPPDQG�RI�WKH�RSHUDWLQJ�V\VWHP�WR�ZULWH�WKH�RXWSXW�WR�DILOH�DQG�UHDG�WKLV�ILOH�LQWR�D�0X3$' VHVVLRQ�YLD LPSRUW��UHDGGDWD�

�����

Page 375: Mupad Tutorial

),

7\SH�6SHFLILHUV

7KH�GDWD�VWUXFWXUH�RI�D�0X3$'� REMHFW�LV�LWV�GRPDLQ�W\SH� ZKLFK�FDQ�EH�UHTXHVWHGE\�PHDQV�RI�WKH�IXQFWLRQ GRPW\SH� 7KH�GRPDLQ�W\SH�UHIOHFWV�WKH�VWUXFWXUH�WKDW�WKH0X3$' NHUQHO�XVHV�LQWHUQDOO\�WR�PDQDJH�WKH�REMHFWV� 7KH�W\SH�FRQFHSW�DOVR�OHDGVWR�D�FODVVLILFDWLRQ�RI�WKH�REMHFWV�DFFRUGLQJ�WR�WKHLU�PDWKHPDWLFDO�PHDQLQJ�QXPEHUV� VHWV� H[SUHVVLRQV� VHULHV�H[SDQVLRQV� SRO\QRPLDOV� HWF�

,Q�WKLV�VHFWLRQ� ZH�GHVFULEH�KRZ�WR�REWDLQ�GHWDLOHG�LQIRUPDWLRQ�DERXW�WKHPDWKHPDWLFDO�VWUXFWXUH�RI�REMHFWV� )RU�H[DPSOH� KRZ�FDQ�\RX�ILQG�RXW�HIILFLHQWO\ZKHWKHU�DQ�LQWHJHU�RI�GRPDLQ�W\SH '20B,17 LV SRVLWLYH RU HYHQ� RU�ZKHWKHU�DOOHOHPHQWV�RI�D�VHW�DUH�HTXDWLRQV"

6XFK�W\SH�FKHFNV�DUH�EDUHO\�UHOHYDQW�ZKHQ�XVLQJ�0X3$' LQWHUDFWLYHO\� \RX�FDQFRQWURO�WKH�PDWKHPDWLFDO�PHDQLQJ�RI�DQ�REMHFW�E\�GLUHFW�LQVSHFWLRQ� 7\SH�FKHFNVDUH�PDLQO\�XVHG�IRU�LPSOHPHQWLQJ�PDWKHPDWLFDO�DOJRULWKPV� L�H�� ZKHQSURJUDPPLQJ�0X3$' SURFHGXUHV��&KDSWHU ���� )RU�H[DPSOH� D�SURFHGXUH�IRUGLIIHUHQWLDWLQJ�H[SUHVVLRQV�KDV�WR�GHFLGH�ZKHWKHU�LWV�LQSXW�LV�D�SURGXFW� DFRPSRVLWLRQ�RI�IXQFWLRQV� D�V\PEROLF�FDOO�RI�D�NQRZQ�IXQFWLRQ� HWF� (DFK�FDVHUHTXLUHV�D�GLIIHUHQW�DFWLRQ� VXFK�DV�VXSSO\LQJ�WKH�SURGXFW�UXOH� WKH�FKDLQ�UXOH� HWF�

Page 376: Mupad Tutorial

*- Mri^�Li^\bÛ^kl

L`]�>mf[lagfk W\SH Yf\ WHVWW\SH

)RU�PRVW�0X3$'� REMHFWV� WKH�IXQFWLRQ W\SH UHWXUQV� OLNH GRPW\SH� WKH�GRPDLQW\SH�

W\SH�>D� E@�� W\SH�^D� E`�� W\SH�DUUD\�������'20B/,67, '20B6(7, '20B$55$<

)RU�H[SUHVVLRQV�RI�GRPDLQ�W\SH '20B(;35��WKH�IXQFWLRQ W\SH \LHOGV�D�ILQHUGLVWLQFWLRQ�DFFRUGLQJ�WR�WKH�PDWKHPDWLFDO�PHDQLQJ�RI�WKH�H[SUHVVLRQ� VXPV�SURGXFWV� IXQFWLRQ�FDOOV� HWF��

W\SH�D � E�� W\SH�D E�� W\SH�DAE�� W\SH�D�E��´BSOXV´, ´BPXOW´, ´BSRZHU´, ´IXQFWLRQ´

W\SH�D E�� W\SH�D � E�� W\SH�D � E�´BHTXDO´, ´BOHVV´, ´BOHHTXDO´

)RU�PRVW�RI�WKHVH��ZLWK�WKH�H[FHSWLRQ�RI D�E��� WKH�UHVXOW�UHWXUQHG�E\ W\SH LV�WKHQDPH�RI�WKH�IXQFWLRQ�WKDW�JHQHUDWHV�WKH�H[SUHVVLRQ��LQWHUQDOO\� D�V\PEROLF�VXP�RUSURGXFW�LV�UHSUHVHQWHG�E\�D�FDOO�RI�WKH�V\VWHP�IXQFWLRQV BSOXV RU BPXOW�UHVSHFWLYHO\�� 0RUH�JHQHUDOO\� WKH�UHVXOW�IRU�PRVW�V\PEROLF�FDOOV�RI�V\VWHPIXQFWLRQV�LV�WKH�LGHQWLILHU�RI�WKH�IXQFWLRQ�DV�D�VWULQJ�

W\SH�OQ�[��� W\SH�GLII�I�[�� [��� W\SH�IDFW�[��´OQ´, ´GLII´, ´IDFW´

<RX�FDQ�XVH�ERWK�WKH�GRPDLQ�W\SHV '20B,17� '20B(;35��HWF��DQG�WKH�VWULQJVUHWXUQHG�E\ W\SH DV W\SH�VSHFLILHUV� 7KHUH�H[LVWV�D�YDULHW\�RI�RWKHU�W\SH�VSHFLILHUVLQ�DGGLWLRQ�WR�WKH�³VWDQGDUG�W\SLQJ´�RI�0X3$' REMHFWV�JLYHQ�E\ W\SH� $Q�H[DPSOHLV 7\SH��1XPHULF� 7KLV�W\SH�FRPSULVHV�DOO�³QXPHULFDO´ REMHFWV��RI�GRPDLQ�W\SH'20B,17� '20B5$7� '20B)/2$7��RU '20B&203/(;��

7KH�FDOO WHVWW\SH�REMHFW� W\SH6SHFLILHU� FKHFNV�ZKHWKHU�DQ�REMHFW�FRPSOLHVZLWK�WKH�VSHFLILHG�W\SH� 7KH�UHVXOW�LV�HLWKHU 758( RU )$/6(��6HYHUDO�W\SH�VSHFLILHUVPD\�FRUUHVSRQG�WR�DQ�REMHFW�

WHVWW\SH����� '20B5$7�� WHVWW\SH����� 7\SH��1XPHULF�758(, 758(

����

Page 377: Mupad Tutorial

Ma^�?ng\mbhgl W\SH Zg] WHVWW\SH

WHVWW\SH�� � [� �BSOXV��� WHVWW\SH�� � [� '20B(;35�758(, 758(

WHVWW\SH�I�[�� �IXQFWLRQ��� WHVWW\SH�I�[�� '20B(;35�758(, 758(

([HUFLVH������ &RQVLGHU�WKH�H[SUHVVLRQ

f(i) =i5/2 + i2 − i1/2 − 1

i5/2 + i2 + 2 i3/2 + 2 i+ i1/2 + 1

+RZ�FDQ�0X3$' GHFLGH�ZKHWKHU�WKH�VHW

6 � ^I�L� � L ���������` XQLRQ ^I�L� � L �������`�

FRQWDLQV�RQO\�UDWLRQDO�QXPEHUV" +LQW� )RU�D�VSHFLILF�LQWHJHU L� XVH�WKH�IXQFWLRQQRUPDO WR�VLPSOLI\�VXEH[SUHVVLRQV�RI I�L� FRQWDLQLQJ�VTXDUH�URRWV�

([HUFLVH������ &RQVLGHU�WKH�H[SUHVVLRQV VLQ(iπ/200) ZLWK�LQWHJHU�YDOXHV�RI iEHWZHHQ 0 DQG 100� :KLFK�RI�WKHP�DUH�VLPSOLILHG�E\�WKH�0X3$' VLQ IXQFWLRQ�ZKLFK�DUH�UHWXUQHG�DV�V\PEROLF�YDOXHV VLQ�·�"

����

Page 378: Mupad Tutorial

*- Mri^�Li^\bÛ^kl

;ge^gjlYZd]�Lqh]�;`][caf_2 l`] 7\SH DaZjYjq

7KH�W\SH�VSHFLILHUV�SUHVHQWHG�DERYH�DUH�XVHIXO�RQO\�IRU�FKHFNLQJ�UHODWLYHO\�VLPSOHVWUXFWXUHV� )RU�PRUH�DGYDQFHG�W\SH�FKHFNLQJ� PRUH�IOH[LEOH�W\SH�VSHFLILFDWLRQV�DUHQHHGHG� )RU�H[DPSOH� KRZ�FDQ�\RX�FKHFN�ZLWKRXW�GLUHFW�LQVSHFWLRQ�ZKHWKHU�WKHREMHFW >���������@ LV�D�OLVW�RI�SRVLWLYH�LQWHJHUV"

)RU�WKDW�SXUSRVH� WKH 7\SH OLEUDU\�SURYLGHV IXUWKHU�W\SH�VSHFLILHUV�DQGFRQVWUXFWRUV� <RX�FDQ�XVH�WKHP�WR�FUHDWH�\RXU�RZQ�W\SH�VSHFLILHUV� ZKLFK�DUHUHFRJQL]HG�E\ WHVWW\SH�

LQIR�7\SH�/LEUDU\ 7\SH� W\SH H[SUHVVLRQV DQG SURSHUWLHV�� ,QWHUIDFH�7\SH��$OJHEUDLF&RQVWDQW� 7\SH��$Q\7\SH�7\SH��$ULWKPHWLFDO� 7\SH��%RROHDQ�7\SH��&RPSOH[� 7\SH��&RQGLWLRQ�7\SH��&RQVWDQW� 7\SH��&RQVWDQW,GHQWV�7\SH��(TXDWLRQ� 7\SH��(YHQ�7\SH��)XQFWLRQ� 7\SH��,PDJLQDU\�7\SH��,QGHS2I� 7\SH��,QGHWHUPLQDWH�7\SH��,QWHJHU� 7\SH��,QWHUVHFWLRQ�7\SH��,QWHUYDO� 7\SH��/LVW2I�7\SH��/LVW3URGXFW� 7\SH��1HJ,QW�7\SH��1HJ5DW� 7\SH��1HJDWLYH�7\SH��1RQ1HJ,QW� 7\SH��1RQ1HJ5DW�7\SH��1RQ1HJDWLYH� 7\SH��1RQ=HUR�7\SH��1XPHULF� 7\SH��2GG�7\SH��3RO\([SU� 7\SH��3RO\2I�7\SH��3RV,QW� 7\SH��3RV5DW�7\SH��3RVLWLYH� 7\SH��3UHGLFDWH�7\SH��3ULPH� 7\SH��3URGXFW�7\SH��3URSHUW\� 7\SH��5DW([SU�7\SH��5DWLRQDO� 7\SH��5HDO�7\SH��5HODWLRQ� 7\SH��5HVLGXH�7\SH��6HTXHQFH2I� 7\SH��6HULHV�7\SH��6HW� 7\SH��6HW2I�7\SH��6LQJOHWRQ� 7\SH��7DEOH2I�7\SH��7DEOH2I(QWU\� 7\SH��7DEOH2I,QGH[�7\SH��8QLRQ� 7\SH��8QNQRZQ�7\SH��=HUR

����

Page 379: Mupad Tutorial

<hf_hkmZ[e^�Mri^�<a^\dbg`3 ma^ 7\SH Eb[kZkr

)RU�H[DPSOH� WKH�W\SH�VSHFLILHU 7\SH��3RV,QW UHSUHVHQWV�WKH�VHW�RI�SRVLWLYHLQWHJHUV n > 0� 7\SH��1RQ1HJ,QW FRUUHVSRQGV�WR�WKH�QRQQHJDWLYH�LQWHJHUV n ≥ 0�7\SH��(YHQ DQG 7\SH��2GG UHSUHVHQW�WKH�HYHQ�DQG�RGG�LQWHJHUV� UHVSHFWLYHO\�7KHVH�W\SH�VSHFLILHUV�DUH�RI�GRPDLQ�W\SH 7\SH�

GRPW\SH�7\SH��(YHQ�7\SH

<RX�FDQ�XVH�VXFK�W\SH�VSHFLILHUV�WR�TXHU\�WKH�PDWKHPDWLFDO�VWUXFWXUH�RI�0X3$'REMHFWV�YLD WHVWW\SH� ,Q�WKH�IROORZLQJ�H[DPSOH� ZH�H[WUDFW�DOO�HYHQ�LQWHJHUV�IURP�DOLVW�RI�LQWHJHUV�YLD VHOHFW �6HFWLRQ �����

VHOHFW�>L � L �����@� WHVWW\SH� 7\SH��(YHQ�[2, 4, 6, 8, 10, 12, 14, 16, 18, 20]

<RX�FDQ�XVH�FRQVWUXFWRUV�VXFK�DV 7\SH��/LVW2I RU 7\SH��6HW2I WR�SHUIRUP�W\SHFKHFNLQJ�IRU�OLVWV�RU�VHWV� )RU�H[DPSOH� D�OLVW�RI�LQWHJHUV�LV�PDWFKHG�E\�WKH�W\SH7\SH��/LVW2I�'20B,17���D�VHW�RI�HTXDWLRQV�FRUUHVSRQGV�WR�WKH�W\SH�VSHFLILHUV7\SH��6HW2I�7\SH��(TXDWLRQ��� �RU 7\SH��6HW2I��BHTXDO���� D�VHW�RI�RGGLQWHJHUV�LV�PDWFKHG�E\�WKH�W\SH 7\SH��6HW2I�7\SH��2GG��

7 � 7\SH��/LVW2I�'20B,17��WHVWW\SH�>��� �@� 7�� WHVWW\SH�^��� �`� 7��WHVWW\SH�>��� ���@� 7�

758(, )$/6(, )$/6(

7KH�FRQVWUXFWRU 7\SH��8QLRQ JHQHUDWHV�W\SH�VSHFLILHUV�FRUUHVSRQGLQJ�WR�WKHXQLRQ�RI�VLPSOHU�W\SHV� )RU�H[DPSOH� WKH�IROORZLQJ�W\SH�VSHFLILHU

7 � 7\SH��8QLRQ�'20B)/2$7� 7\SH��1HJ,QW� 7\SH��(YHQ��

PDWFKHV�UHDO�IORDWLQJ�SRLQW�QXPEHUV�DV�ZHOO�DV�QHJDWLYH�LQWHJHUV�DV�ZHOO�DV�HYHQLQWHJHUV�

WHVWW\SH�������� 7�� WHVWW\SH���� 7��WHVWW\SH��� 7�� WHVWW\SH��� 7�

758(, 758(, 758(, )$/6(

:H�GHVFULEH�DQ�DSSOLFDWLRQ�RI�W\SH�FKHFNLQJ�IRU�WKH�LPSOHPHQWDWLRQ�RI�0X3$'SURFHGXUHV�LQ�6HFWLRQ �����

����

Page 380: Mupad Tutorial

*- Mri^�Li^\bÛ^kl

([HUFLVH������ +RZ�FDQ�\RX�FRPSXWH�WKH�LQWHUVHFWLRQ�RI�D�VHW�ZLWK�WKH�VHW�RISRVLWLYH�LQWHJHUV"

([HUFLVH������ 8VH "7\SH��/LVW2I WR�FRQVXOW�WKH�KHOS�SDJH�IRU�WKLV�W\SHFRQVWUXFWRU� &RQVWUXFW�D�W\SH�VSHFLILHU�FRUUHVSRQGLQJ�WR�D�OLVW�RI�WZR�HOHPHQWVVXFK�WKDW�HDFK�HOHPHQW�LV�DJDLQ�D�OLVW�ZLWK�WKUHH�DUELWUDU\�HOHPHQWV�

����

Page 381: Mupad Tutorial

)-

/RRSV

/RRSV�DUH�LPSRUWDQW�HOHPHQWV�RI�WKH�0X3$'� SURJUDPPLQJ�ODQJXDJH� 7KHIROORZLQJ�H[DPSOH�LOOXVWUDWHV�WKH�VLPSOHVW�IRUP�RI�D IRU ORRS�

IRU L IURP � WR � GR[ � LA��SULQW��7KH VTXDUH RI�� L� �LV�� [�

HQGBIRU��7KH VTXDUH RI�� �� �LV�� �

�7KH VTXDUH RI�� �� �LV�� �

�7KH VTXDUH RI�� �� �LV�� �

�7KH VTXDUH RI�� �� �LV�� ��

7KH�ORRS�YDULDEOH L DXWRPDWLFDOO\�UXQV�WKURXJK�WKH�YDOXHV 1, 2, 3, 4� )RU�HDFK�YDOXHRI L� DOO�FRPPDQGV�EHWZHHQ GR DQG HQGBIRU DUH�H[HFXWHG� 7KHUH�PD\�EHDUELWUDULO\�PDQ\�FRPPDQGV� VHSDUDWHG�E\�VHPLFRORQV�RU�FRORQV� 7KH�V\VWHP�GRHVQRW�SULQW�WKH�UHVXOWV�FRPSXWHG�LQ�HDFK�ORRS�LWHUDWLRQ�RQ�WKH�VFUHHQ� HYHQ�LI�\RXWHUPLQDWH�WKH�FRPPDQGV�E\�VHPLFRORQV� )RU�WKDW�UHDVRQ� ZH�XVHG�WKH SULQWFRPPDQG�WR�JHQHUDWH�DQ�RXWSXW�LQ�WKH�DERYH�H[DPSOH�

Page 382: Mupad Tutorial

*. Ehhil

7KH�IROORZLQJ�YDULDQW�FRXQWV�EDFNZDUGV� :H�XVH�WKH�WRROV�IURP�6HFWLRQ ���� WRPDNH�WKH�RXWSXW�ORRN�PRUH�DSSHDOLQJ�

IRU M IURP � GRZQWR � GRSULQW�8QTXRWHG�

�7KH VTXDUH RI ��H[SU�WH[W�M��� LV ��H[SU�WH[W�MA���

HQGBIRU�7KH VTXDUH RI � LV ��

7KH VTXDUH RI � LV �

7KH VTXDUH RI � LV �

<RX�FDQ�XVH�WKH�NH\ZRUG VWHS WR�LQFUHPHQW�RU�GHFUHPHQW�WKH�ORRS�YDULDEOH�LQELJJHU�RU�VPDOOHU�VWHSV�

IRU [ IURP � WR � VWHS � GR SULQW�[� [A�� HQGBIRU��� �

�� ��

�� ��

1RWH�WKDW�DW�WKH�HQG�RI�WKH�LWHUDWLRQ�ZLWK x = 7 WKH�YDOXH�RI x LV�LQFUHPHQWHG�WR 9�7KLV�H[FHHGV�WKH�XSSHU�ERXQG 8� DQG�WKH�ORRS�WHUPLQDWHV� +HUH�LV�DQRWKHU�YDULDQWRI�WKH IRU ORRS�

IRU L LQ >�� ��� \@ GR SULQW�L� LA�� HQGBIRU��� ��

��� ���

�\� \

7KH�ORRS�YDULDEOH�RQO\�UXQV�WKURXJK�WKH�YDOXHV�IURP�WKH�OLVW [5, 27, y]� $V�\RX�FDQVHH� VXFK�D�OLVW�PD\�FRQWDLQ�V\PEROLF�HOHPHQWV�VXFK�DV�WKH�YDULDEOH y�

����

Page 383: Mupad Tutorial

Ehhil

,Q�D IRU ORRS� D�ORRS�YDULDEOH�FKDQJHV�DFFRUGLQJ�WR�IL[HG�UXOHV��W\SLFDOO\� LW�LVLQFUHPHQWHG�RU�GHFUHPHQWHG�� 7KH UHSHDW ORRS LV�D�PRUH�IOH[LEOH�DOWHUQDWLYH�ZKHUH�\RX�FDQ�DUELWUDULO\�PRGLI\�PDQ\�YDULDEOHV�LQ�HDFK�VWHS� ,Q�WKH�IROORZLQJH[DPSOH� ZH�FRPSXWH�WKH�VTXDUHV�RI�WKH�LQWHJHUV i = 2, 22, 24, 28, . . . XQWLO i2 > 100KROGV�IRU�WKH�ILUVW�WLPH�

[ � ��UHSHDW

L � [� [ � LA�� SULQW�L� [�XQWLO [ ! ��� HQGBUHSHDW�

�� �

�� ��

��� ���

7KH�V\VWHP�H[HFXWHV�WKH�FRPPDQGV�EHWZHHQ UHSHDW DQG XQWLO UHSHDWHGO\� 7KHORRS�WHUPLQDWHV�ZKHQ�WKH�FRQGLWLRQ�EHWZHHQ XQWLO DQG HQGBUHSHDW KROGV�WUXH� ,QWKH�DERYH�H[DPSOH� ZH�KDYH i = 4 DQG x = 16 DW�WKH�HQG�RI�WKH�VHFRQG�VWHS� +HQFHWKH�WKLUG�VWHS�LV�H[HFXWHG� DQG�DIWHUZDUGV�ZH�KDYH i = 16� x = 256� 1RZ�WKHWHUPLQDWLRQ�FRQGLWLRQ x > 100 LV�VDWLVILHG�DQG�WKH�ORRS�WHUPLQDWHV�

$QRWKHU�ORRS�YDULDQW�LV�WKH ZKLOH ORRS�

[ � ��ZKLOH [ � ��� GR

L � [� [ � LA�� SULQW�L� [�HQGBZKLOH�

�� �

�� ��

��� ���

,Q�D UHSHDW ORRS� WKH�V\VWHP�FKHFNV�WKH�WHUPLQDWLRQ�FRQGLWLRQ DIWHU HDFK�ORRSLWHUDWLRQ� ,Q�D ZKLOH ORRS� WKLV�FRQGLWLRQ�LV�FKHFNHG EHIRUH HDFK�LWHUDWLRQ� $V�VRRQDV�WKH�FRQGLWLRQ�HYDOXDWHV�WR )$/6(��WKH�V\VWHP�WHUPLQDWHV�WKH ZKLOH ORRS�

����

Page 384: Mupad Tutorial

*. Ehhil

<RX�FDQ�XVH EUHDN WR�DERUW�D�ORRS�H[SOLFLWO\� 7\SLFDOO\�WKLV�LV�GRQH�ZLWKLQ�DQ LIFRQVWUXFWLRQ��&KDSWHU ����

IRU L IURP � WR ��� GRSULQW�L��LI LA� ! �� WKHQ EUHDN HQGBLI

HQGBIRU��

$IWHU�D�FDOO�WR QH[W� WKH�V\VWHP�VNLSV�DOO�FRPPDQGV�XS�WR HQGBIRU� ,W�UHWXUQVLPPHGLDWHO\�WR�WKH�EHJLQQLQJ�RI�WKH�ORRS�DQG�VWDUWV�WKH�QH[W�LWHUDWLRQ�ZLWK�WKHQH[W�YDOXH�RI�WKH�ORRS�YDULDEOH�

IRU L IURP � WR � GR[ � L�LI L ! � WKHQ QH[W HQGBLI�\ � L�SULQW�[� \�

HQGBIRU��� �

�� �

)RU i > 3� RQO\�WKH�ILUVW�DVVLJQPHQW [� L LV�H[HFXWHG�

[� \5, 3

����

Page 385: Mupad Tutorial

Ehhil

:H�UHFDOO�WKDW�HYHU\�0X3$' FRPPDQG�UHWXUQV�DQ�REMHFW� )RU�D�ORRS� WKLV�LV�WKHUHWXUQ�YDOXH�RI�WKH�PRVW�UHFHQWO\�H[HFXWHG�FRPPDQG� ,I�\RX�GR�QRW�WHUPLQDWH�WKHORRS�FRPPDQG�ZLWK�D�FRORQ�DV�LQ�DOO�RI�WKH�DERYH�H[DPSOHV� WKHQ�0X3$' GLVSOD\VWKLV�YDOXH�

GHOHWH [� IRU L IURP � WR � GR [�L � LA� HQGBIRU9

<RX�PD\�SURFHVV�WKLV�YDOXH�IXUWKHU� ,Q�SDUWLFXODU� \RX�FDQ�DVVLJQ�LW�WR�DQ�LGHQWLILHURU�XVH�LW�DV�WKH�UHWXUQ�YDOXH�RI�D�0X3$' SURFHGXUH��&KDSWHU ����

IDFWRULDO � SURF�Q�ORFDO UHVXOW�EHJLQ

UHVXOW � ��IRU L IURP � WR Q GR

UHVXOW � UHVXOW LHQGBIRU

HQGBSURF�

7KH�UHWXUQ�YDOXH�RI�WKH�DERYH�SURFHGXUH�LV�WKH�UHWXUQ�YDOXH�RI�WKH IRU ORRS� ZKLFKLQ�WXUQ�LV�WKH�YDOXH�RI�WKH�ODVW�DVVLJQPHQW�WR UHVXOW�

,QWHUQDOO\� ORRSV�DUH�V\VWHP�IXQFWLRQ�FDOOV� )RU�H[DPSOH� 0X3$' SURFHVVHV�D IRUORRS�E\�HYDOXDWLQJ�WKH�IXQFWLRQ BIRU�

BIRU�L� ILUVWBL� ODVWBL� LQFUHPHQW� FRPPDQG��

7KLV�LV�HTXLYDOHQW�WR

IRU L IURP ILUVWBL WR ODVWBL VWHS LQFUHPHQW GRFRPPDQG

HQGBIRU�

����

Page 386: Mupad Tutorial
Page 387: Mupad Tutorial

).

%UDQFKLQJ� LI�WKHQ�HOVH DQG FDVH

%UDQFKLQJ�LQVWUXFWLRQV DUH�DQ�LPSRUWDQW�HOHPHQW�RI�HYHU\�SURJUDPPLQJODQJXDJH� 'HSHQGLQJ�RQ�WKH�YDOXH�RU�WKH�PHDQLQJ�RI�YDULDEOHV� GLIIHUHQWFRPPDQGV�DUH�H[HFXWHG� 7KH�VLPSOHVW�YDULDQW�LQ�0X3$'� LV�WKH LI VWDWHPHQW�

IRU L IURP � WR � GRLI LVSULPH�L�

WKHQ SULQW�H[SU�WH[W�L��� LV SULPH��HOVH SULQW�H[SU�WH[W�L��� LV QRW SULPH��

HQGBLIHQGBIRU�

�� LV SULPH�

�� LV SULPH�

�� LV QRW SULPH�

+HUH� WKH�SULPDOLW\�WHVW LVSULPH�L� UHWXUQV�HLWKHU 758( RU )$/6(��,I�WKH�YDOXH�LV758(��WKH�V\VWHP�H[HFXWHV�WKH�FRPPDQGV�EHWZHHQ WKHQ DQG HOVH �LQ�WKLV�FDVH�RQO\�RQH SULQW FRPPDQG�� ,I�LW�LV )$/6(��WKH�FRPPDQGV�EHWZHHQ HOVH DQG HQGBLIDUH�H[HFXWHG� 7KH HOVH EUDQFK�LV�RSWLRQDO�

IRU L IURP � WR � GRLI LVSULPH�L�

WKHQ WH[W � H[SU�WH[W�L��� LV SULPH��SULQW�WH[W�

HQGBLIHQGBIRU�

�� LV SULPH�

�� LV SULPH�

Page 388: Mupad Tutorial

*/ ;kZg\abg`3 LI�WKHQ�HOVH Zg] FDVH

+HUH� WKH WKHQ EUDQFK�FRPSULVHV�WZR�FRPPDQGV�VHSDUDWHG�E\�D�VHPLFRORQ��RU�DOWHUQDWLYHO\� D�FRORQ�� <RX�PD\�QHVW�FRPPDQGV� ORRSV� DQG�EUDQFKLQJ�VWDWHPHQWVDUELWUDULO\�

SULPHV � >@� HYHQ1XPEHUV � >@�IRU L IURP �� WR �� GR

LI LVSULPH�L�WKHQ SULPHV � SULPHV�>L@HOVH LI WHVWW\SH�L�7\SH��(YHQ�

WKHQ HYHQ1XPEHUV � HYHQ1XPEHUV�>L@HQGBLI

HQGBLIHQGBIRU�

,Q�WKLV�H[DPSOH� ZH�LQVSHFW�WKH�LQWHJHUV�EHWZHHQ 30 DQG 50� ,I�ZH�HQFRXQWHU�DSULPH� WKHQ�ZH�DSSHQG�LW�WR�WKH�OLVW SULPHV� 2WKHUZLVH� ZH�XVH WHVWW\SH WR�FKHFNZKHWKHU L LV�HYHQ��VHH�6HFWLRQV ���� DQG ������ ,Q�WKDW�FDVH� ZH�DSSHQG L WR�WKH�OLVWHYHQ1XPEHUV� 8SRQ�WHUPLQDWLRQ� WKH�OLVW SULPHV FRQWDLQV�DOO�SULPH�QXPEHUVEHWZHHQ 30 DQG 50� ZKLOVW HYHQ1XPEHUV FRQWDLQV�DOO�HYHQ�LQWHJHUV�LQ�WKLV�UDQJH�

SULPHV� HYHQ1XPEHUV[31, 37, 41, 43, 47] , [30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50]

<RX�FDQ�FUHDWH�PRUH�FRPSOH[�FRQGLWLRQV�IRU�WKH LI VWDWHPHQW�E\�XVLQJ�WKH%RROHDQ�RSHUDWRUV DQG� RU� DQG QRW �6HFWLRQ ������ 7KH�IROORZLQJ IRU ORRS�SULQWVDOO�SULPH�WZLQV [i, i+ 2] ZLWK i ≤ 100� 7KH�DOWHUQDWLYH�FRQGLWLRQ QRW �L!�� \LHOGVWKH�DGGLWLRQDO�SDLU [2, 4]�

IRU L IURP � WR ��� GRLI �LVSULPH�L� DQG LVSULPH�L���� RU QRW �L!��

WKHQ SULQW�>L�L��@�HQGBLI

HQGBIRU�>�� �@

>�� �@

>�� �@

���

����

Page 389: Mupad Tutorial

;kZg\abg`3 LI�WKHQ�HOVH Zg] FDVH

,QWHUQDOO\� DQ LI VWDWHPHQW�LV�MXVW�D�FDOO�RI�WKH�V\VWHP�IXQFWLRQ BLI�

BLI�FRQGLWLRQ� FRPPDQG�� FRPPDQG���

LV�HTXLYDOHQW�WR

LI FRQGLWLRQ WKHQ FRPPDQG� HOVH FRPPDQG� HQGBLI�

7KH�UHWXUQ�YDOXH�RI�DQ LI VWDWHPHQW�LV� DV�IRU�DQ\�0X3$' SURFHGXUH� WKH�UHVXOW�RIWKH�ODVW�H[HFXWHG�FRPPDQG��

[ � ��� LI [ ! � WKHQ [ HOVH �[ HQGBLI2

)RU�H[DPSOH� \RX�FDQ�XVH�WKH�DUURZ�RSHUDWRU �! �6HFWLRQ ������WR�LPSOHPHQW�WKHDEVROXWH�YDOXH�RI�QXPEHUV�DV�IROORZV�

$EV � \ �! �LI \ ! � WKHQ \ HOVH �\ HQGBLI��$EV����� $EV������� $EV�����

2,23, 3.5

$V�\RX�FDQ�VHH� \RX�FDQ�XVH LI FRPPDQGV�ERWK�DW�WKH�LQWHUDFWLYH�OHYHO�DQG�ZLWKLQSURFHGXUHV� 7KH�W\SLFDO�DSSOLFDWLRQ�LV�LQ�SURJUDPPLQJ�0X3$' SURFHGXUHV� ZKHUHLI VWDWHPHQWV�DQG�ORRSV�FRQWURO�WKH�IORZ�RI�WKH�DOJRULWKP� $ VLPSOH�H[DPSOH�LV�WKHDERYH�IXQFWLRQ $EV� <RX�ILQG�PRUH�H[DPSOHV�LQ�&KDSWHU ���

,I�\RX�KDYH�VHYHUDO�QHVWHG LI « HOVH LI « FRQVWUXFWLRQV� \RX�FDQ�DEEUHYLDWH�WKLV�E\XVLQJ�WKH HOLI VWDWHPHQW�

LI FRQGLWLRQ� WKHQVWDWHPHQWV�

HOLI FRQGLWLRQ� WKHQVWDWHPHQWV�

HOLI ���HOVH

VWDWHPHQWVHQGBLI�

�,I�QR�FRPPDQG�LV�H[HFXWHG�WKHQ�WKH�UHVXOW�LV QXOO���

����

Page 390: Mupad Tutorial

*/ ;kZg\abg`3 LI�WKHQ�HOVH Zg] FDVH

7KLV�LV�HTXLYDOHQW�WR�WKH�IROORZLQJ�QHVWHG LI VWDWHPHQW�

LI FRQGLWLRQ� WKHQVWDWHPHQWV�

HOVH LI FRQGLWLRQ� WKHQVWDWHPHQWV�

HOVH LI ���HOVH

VWDWHPHQWVHQGBLI

HQGBLIHQGBLI�

$ W\SLFDO�DSSOLFDWLRQ�LV�IRU�W\SH�FKHFNLQJ�ZLWKLQ�SURFHGXUHV��&KDSWHU ���� 7KHIROORZLQJ�YHUVLRQ�RI $EV FRPSXWHV�WKH�DEVROXWH�YDOXH�LI�WKH�LQSXW�LV�DQ�LQWHJHU� DUDWLRQDO�QXPEHU� D�UHDO�IORDWLQJ�SRLQW�QXPEHU� RU�D�FRPSOH[�QXPEHUD� 2WKHUZLVH�LW�UDLVHV�DQ�HUURU�

$EV � SURF�\� EHJLQLI �GRPW\SH�\� '20B,17� RU �GRPW\SH�\� '20B5$7�

RU �GRPW\SH�\� '20B)/2$7� WKHQLI \ ! � WKHQ \ HOVH �\ HQGBLI�

HOLI �GRPW\SH�\� '20B&203/(;� WKHQVTUW�5H�\�A� � ,P�\�A���

HOVH �,QYDOLG DUJXPHQW W\SH� HQGBLI�HQGBSURF�GHOHWH [� $EV����� $EV������ $EV���� ,�� $EV�[�

3, 5.0,√

5, ´,QYDOLG�DUJXPHQW�W\SH´

����

Page 391: Mupad Tutorial

;kZg\abg`3 LI�WKHQ�HOVH Zg] FDVH

,Q�RXU�H[DPSOH� ZH�GLVWLQJXLVK�VHYHUDO�FDVHV�DFFRUGLQJ�WR�WKH�HYDOXDWLRQ�RI�DVLQJOH�H[SUHVVLRQ� :H�FDQ�DOVR�LPSOHPHQW�WKLV�E\�XVLQJ�D FDVH VWDWHPHQW� ZKLFK�LVRIWHQ�HDVLHU�WR�UHDG�

FDVH GRPW\SH�\�RI '20B,17 GRRI '20B5$7 GRRI '20B)/2$7 GR

LI \ ! � WKHQ \ HOVH �\ HQGBLI�EUHDN�

RI '20B&203/(; GRVTUW�5H�\�A� � ,P�\�A���EUHDN�

RWKHUZLVH�,QYDOLG DUJXPHQW W\SH��

HQGBFDVH�

7KH�NH\ZRUGV FDVH DQG HQGBFDVH LQGLFDWH�WKH�EHJLQQLQJ�DQG�WKH�HQG�RI�WKHVWDWHPHQW� UHVSHFWLYHO\� 0X3$' HYDOXDWHV�WKH�H[SUHVVLRQ�DIWHU FDVH� ,I�WKH�UHVXOWPDWFKHV�RQH�RI�WKH�H[SUHVVLRQV�EHWZHHQ RI DQG GR� WKH�V\VWHP�H[HFXWHV�DOOFRPPDQGV�IURP�WKH�ILUVW�PDWFKLQJ RI RQ�XQWLO�LW�HQFRXQWHUV�HLWKHU�D EUHDN RU�WKHNH\ZRUG HQGBFDVH�

:DUQLQJ� 1RWH�WKDW� LI�QR EUHDN VWDWHPHQW�XVHG�LQ�D�EUDQFK� WKH�IROORZLQJEUDQFKHV�DUH�HQWHUHG�DQG�H[HFXWHG� WRR� 7KLV�LV�LQ�WKH�VDPH�VW\OH�DV�WKH VZLWFKVWDWHPHQW�LQ�WKH�& SURJUDPPLQJ�ODQJXDJH� ,W�DOORZV�VHYHUDO�EUDQFKHV�WR�VKDUH�WKHVDPH�FRGH�

,I�QRQH�RI�WKH RI EUDQFKHV�DSSOLHV�DQG�WKHUH�LV�DQ RWKHUZLVH EUDQFK� WKH�FRGHEHWZHHQ RWKHUZLVH DQG HQGBFDVH LV�H[HFXWHG� 7KH�UHWXUQ�YDOXH�RI�D FDVHVWDWHPHQW�LV�WKH�YDOXH�RI�WKH�ODVW�H[HFXWHG�FRPPDQG� :H�UHIHU�WR�WKHFRUUHVSRQGLQJ�KHOS�SDJH "FDVH IRU�D�PRUH�GHWDLOHG�GHVFULSWLRQ�

����

Page 392: Mupad Tutorial

*/ ;kZg\abg`3 LI�WKHQ�HOVH Zg] FDVH

$V�IRU�ORRSV�DQG LI VWDWHPHQWV� WKHUH�LV�D�IXQFWLRQDO�HTXLYDOHQW�IRU�D FDVHVWDWHPHQW� WKH�V\VWHP�IXQFWLRQ BFDVH� ,QWHUQDOO\� 0X3$' FRQYHUWV�WKH�DERYHFDVH VWDWHPHQW�WR�WKH�IROORZLQJ�HTXLYDOHQW�IRUP�

BFDVH�GRPW\SH�\��'20B,17� 1,/�'20B5$7� 1,/�'20B)/2$7�

�LI \ ! � WKHQ \ HOVH �\ HQGBLI� EUHDN��'20B&203/(;� �VTUW�5H�\�A� � ,P�\�A��� EUHDN���,QYDOLG DUJXPHQW W\SH���

([HUFLVH������ ,Q LI VWDWHPHQWV�RU�WHUPLQDWLRQ�FRQGLWLRQV�RI ZKLOH DQG UHSHDWORRSV� WKH�V\VWHP�HYDOXDWHV�FRPSRVLWH�FRQGLWLRQV�ZLWK�%RROHDQ�RSHUDWRUV�RQHDIWHU�WKH�RWKHU� 7KH�HYDOXDWLRQ�URXWLQH�VWRSV�SUHPDWXUHO\�LI�LW�FDQ�GHFLGH�ZKHWKHUWKH�ILQDO�UHVXOW�LV 758( RU )$/6( �³OD]\�HYDOXDWLRQ´�� $UH�WKHUH�SUREOHPV�ZLWK�WKHIROORZLQJ�VWDWHPHQWV" :KDW�KDSSHQV�ZKHQ�WKH�FRQGLWLRQV�DUH�HYDOXDWHG"

$ � [��[ � �� ! �� [ � ���LI [ �! � DQG $ WKHQ ULJKW HOVH ZURQJ HQGBLI���LI [ � RU $ WKHQ ULJKW HOVH ZURQJ HQGBLI�

����

Page 393: Mupad Tutorial

)/

0X3$'� 3URFHGXUHV

0X3$'� SURYLGHV�WKH�HVVHQWLDO�FRQVWUXFWV�RI�D�SURJUDPPLQJ�ODQJXDJH� 7KH�XVHUFDQ�LPSOHPHQW�FRPSOH[�DOJRULWKPV�FRPIRUWDEO\�LQ�0X3$'��,QGHHG� PRVW�RI0X3$'¶V�PDWKHPDWLFDO�LQWHOOLJHQFH�LV�QRW�LPSOHPHQWHG�LQ�& RU�&���ZLWKLQ�WKHNHUQHO� EXW�LQ�WKH�0X3$' SURJUDPPLQJ�ODQJXDJH�DW�WKH�OLEUDU\�OHYHO� 7KHSURJUDPPLQJ�IHDWXUHV�DUH�PRUH�H[WHQVLYH�WKDQ�LQ�RWKHU�ODQJXDJHV�VXFK�DV�&�3DVFDO� RU�)RUWUDQ� VLQFH�WKH�0X3$' ODQJXDJH�RIIHUV�PRUH�JHQHUDO�DQG�PRUHIOH[LEOH�FRQVWUXFWV�

:H�KDYH�DOUHDG\�SUHVHQWHG�EDVLF�VWUXFWXUHV�VXFK�DV�ORRSV��&KDSWHU ���� EUDQFKLQJLQVWUXFWLRQV��&KDSWHU ���� DQG�³VLPSOH´�IXQFWLRQV��SDJH ������

,Q�WKLV�FKDSWHU� ZH�UHJDUG�³SURJUDPPLQJ´�DV�ZULWLQJ�FRPSOH[�0X3$' SURFHGXUHV�,Q�SULQFLSOH�WKH�XVHU�UHFRJQL]HV�QR�GLIIHUHQFHV�EHWZHHQ�³VLPSOH�IXQFWLRQV´JHQHUDWHG�YLD �! �SDJH ������DQG�³PRUH�FRPSOH[�SURFHGXUHV´�DV�SUHVHQWHG�LQ�WKLVFKDSWHU� 3URFHGXUHV� OLNH�IXQFWLRQV� UHWXUQ�YDOXHV� 2QO\�WKH�ZD\�RI�JHQHUDWLQJVXFK�SURFHGXUH�REMHFWV�YLD SURF« HQGBSURF LV�D�OLWWOH�PRUH�FRPSOLFDWHG�3URFHGXUHV�SURYLGH�DGGLWLRQDO�IXQFWLRQDOLW\� WKHUH�LV�D�GLVWLQFWLRQ�EHWZHHQ�ORFDODQG�JOREDO�YDULDEOHV� \RX�FDQ�XVH�DUELWUDULO\�PDQ\�FRPPDQGV�LQ�D�FOHDU�DQGFRQYHQLHQW�ZD\�HWF�

$V�VRRQ�DV�D�SURFHGXUH�LV�LPSOHPHQWHG�DQG�DVVLJQHG�WR�DQ�LGHQWLILHU� \RX�PD\�FDOOLW�LQ�WKH�IRUP SURFHGXUH1DPH�DUJXPHQWV� OLNH�DQ\�RWKHU�0X3$' IXQFWLRQ� $IWHUH[HFXWLQJ�WKH�LPSOHPHQWHG�DOJRULWKP� LW�UHWXUQV�DQ�RXWSXW�YDOXH�

<RX�FDQ�GHILQH�DQG�XVH�0X3$' SURFHGXUHV�ZLWKLQ�DQ�LQWHUDFWLYH�VHVVLRQ� OLNH�DQ\RWKHU�0X3$' REMHFW� 7\SLFDOO\� KRZHYHU� \RX�ZDQW�WR�XVH�WKHVH�SURFHGXUHV�DJDLQLQ�ODWHU�VHVVLRQV� LQ�SDUWLFXODU�ZKHQ�WKH\�LPSOHPHQW�PRUH�FRPSOH[�DOJRULWKPV�7KHQ�LW�LV�XVHIXO�WR�ZULWH�WKH�SURFHGXUH�GHILQLWLRQ�LQWR�D�WH[W�ILOH�XVLQJ�\RXUIDYRULWH�WH[W�HGLWRU��VXFK�DV� H�J�� WKH�0X3$' VRXUFH�FRGH�HGLWRU�� DQG�UHDG�LW�LQWR

Page 394: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

D�0X3$' VHVVLRQ�YLD UHDG �SDJH ������RU�ZLWK�WKH�HQWU\�µ5HDG�FRPPDQGV «¶�IURPWKH�µ1RWHERRN¶�PHQX� $SDUW�IURP�WKH�HYDOXDWLRQ�OHYHO� WKH�0X3$' NHUQHOSURFHVVHV�WKH�FRPPDQGV�LQ�WKH�ILOH�H[DFWO\�LQ�WKH�VDPH�ZD\�DV�LI�WKH\�ZHUH�HQWHUHGLQWHUDFWLYHO\� 0X3$' LQFOXGHV�DQ�HGLWRU�ZLWK�V\QWD[�KLJKOLJKWLQJ�IRU�0X3$'SURJUDPV�

����

Page 395: Mupad Tutorial

=^Ûgbg`�Ikh\^]nk^l

<]Úfaf_�Hjg[]\mj]k

7KH�IROORZLQJ�IXQFWLRQ� ZKLFK�FRPSDUHV�WZR�QXPEHUV�DQG�UHWXUQV�WKHLUPD[LPXP� LV�DQ�H[DPSOH�RI�D�SURFHGXUH�GHILQLWLRQ�YLD SURF« HQGBSURF�

PD[LPXP � SURF�D� E� � FRPPHQW� PD[LPXP RI D DQG E �EHJLQ

LI D�E WKHQ UHWXUQ�E� HOVH UHWXUQ�D� HQGBLI�HQGBSURF�

7KH�WH[W�HQFORVHG�EHWZHHQ � DQG � LV�D�FRPPHQW� DQG�FRPSOHWHO\�LJQRUHG�E\�WKHV\VWHP� 7KLV�LV�D�XVHIXO�WRRO�IRU�GRFXPHQWLQJ�WKH�VRXUFH�FRGH�ZKHQ�\RX�ZULWH�WKHSURFHGXUH�GHILQLWLRQ�LQ�D�WH[W�ILOH�

7KH�DERYH�VDPSOH�SURFHGXUH�FRQWDLQV�DQ LI VWDWHPHQW�DV�WKH�RQO\�FRPPDQG�0RUH�UHDOLVWLF�SURFHGXUHV�FRQWDLQ�PDQ\�FRPPDQGV��VHSDUDWHG�E\�FRORQV�RUVHPLFRORQV�� 7KH�FRPPDQG UHWXUQ WHUPLQDWHV�D�SURFHGXUH�DQG�SDVVHV�LWVDUJXPHQW�DV�RXWSXW�YDOXH�WR�WKH�V\VWHP�

$ 0X3$'� REMHFW�JHQHUDWHG�YLD SURF« HQGBSURF LV�RI�GRPDLQ�W\SH '20B352&�

GRPW\SH�PD[LPXP�'20B352&

<RX�FDQ�GHFRPSRVH�DQG�PDQLSXODWH�D�SURFHGXUH�OLNH�DQ\�RWKHU�0X3$' REMHFW� ,QSDUWLFXODU� \RX�PD\�DVVLJQ�LW�WR�DQ�LGHQWLILHU� DV�DERYH� 7KH�V\QWD[�RI�WKH�IXQFWLRQFDOO�LV�WKH�VDPH�DV�IRU�RWKHU�0X3$' IXQFWLRQV�

PD[LPXP����� ����37

7KH�VWDWHPHQWV�EHWZHHQ EHJLQ DQG HQGBSURFPD\�EH�DUELWUDU\�0X3$'FRPPDQGV� ,Q�SDUWLFXODU� \RX�PD\�FDOO�V\VWHP�IXQFWLRQV�RU�RWKHU�SURFHGXUHV�IURPZLWKLQ�D�SURFHGXUH� $ SURFHGXUH�PD\�HYHQ�FDOO�LWVHOI� ZKLFK�LV�KHOSIXO�IRULPSOHPHQWLQJ�UHFXUVLYH�DOJRULWKPV�

�$OWHUQDWLYHO\� \RX�FDQ�VWDUW�D�FRPPHQW�E\ ��� $ FRPPHQW�VWDUWHG�ZLWK �� DXWRPDWLFDOO\�HQGV�DW�WKHHQG�RI�WKH�OLQH�

����

Page 396: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

7KH�IDYRULWH�H[DPSOH�IRU�D�UHFXUVLYH�DOJRULWKP�LV�WKH�FRPSXWDWLRQ�RI�WKH�IDFWRULDOn! = 1 · 2 · · · · · n RI�D�QRQQHJDWLYH�LQWHJHU� ZKLFK�PD\�EH�GHILQHG�E\�WKH�UXOHn! = n · (n− 1)! WRJHWKHU�ZLWK�WKH�LQLWLDO�FRQGLWLRQ 0! = 1� 7KH�UHDOL]DWLRQ�DV�DUHFXUVLYH�SURFHGXUH�PLJKW�ORRN�DV�IROORZV�

IDFWRULDO � SURF�Q� EHJLQLI Q � WKHQ

UHWXUQ���HOVH UHWXUQ�Q IDFWRULDO�Q � ���HQGBLI

HQGBSURF�IDFWRULDO����

3628800

7KH�HQYLURQPHQW�YDULDEOH 0$;'(37+ GHWHUPLQHV�WKH�PD[LPDO�QXPEHU�RI�QHVWHGSURFHGXUH�FDOOV� ,WV�GHIDXOW�YDOXH�LV 500� :LWK�WKLV�YDOXH� WKH�DERYH�IDFWRULDOIXQFWLRQ�ZRUNV�RQO\�IRU n ≤ 500� )RU�ODUJHU�YDOXHV� DIWHU 0$;'(37+ VWHSV�0X3$'DVVXPHV�WKDW�WKHUH�LV�DQ�LQILQLWH�UHFXUVLRQ�DQG�DERUWV�ZLWK�DQ�HUURU�PHVVDJH� ,W�LVDGYLVDEOH�WR�DYRLG�GHHS�UHFXUVLRQV� WR�VDYH�RQ�PHPRU\� LQFUHDVH�HIILFLHQF\�DQG�WRDYRLG�WKHVH�SUREOHPV�

����

Page 397: Mupad Tutorial

Ma^�K^mnkg�OZen^�h_�Z�Ikh\^]nk^

L`]�J]lmjf�NYdm]�g^�Y�Hjg[]\mj]

:KHQ�\RX�FDOO�D�SURFHGXUH� WKH�V\VWHP�H[HFXWHV�LWV�ERG\� L�H�� WKH�VHTXHQFH�RIVWDWHPHQWV�EHWZHHQ EHJLQ DQG HQGBSURF� (YHU\�SURFHGXUH�UHWXUQV�VRPH�YDOXH�HLWKHU�H[SOLFLWO\�YLD UHWXUQ RU�RWKHUZLVH WKH�YDOXH�RI�WKH�ODVW�FRPPDQG�H[HFXWHGZLWKLQ�WKH�SURFHGXUH�� 7KXV� \RX�FDQ�LPSOHPHQW�WKH�DERYH�IDFWRULDO�IXQFWLRQZLWKRXW�XVLQJ UHWXUQ�

IDFWRULDO � SURF�Q� EHJLQLI Q � WKHQ � HOVH Q IDFWRULDO�Q � �� HQGBLI�

HQGBSURF�

7KH LI VWDWHPHQW�UHWXUQV�HLWKHU 1 RU n (n− 1)! � 6LQFH�WKH�HQG�RI�WKH�SURFHGXUH�LVUHDFKHG�GLUHFWO\�DIWHU�WKH LI VWDWHPHQW� WKLV�LV�WKH�UHWXUQ�YDOXH�RI�WKH�FDOOIDFWRULDO�Q��

$V�VRRQ�DV�WKH�V\VWHP�HQFRXQWHUV�D UHWXUQ VWDWHPHQW� LW�WHUPLQDWHV�WKH�SURFHGXUH�

IDFWRULDO � SURF�Q� EHJLQLI Q � WKHQ UHWXUQ��� HQGBLI�Q IDFWRULDO�Q � ���

HQGBSURF�

)RU n = 0� 0X3$'� GRHV�QRW�H[HFXWH�WKH�ODVW�VWDWHPHQW��WKH�UHFXUVLYH�FDOO�RIQ IDFWRULDO�Q�����DIWHU�UHWXUQLQJ 1� )RU n ̸= 0� WKH�PRVW�UHFHQWO\�FRPSXWHGYDOXH�LV Q IDFWRULDO�Q���� ZKLFK�LV�WKHQ�WKH�UHWXUQ�YDOXH�RI�WKH�FDOOIDFWRULDO�Q��

�,I�WKHUH�LV�QR�FRPPDQG�WR�H[HFXWH� WKH�UHWXUQ�YDOXH�LV 1,/�

����

Page 398: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

$ SURFHGXUH�PD\�UHWXUQ�DQ�DUELWUDU\�0X3$' REMHFW� VXFK�DV�DQ�H[SUHVVLRQ� DVHTXHQFH� D�VHW� D�OLVW� RU�HYHQ�D�SURFHGXUH� +RZHYHU� LI�WKH�UHWXUQHG�SURFHGXUHXVHV�ORFDO�YDULDEOHV�RI�WKH�RXWHU�SURFHGXUH� \RX�KDYH�WR�GHFODUH�WKH�ODWWHU�ZLWK�WKHRSWLRQ HVFDSH� 2WKHUZLVH� WKLV�OHDGV�WR�D�0X3$' ZDUQLQJ�PHVVDJH�RU�WRXQZDQWHG�HIIHFWV� 7KH�IROORZLQJ�SURFHGXUH�UHWXUQV�D�IXQFWLRQ�WKDW�XVHV�WKHSDUDPHWHU SRZHU RI�WKH�RXWHU�SURFHGXUH�

JHQHUDWH3RZHU)XQFWLRQ � SURF�SRZHU�RSWLRQ HVFDSH�EHJLQ

[ �! �[ASRZHU�HQGBSURF�

I � JHQHUDWH3RZHU)XQFWLRQ����J � JHQHUDWH3RZHU)XQFWLRQ����I�D�� J�E�a2, b5

����

Page 399: Mupad Tutorial

K^mnkgbg`�Lrf[heb\�?ng\mbhg�<Zeel

J]lmjfaf_�KqeZgda[�>mf[lagf�;Yddk

0DQ\�V\VWHP�IXQFWLRQV�UHWXUQ�³WKHPVHOYHV´�DV�V\PEROLF�IXQFWLRQ�FDOOV�LI�WKH\FDQQRW�ILQG�D�VLPSOH�UHSUHVHQWDWLRQ�RI�WKH�UHTXHVWHG�UHVXOW�

VLQ�[�� PD[�D� E�� LQW�H[S�[A��� [�

VLQ(x) , PD[(a, b) ,∫Hx

3Gx

<RX�DFKLHYH�WKH�VDPH�EHKDYLRU�LQ�\RXU�RZQ�SURFHGXUHV�ZKHQ�\RX�HQFDSVXODWH�WKHSURFHGXUH�QDPH�LQ�D KROG XSRQ�UHWXUQ� 7KH KROG �SDJH �����SUHYHQWV�WKH�IXQFWLRQIURP�FDOOLQJ�LWVHOI�UHFXUVLYHO\�DQG�HQGLQJ�XS�LQ�DQ�LQILQLWH�UHFXUVLRQ� 7KHIROORZLQJ�IXQFWLRQ�FRPSXWHV�WKH�DEVROXWH�YDOXH�IRU�QXPHULFDO�LQSXWV��LQWHJHUV�UDWLRQDO�QXPEHUV� UHDO�IORDWLQJ�SRLQW�QXPEHUV� DQG�FRPSOH[�QXPEHUV�� )RU�DOORWKHU�NLQGV�RI�LQSXWV� LW�UHWXUQV�LWVHOI�V\PEROLFDOO\�

$EV � SURF�[� EHJLQLI WHVWW\SH�[� 7\SH��1XPHULF� WKHQ

LI GRPW\SH�[� '20B&203/(; WKHQUHWXUQ�VTUW�5H�[�A� � ,P�[�A���

HOVH LI [ ! � WKHQUHWXUQ�[�

HOVH UHWXUQ��[�HQGBLI

HQGBLIHQGBLI�KROG�$EV��[�

HQGBSURF�$EV����� $EV������� $EV�������� $EV�� � ,����$EV�[ � ��

1,23, 1.234,

√373, $EV(x+ 1)

����

Page 400: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

$ PRUH�HOHJDQW�ZD\�LV�WR�XVH�WKH�0X3$'� REMHFW SURFQDPH� ZKLFK�UHWXUQV�WKHQDPH�RI�WKH�FDOOLQJ�SURFHGXUH�

$EV � SURF�[� EHJLQLI WHVWW\SH�[� 7\SH��1XPHULF� WKHQ

LI GRPW\SH�[� '20B&203/(; WKHQUHWXUQ�VTUW�5H�[�A� � ,P�[�A���

HOVH LI [ ! � WKHQUHWXUQ�[�

HOVH UHWXUQ��[�HQGBLI

HQGBLIHQGBLI�SURFQDPH�DUJV���

HQGBSURF�$EV����� $EV������� $EV�������� $EV�� � ,����$EV�[ � ��

1,23, 1.234,

√373, $EV(x+ 1)

+HUH� ZH�XVH�WKH�H[SUHVVLRQ DUJV��� ZKLFK�UHWXUQV�WKH�VHTXHQFH�RI�DUJXPHQWVSDVVHG�WR�WKH�SURFHGXUH��FI� SDJH �������

����

Page 401: Mupad Tutorial

Eh\Ze�Zg]�@eh[Ze�OZkbZ[e^l

Dg[Yd�Yf\�?dgZYd�NYjaYZd]k

<RX�FDQ�XVH�DUELWUDU\�LGHQWLILHUV�LQ�SURFHGXUHV� 7KH\�DUH�DOVR�FDOOHG JOREDOYDULDEOHV�

D � E� I � SURF�� EHJLQ D � DA� � � HQGBSURF�I��� I��� I��b2 + 1(b2 + 1

)2 + 1

((b2 + 1

)2 + 1)2

+ 1

7KH�SURFHGXUH IPRGLILHV�WKH�YDOXH�RI D� ZKLFK�KDV�EHHQ�VHW�RXWVLGH�WKHSURFHGXUH� :KHQ�WKH�SURFHGXUH�WHUPLQDWHV� D KDV�D�QHZ�YDOXH� ZKLFK�LV�DJDLQFKDQJHG�E\�IXUWKHU�FDOOV�WR I�

7KH�NH\ZRUG ORFDO GHFODUHV�LGHQWLILHUV�DV ORFDO�YDULDEOHV WKDW�DUH�RQO\�YDOLGZLWKLQ�WKH�SURFHGXUH�

D � E� I � SURF�� ORFDO D� EHJLQ D � � HQGBSURF�I��� Db

'HVSLWH�WKH�HTXDO�QDPHV� WKH�DVVLJQPHQW D� � RI�WKH�ORFDO�YDULDEOH�GRHV�QRW�DIIHFWWKH�YDOXH�RI�WKH�JOREDO�LGHQWLILHU D WKDW�KDV�EHHQ�GHILQHG�RXWVLGH�WKH�SURFHGXUH�<RX�FDQ�GHFODUH�DQ�DUELWUDU\�QXPEHU�RI�ORFDO�YDULDEOHV�E\�VSHFLI\LQJ�D�VHTXHQFH�RILGHQWLILHUV�DIWHU ORFDO�I � SURF�[� \� ]�

ORFDO $� %� &�EHJLQ

$� �� %� �� &� �� $ % & �[ � \ � ]�HQGBSURF�

I�$� %� &�6A+ 6B + 6C

����

Page 402: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

/RFDO�YDULDEOHV�RI�D�SURFHGXUH�KDYH�D�VSHFLDO�GRPDLQ�W\SH '20B9$5��7KH\�GR�QRWJHW�PL[HG�XS�ZLWK�JOREDO�YDULDEOHV�ZKLFK�DUH�LGHQWLILHUV�RI�W\SH '20B,'(17��1RWHWKDW�DOVR�ORFDO�YDULDEOHV�GHFODUHG�E\�WKH�VDPH�QDPH�LQ�WKH�VRXUFH�FRGH�RI�GLIIHUHQWIXQFWLRQV�KDYH�QR�UHIHUHQFH�WR�RQH�DQRWKHU� $OVR� ZKHQ�FDOOLQJ�D�IXQFWLRQ�VHYHUDOWLPHV� D�ORFDO�YDULDEOH�UHIHUV�WR�D�GLIIHUHQW�YDOXH�LQ�HDFK�FDOO�

I � SURF�[� ORFDO D� E�EHJLQ

D � [�LI [ ! � WKHQ E � I�[ � ���HOVH E � ��HQGBLI�SULQW�D� [��E � D�

HQG�I���

�� �

�� �

�� �

4

:H�UHFRPPHQG�WR�WDNH�LQWR�DFFRXQW�WKH�IROORZLQJ�UXOH�RI�WKXPE�

8VLQJ�JOREDO�YDULDEOHV�LV�JHQHUDOO\�FRQVLGHUHG�EDG�SURJUDPPLQJ�VW\OH�8VH�ORFDO�YDULDEOHV�ZKHQHYHU�SRVVLEOH�

7KH�UHDVRQ�IRU�WKLV�SULQFLSOH�LV�WKDW�SURFHGXUHV�LPSOHPHQW�PDWKHPDWLFDOIXQFWLRQV� ZKLFK�VKRXOG�UHWXUQ�D�XQLTXH�RXWSXW�YDOXH�IRU�D�JLYHQ�VHW�RI�LQSXWYDOXHV� ,I�\RX�XVH�JOREDO�YDULDEOHV�WKHQ� GHSHQGLQJ�RQ�WKHLU�YDOXHV� WKH�VDPHSURFHGXUH�FDOO�PD\�OHDG�WR�GLIIHUHQW�UHVXOWV�

D � �� I � SURF�E� EHJLQ D � D � �� D � E HQGBSURF�I���� I���� I���

3, 4, 5

0RUHRYHU� D�SURFHGXUH�FDOO�FDQ�FKDQJH�WKH�FDOOLQJ�HQYLURQPHQW�LQ�D�VXEWOH�ZD\�E\UHGHILQLQJ�JOREDO�YDULDEOHV��³VLGH�HIIHFW´�� ,Q�PRUH�FRPSOH[�SURJUDPV� WKLV�PD\OHDG�WR�XQZDQWHG�HIIHFWV�WKDW�DUH�GLIILFXOW�WR�GHEXJ�

�����

Page 403: Mupad Tutorial

Eh\Ze�Zg]�@eh[Ze�OZkbZ[e^l

$Q�LPSRUWDQW�GLIIHUHQFH�EHWZHHQ�JOREDO�DQG�ORFDO�YDULDEOHV�LV�WKDW�DQ�XQLQLWLDOL]HGJOREDO�YDULDEOH�LV�UHJDUGHG�DV�D V\PERO� ZKRVH�YDOXH�LV�LWV�RZQ�QDPH� ZKLOH�WKHYDOXH�RI�DQ�XQLQLWLDOL]HG�ORFDO�YDULDEOH�LV 1,/��8VLQJ�D�ORFDO�YDULDEOH�ZLWKRXWLQLWLDOL]DWLRQ�OHDGV�WR�D�ZDUQLQJ�PHVVDJH�LQ�0X3$'� DQG�VKRXOG�EH�DYRLGHG�

,$P*OREDO � �,$P*OREDO+ 1

I � SURF��ORFDO ,$P/RFDO�EHJLQ

,$P/RFDO � �HQGBSURF�

I��:DUQLQJ� 8QLQLWLDOL]HG YDULDEOH ,$P/RFDO XVHG�GXULQJ HYDOXDWLRQ RI I(UURU� ,OOHJDO RSHUDQG >BSOXV@�GXULQJ HYDOXDWLRQ RI I

7KH�UHDVRQ�IRU�WKH�HUURU�LV�WKDW�0X3$' FDQQRW�DGG�WKH�YDOXH 1,/ RI�WKH�ORFDOYDULDEOH�WR�WKH�QXPEHU 1�

:H�QRZ�SUHVHQW�D�UHDOLVWLF�H[DPSOH�RI�D�PHDQLQJIXO�SURFHGXUH� ,I�ZH�XVH�DUUD\V�RIGRPDLQ�W\SH '20B$55$< WR�UHSUHVHQW�PDWULFHV� ZH�DUH�IDFHG�ZLWK�WKH�SUREOHP�WKDWWKHUH�LV�QR�GLUHFW�ZD\�WR�SHUIRUP�PDWUL[�PXOWLSOLFDWLRQ�ZLWK�VXFK�DUUD\V�� 7KHIROORZLQJ�SURFHGXUH�VROYHV�WKLV�SUREOHP� \RX�FDQ�FRPSXWH�WKH�PDWUL[�SURGXFWC = A ·B ZLWK�WKH�FRPPDQG &� 0DWUL[3URGXFW�$�%���:H�ZDQW�WKH�SURFHGXUH�WRZRUN�IRU�DUELWUDU\�GLPHQVLRQV�RI�WKH�PDWULFHV A DQG B� SURYLGHG�WKH�UHVXOW�LVGHILQHG�PDWKHPDWLFDOO\� ,I A LV�DQm× nPDWUL[� WKHQ B PD\�EH�DQ n× r PDWUL[�ZKHUHm,n, r DUH�DUELWUDU\�SRVLWLYH�LQWHJHUV� 7KH�UHVXOW�LV�WKH m× r PDWUL[ C ZLWKWKH�HQWULHV

Cij =n∑

k=1

Aik Bkj , i = 1, . . . ,m, j = 1, . . . , r.

7KH�PXOWLSOLFDWLRQ�SURFHGXUH�EHORZ�DXWRPDWLFDOO\�H[WUDFWV�WKH�GLPHQVLRQSDUDPHWHUVm,n, r IURP�WKH�DUJXPHQWV� QDPHO\�IURP�WKH 0�WK�RSHUDQGV�RI�WKHLQSXW�DUUD\V��SDJH ������ ,I B LV�D q× rPDWUL[�ZLWK q ̸= n� WKH�PXOWLSOLFDWLRQ�LV�QRWGHILQHG�PDWKHPDWLFDOO\� ,Q�WKLV�FDVH� WKH�SURFHGXUH�WHUPLQDWHV�ZLWK�DQ�HUURU

�,I�\RX�XVH�WKH�GDWD�W\SH 'RP��0DWUL[�� LQVWHDG� \RX�FDQ�GLUHFWO\�XVH�WKH�VWDQGDUG�RSHUDWRUV �� �� � A� � IRU�DULWKPHWLF�ZLWK�PDWULFHV��FI� SDJH ���� II���

�����

Page 404: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

PHVVDJH� )RU�WKDW�SXUSRVH� ZH�HPSOR\�WKH�IXQFWLRQ HUURU� ZKLFK�DERUWV�WKHFDOOLQJ�SURFHGXUH�DQG�ZULWHV�WKH�VWULQJ�SDVVHG�DV�DUJXPHQW�RQ�WKH�VFUHHQ� :HVWRUH�WKH�UHVXOW�FRPSRQHQW�E\�FRPSRQHQW�LQ�WKH�ORFDO�YDULDEOH &��:H�LQLWLDOL]H�WKLVYDULDEOH�DV�DQ�DUUD\�RI�GLPHQVLRQm× r� VR�WKDW�WKH�UHVXOW�RI�RXU�SURFHGXUH�LV�RIWKH�GHVLUHG�GDWD�W\SH '20B$55$<��:H�PLJKW�LPSOHPHQW�WKH�VXP�RYHU k LQ�WKHFRPSXWDWLRQ�RI Cij DV�D�ORRS�RI�WKH�IRUP IRU�N�IURP���WR�Q�GR��� ,QVWHDG� ZHXVH�WKH�V\VWHP�IXQFWLRQ BSOXV ZKLFK�UHWXUQV�WKH�VXP�RI�LWV�DUJXPHQWV� :HJHQHUDOO\�UHFRPPHQG�WR�XVH�WKHVH�V\VWHP�IXQFWLRQV� LI�SRVVLEOH� VLQFH�WKH\�ZRUNTXLWH�HIILFLHQWO\� 7KH�UHWXUQ�YDOXH�RI 0DWUL[3URGXFW LV�WKH�ILQDO�H[SUHVVLRQ &�

0DWUL[3URGXFW � � PXOWLSOLFDWLRQ & $% RI DQ P [ Q �SURF�$� %� � PDWUL[ $ E\ DQ Q [ U 0DWUL[ % �ORFDO P� Q� U� L� M� N� &� � ZLWK DUUD\V $� % RI �EHJLQ � GRPDLQ W\SH '20B$55$< �

P � RS�$� >�� �� �@��Q � RS�$� >�� �� �@��LI Q �! RS�%� >�� �� �@� WKHQ

HUURU��LQFRPSDWLEOH PDWUL[ GLPHQVLRQV��HQGBLI�U � RS�%� >�� �� �@��& � DUUD\����P� ���U�� � LQLWLDOL]DWLRQ �

IRU L IURP � WR P GRIRU M IURP � WR U GR

&>L� M@ � BSOXV�$>L� N@ %>N� M@ � N ���Q�HQGBIRU

HQGBIRU�&

HQGBSURF�

$ JHQHUDO�UHPDUN�DERXW�0X3$' SURJUDPPLQJ�VW\OH� \RX�VKRXOG�DOZD\V�SHUIRUPDUJXPHQW�FKHFNLQJ�LQ�SURFHGXUHV�PHDQW�IRU�LQWHUDFWLYH�XVH� ,I�\RX�LPSOHPHQW�DSURFHGXUH� \RX�XVXDOO\�NQRZ�ZKLFK�W\SHV�RI�LQSXWV�DUH�YDOLG��VXFK�DV '20B$55$< LQWKH�DERYH�H[DPSOH�� ,I�VRPHERG\�SDVVHV�SDUDPHWHUV�RI�WKH�ZURQJ�W\SH�E\�PLVWDNH�WKLV�XVXDOO\�OHDGV�WR�V\VWHP�IXQFWLRQV�FDOOV�ZLWK�LQYDOLG�DUJXPHQWV� DQG�\RXUSURFHGXUH�DERUWV�ZLWK�DQ�HUURU�PHVVDJH�RULJLQDWLQJ�IURP�D�V\VWHP�IXQFWLRQ� ,Q�WKHDERYH�H[DPSOH� WKH�IXQFWLRQ RS UHWXUQV�WKH�YDOXH )$,/ ZKHQ�DFFHVVLQJ�WKH 0�WKRSHUDQG�RI $ RU % DQG�RQH�RI�WKHP�LV�QRW�RI�W\SH '20B$55$<��7KHQ�WKLV�YDOXH�LVDVVLJQHG�WR P� Q RU U� DQG�WKH�IROORZLQJ IRU ORRS�DERUWV�ZLWK�DQ�HUURU�PHVVDJH�VLQFH )$,/ LV�QRW�DOORZHG�DV�D�YDOXH�IRU�WKH�HQGSRLQW�RI�WKH�ORRS�

�����

Page 405: Mupad Tutorial

Eh\Ze�Zg]�@eh[Ze�OZkbZ[e^l

,Q�VXFK�D�VLWXDWLRQ� LW�LV�RIWHQ�GLIILFXOW�WR�ORFDWH�WKH�VRXUFH�RI�WKH�HUURU� +RZHYHU�DQ�HYHQ�ZRUVH�VFHQDULR�PLJKW�KDSSHQ� LI�WKH�SURFHGXUH�GRHV�QRW�DERUW� WKH�UHVXOWLV�OLNHO\�WR�EH�ZURQJ� 7KXV� W\SH�FKHFNLQJ�KHOSV�WR�DYRLG�HUURUV�

,Q�WKH�DERYH�H[DPSOH� ZH�PLJKW�DGG�D�W\SH�FKHFN�RI�WKH�IRUP

LI GRPW\SH�$� �! '20B$55$< RU GRPW\SH�%� �! '20B$55$<WKHQ HUURU��DUJXPHQWV PXVW EH RI W\SH '20B$55$<��

HQGBLI

WR�WKH�SURFHGXUH�ERG\� 6WDUWLQJ�RQ�SDJH ������ ZH�GLVFXVV�D�VLPSOHU�W\SH�FKHFNLQJFRQFHSW�

�����

Page 406: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

KmZhjg[]\mj]k

2IWHQ�WDVNV�RFFXU�IUHTXHQWO\�ZLWKLQ�D�SURFHGXUH�DQG�\RX�ZDQW�WR�LPSOHPHQW�WKHPDJDLQ�LQ�WKH�IRUP�RI�D�SURFHGXUH� 7KLV�VWUXFWXUHV�DQG�VLPSOLILHV�WKH�SURJUDP�FRGH�,Q�PDQ\�FDVHV� VXFK�D�SURFHGXUH�LV�XVHG�RQO\�IURP�ZLWKLQ�D�VLQJOH�SURFHGXUH�7KHQ�LW�LV�UHDVRQDEOH�WR�GHILQH�WKLV�SURFHGXUH�ORFDOO\�DV�D�VXESURFHGXUH�RQO\�LQ�WKHVFRSH�RI�WKH�FDOOLQJ�SURFHGXUH� ,Q�0X3$'� \RX�FDQ�XVH�ORFDO�YDULDEOHV�WRLPSOHPHQW�VXESURFHGXUHV� ,I�\RX�ZDQW�WR�PDNH

J � SURF�� EHJLQ ��� HQGBSURF�

D�ORFDO�SURFHGXUH�RI

I � SURF�� EHJLQ ��� HQGBSURF�

GHILQH I DV�IROORZV�I � SURF��ORFDO J�EHJLQ

J � SURF�� EHJLQ ��� HQGBSURF� � VXESURFHGXUH �

� PDLQ SDUW RI I� ZKLFK FDOOV J����� ����

HQGBSURF�

1RZ� J LV�D ORFDO SURFHGXUH�RI I DQG�\RX�FDQ�XVH�LW�RQO\�IURP�ZLWKLQ I�

:H�JLYH�DQ�H[DPSOH� <RX�FDQ�LPSOHPHQW�PDWUL[�PXOWLSOLFDWLRQ�E\�PHDQV�RIVXLWDEOH�FROXPQ×URZ�PXOWLSOLFDWLRQV�

(2 15 3

)·(

4 62 3

)=

⎜⎜⎜⎝

(2, 1) ·(

42

)(2, 1) ·

(63

)

(5, 3) ·(

42

)(5, 3) ·

(63

)

⎟⎟⎟⎠=(

10 1526 39

).

0RUH�JHQHUDOO\� LI�ZH�SDUWLWLRQ�WKH�LQSXW�PDWULFHV�E\�URZV ai DQG�FROXPQV bj �UHVSHFWLYHO\� WKHQ

⎜⎝a1���am

⎟⎠ · (b1, . . . , bn) =

⎜⎝a1 · b1 . . . a1 · bn���

� � ����

am · b1 . . . am · bn

⎟⎠ ,

�����

Page 407: Mupad Tutorial

Ln[ikh\^]nk^l

ZLWK�WKH�LQQHU�SURGXFWai · bj =

r

(ai)r (bj)r .

:H�QRZ�ZULWH�D�SURFHGXUH 0DWUL[0XOW WKDW�H[SHFWV�LQSXW�DUUD\V A DQG B RI�WKHIRUP DUUD\����P����N� DQG DUUD\����N����Q�� DQG�UHWXUQV�WKHm× nPDWUL[SURGXFW A ·B� $ FDOO�RI�WKH�VXESURFHGXUH 5RZ7LPHV&ROXPQ ZLWK�DUJXPHQWV i, jH[WUDFWV�WKH i�WK�URZ�DQG�WKH j�WK�FROXPQ�IURP�WKH�LQSXW�PDWULFHV A DQG B�UHVSHFWLYHO\� DQG�FRPSXWHV�WKH�LQQHU�SURGXFW�RI�WKH�URZ�DQG�WKH�FROXPQ� 7KHVXESURFHGXUH�XVHV�WKH�DUUD\V $� % DV�ZHOO�DV�WKH�ORFDOO\�GHFODUHG�GLPHQVLRQSDUDPHWHUV P� Q� DQG N DV�³JOREDO´�YDULDEOHV�

0DWUL[0XOW � SURF�$� %�ORFDO P� Q� N� .� � ORFDO YDULDEOHV �

5RZ7LPHV&ROXPQ� � ORFDO VXESURFHGXUH �EHJLQ

� VXESURFHGXUH �5RZ7LPHV&ROXPQ � SURF�L� M�ORFDO URZ� FROXPQ� U�EHJLQ

� LWK URZ RI $� �URZ � DUUD\����N� >$>L�U@ � U ���N@��� MWK FROXPQ RI %� �FROXPQ � DUUD\����N� >%>U�M@ � U ���N@��� URZ WLPHV FROXPQ �BSOXV�URZ>U@ FROXPQ>U@ � U ���N�

HQGBSURF�

� PDLQ SDUW RI WKH SURFHGXUH 0DWUL[0XOW� �P � RS�$� >�� �� �@�� � QXPEHU RI URZV RI $ �N � RS�$� >�� �� �@�� � QXPEHU RI FROXPQV RI $ �. � RS�%� >�� �� �@�� � QXPEHU RI URZV RI % �Q � RS�%� >�� �� �@�� � QXPEHU RI FROXPQV RI % �

LI N �! . WKHQHUURU��� RI FROXPQV RI $ �! � RI URZV RI %��

HQGBLI�

� PDWUL[ $ %� �DUUD\����P� ���Q�

>>5RZ7LPHV&ROXPQ�L� M� � M ���Q@ � L ���P@�HQGBSURF�

�����

Page 408: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

7KH�IROORZLQJ�H[DPSOH�UHWXUQV�WKH�GHVLUHG�UHVXOW�

$ � DUUD\������ ����� >>�� �@� >�� �@@��% � DUUD\������ ����� >>�� �@� >�� �@@��0DWUL[0XOW�$� %�(

10 1526 39

)

�����

Page 409: Mupad Tutorial

L\hi^�h_�OZkbZ[e^l

K[gh]�g^�NYjaYZd]k

7KH�0X3$'� SURJUDPPLQJ�ODQJXDJH�LPSOHPHQWV OH[LFDO�VFRSLQJ� 7KLVHVVHQWLDOO\�PHDQV�WKDW�WKH�VFRSH�RI�D�SURFHGXUH¶V�ORFDO�YDULDEOHV�DQG�SDUDPHWHUVFDQ�DOUHDG\�EH�GHWHUPLQHG�ZKHQ�WKH�SURFHGXUH�LV�GHILQHG� :H�VWDUW�ZLWK�D�VLPSOHH[DPSOH�WR�H[SODLQ�WKLV�FRQFHSW�

S � SURF�� EHJLQ [ HQGBSURF�[ � �� S��� [ � �� S��

3

4

T � SURF�� ORFDO [� EHJLQ [ � �� S��� HQGBSURF�T��

4

)LUVW� D�SURFHGXUH S ZLWKRXW�DUJXPHQWV�LV�GHILQHG� ,W�XVHV�D�YDULDEOH [� ZKLFK�LVQRW�GHFODUHG�DV�D�ORFDO�YDULDEOH�RI S� 7KXV� WKH�FDOO S�� UHWXUQV�WKH�YDOXH�RI�WKHJOREDO�YDULDEOH [� DV�VKRZQ�LQ�WKH�WZR�VXEVHTXHQW�FDOOV� ,Q�WKH�SURFHGXUH T�KRZHYHU� WKH�YDULDEOH [ LV�GHFODUHG�ORFDO� DQG�WKH�YDOXH 5 LV�DVVLJQHG�WR�LW� 7KHJOREDO�YDULDEOH [ LV�QRW�YLVLEOH�IURP�ZLWKLQ�WKH�SURFHGXUH T� RQO\�WKH�ORFDO�YDULDEOH[ FDQ�EH�DFFHVVHG� 1HYHUWKHOHVV� WKH�FDOO T�� UHWXUQV�WKH�YDOXH�RI�WKH JOREDOYDULDEOH [� DQG QRW WKH�FXUUHQW�YDOXH�RI�WKH�ORFDO�YDULDEOH [ ZLWKLQ S� :H�PLJKW� IRUH[DPSOH� GHILQH�D�VXESURFHGXUH�ZLWKLQ T WR�DFKLHYH�WKH�ODWWHU�EHKDYLRU�

[ � ��T � SURF��

ORFDO [� S�EHJLQ

[ � ��S � SURF�� EHJLQ [� HQGBSURF�[ � ��S��

HQGBSURF�T��� S��

6, 4

�����

Page 410: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

7KH�SUHYLRXVO\�GHILQHG�JOREDO�SURFHGXUH S LV�QRW�DFFHVVHG�IURP�ZLWKLQ T EHFDXVH�DORFDO�YDULDEOH S LV�GHFODUHG� 7KH�FDOO�RI�WKH�ORFDO�SURFHGXUH S IURP�ZLWKLQ T QRZLQGHHG�UHWXUQV�WKH�FXUUHQW�YDOXH�RI�WKH�ORFDO�YDULDEOH [� DV�VKRZQ�E\�WKH�FDOO T���7KH�ODVW�FRPPDQG S��� KRZHYHU� H[HFXWHV�WKH�JOREDO�SURFHGXUH S GHILQHG�EHIRUH�ZKLFK�VWLOO�UHWXUQV�WKH�FXUUHQW�YDOXH 4 RI�WKH�JOREDO�YDULDEOH [�

+HUH�LV�DQRWKHU�H[DPSOH�

S � SURF�[� EHJLQ � FRV�[� � �� HQGBSURF�T � SURF�\�

ORFDO FRV�EHJLQ

FRV � SURF�]� EHJLQ ] � �� HQGBSURF�S�\� FRV�\�

HQGBSURF�S�3,�� T�3,�−1, −π − 1

7KH�SURFHGXUH S XVHV�0X3$'¶V�JOREDOO\�GHILQHG�FRVLQH�IXQFWLRQ� :LWKLQ T� ZHKDYH�GHILQHG�D�ORFDO�VXESURFHGXUH FRV� &DOOLQJ T� WKH�LQWHUQDO�FDOO FRV�\� UHIHUV�WRWKH�ORFDO�IXQFWLRQ FRV� (YHQ�ZKHQ�FDOOHG�IURP�ZLWKLQ T� WKH�SURFHGXUH S VWLOO�XVHVWKH�0X3$' JOREDO�FRVLQH�IXQFWLRQ�

:KHQ�XVLQJ�WKH�RSWLRQ HVFDSH� D�ORFDO�YDULDEOH�FDQ�EH�DFFHVVHG�HYHQ�ZKHQ�LW�KDVHVFDSHG�WKH�VFRSH�RI�WKH�SURFHGXUH�WKDW�LW�ZDV�ORFDO�WR� ,Q�WKH�IROORZLQJ�H[DPSOH�WKH�³FRXQWHU´�UHWXUQHG�E\ I UHIHUV�WR�WKH�ORFDO�YDULDEOH [ RI I� 7KH�SURFHGXUHVFRXQWHU�� FRXQWHU� HWF� DUH�QRW�ORFDO�WR I DQ\PRUH�EXW�VWLOO�XVH�WKH�UHIHUHQFH�WR [WR�V\QFKURQL]H�VXEVHTXHQW�FDOOV�RI�HDFK�FRXQWHU� 1RWH�WKDW�ERWK�FRXQWHUV�UHIHU�WRGLIIHUHQW LQGHSHQGHQW�LQVWDQFHV�RI [� VR�WKDW�WKH\�FRXQW�LQGHSHQGHQWO\�

I � SURF�� ORFDO [�RSWLRQ HVFDSH�

EHJLQ[ � ���� 7KH IROORZLQJ SURFHGXUH�� LV WKH UHWXUQ YDOXH RI I�SURF�� EHJLQ [ � [ � � HQG�

HQG�FRXQWHU� � I��� FRXQWHU� � I���

�����

Page 411: Mupad Tutorial

L\hi^�h_�OZkbZ[e^l

FRXQWHU���� FRXQWHU���� FRXQWHU����FRXQWHU���� FRXQWHU����FRXQWHU���� FRXQWHU���� FRXQWHU����

2, 3, 4

2, 3

5, 4, 6

�����

Page 412: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

Lqh]�<][dYjYlagf

0X3$'� SURYLGHV�HDV\�WR�XVH�W\SH�FKHFNLQJ�IRU�SURFHGXUH�DUJXPHQWV� )RUH[DPSOH� \RX�FDQ�UHVWULFW�WKH�DUJXPHQWV�RI 0DWUL[3URGXFW� D�SURFHGXUH�IURP�WKH6HFWLRQ�VWDUWLQJ�RQ�SDJH ����� WR�WKH�GRPDLQ�W\SH '20B$55$< DV�IROORZV�

0DWUL[3URGXFW � SURF�$� '20B$55$<� %� '20B$55$<�ORFDO P� Q� U� L� M� N� &�

EHJLQ ���

,I�\RX�GHFODUH�WKH�W\SH�RI�WKH�SDUDPHWHUV�RI�D�SURFHGXUH�LQ�WKH�IRUP�XVHG�DERYH�DUJXPHQW�W\SH6SHFLILHU� D�FDOO�RI�WKH�SURFHGXUH�ZLWK�SDUDPHWHUV�RI�DQLQFRPSDWLEOH�W\SH�OHDGV�WR�DQ�HUURU�PHVVDJH� ,Q�WKH�H[DPSOH�DERYH� ZH�XVHG�WKHGRPDLQ�W\SH '20B$55$< DV�D�W\SH�VSHFLILHU�

:H�KDYH�GLVFXVVHG�0X3$'¶V�W\SH�FRQFHSW�LQ�&KDSWHU ��� 7KH 7\SH OLEUDU\�RIIHUV7\SH��1RQ1HJ,QW WR�UHSUHVHQW�WKH�VHW�RI�QRQQHJDWLYH�LQWHJHUV� ,I�ZH�XVH�LW�LQ�WKHIROORZLQJ�YDULDQW�RI�WKH�IDFWRULDO�IXQFWLRQ

IDFWRULDO � SURF�Q� 7\SH��1RQ1HJ,QW� EHJLQLI Q � WKHQ

UHWXUQ���HOVH Q IDFWRULDO�Q � ��HQGBLI

HQGBSURF�

WKHQ�RQO\�QRQQHJDWLYH�LQWHJHUV�DUH�SHUPLWWHG�IRU�WKH�DUJXPHQW Q�

IDFWRULDO���24

IDFWRULDO�����(UURU� :URQJ W\SH RI �� DUJXPHQW �W\SH 7\SH��1RQ1HJ?,QW H[SHFWHG�

JRW DUJXPHQW �����GXULQJ HYDOXDWLRQ RI IDFWRULDO

IDFWRULDO����(UURU� :URQJ W\SH RI DUJXPHQW Q �W\SH 7\SH��1RQ1HJ?,QW H[SHFWHG�

JRW DUJXPHQW ����GXULQJ HYDOXDWLRQ RI IDFWRULDO

�����

Page 413: Mupad Tutorial

Ikh\^]nk^l�pbma�Z�OZkbZ[e^�Gnf[^k�h_�:k`nf^gml

Hjg[]\mj]k�oal`�Y�NYjaYZd]�FmeZ]j�g^�9j_me]flk

7KH�V\VWHP�IXQFWLRQ PD[ FRPSXWHV�WKH�PD[LPXP�RI�LWV�DUJXPHQWV� <RX�PD\�FDOO�LWZLWK�DUELWUDULO\�PDQ\�DUJXPHQWV�

PD[���� PD[����� ������ PD[���� �� �� �� ���� ����

1,920, 7.5

<RX�FDQ�LPSOHPHQW�WKLV�EHKDYLRU�LQ�\RXU�RZQ�SURFHGXUHV�DV�ZHOO� 7KH�IXQFWLRQDUJV UHWXUQV�WKH�DUJXPHQWV�SDVVHG�WR�WKH�FDOOLQJ�SURFHGXUH�

DUJV��� � WKH�QXPEHU�RI�DUJXPHQWV�

DUJV�L� � WKH i�WK�DUJXPHQW� 1 ≤ i ≤ DUJV����DUJV�L��M� � WKH�VHTXHQFH�RI�DUJXPHQWV�IURP i WR j�

1 ≤ i ≤ j ≤ DUJV����DUJV�� � WKH�VHTXHQFH DUJV����DUJV������� RI�DOO

DUJXPHQWV�

7KH�IROORZLQJ�IXQFWLRQ�VLPXODWHV�WKH�EHKDYLRU�RI�WKH�V\VWHP�IXQFWLRQ PD[�

PD[LPXP � SURF�� ORFDO P� L� EHJLQP � DUJV����IRU L IURP � WR DUJV��� GR

LI P � DUJV�L� WKHQ P � DUJV�L� HQGBLIHQGBIRU�P

HQGBSURF�PD[LPXP���� PD[LPXP����� ������ PD[LPXP���� �� �� �� ���� ����

1,920, 7.5

+HUH� ZH�LQLWLDOL]H P ZLWK�WKH�ILUVW�DUJXPHQW� 7KHQ� ZH�WHVW�IRU�HDFK�RI�WKHUHPDLQLQJ�DUJXPHQWV�ZKHWKHU�LW�LV�JUHDWHU�WKDQ P� DQG�LI�VR� UHSODFH P E\�WKHFRUUHVSRQGLQJ�DUJXPHQW� 7KXV� P FRQWDLQV�WKH�PD[LPXP�DW�WKH�HQG�RI�WKH�ORRS�1RWH�WKDW�LI�\RX�FDOO PD[LPXP ZLWK�RQO\�RQH�DUJXPHQW��VR�WKDW DUJV��� ��� WKHQWKH�ORRS IRU�L�IURP���WR���GR���� LV�QRW�H[HFXWHG�DW�DOO�

�����

Page 414: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

<RX�PD\�XVH�ERWK�IRUPDO�SDUDPHWHUV�DQG�DFFHVVHV�YLD DUJV LQ�D�SURFHGXUH�

I � SURF�[� \� EHJLQLI DUJV��� � WKHQ

[A� � \A� � DUJV���A�HOVH [A� � \A�HQGBLI

HQGBSURF�I�D� E�� I�D� E� F�a2 + b3, a2 + b3 + c4

7KH�IROORZLQJ�H[DPSOH�LV�D�WULYLDO�IXQFWLRQ�UHWXUQLQJ�LWVHOI�V\PEROLFDOO\�IRU�DQ\QXPEHU�RI�DUJXPHQWV�WKDW�DUH�DFWXDOO\�SDVVHG�

I � SURF�� EHJLQ SURFQDPH�DUJV��� HQGBSURF�I���� I��� ��� I��� �� ��� I�D�� E�� F�� G��f(1) , f(1, 2) , f(1, 2, 3) , f(D�,E�, F�,G�)

�����

Page 415: Mupad Tutorial

Himbhgl3 ma^�K^f^f[^k�MZ[e^

Ghlagfk2 l`]�J]e]eZ]j�LYZd]

:KHQ�GHFODULQJ�0X3$'� SURFHGXUHV� \RX�FDQ�VSHFLI\ RSWLRQV WKDW�DIIHFW�WKHH[HFXWLRQ�RI�D�SURFHGXUH�FDOO� %HVLGHV�WKH�RSWLRQ HVFDSH DOUHDG\�PHQWLRQHG� WKHRSWLRQ UHPHPEHUPD\�EH�RI�LQWHUHVW�WR�WKH�XVHU� ,Q�WKLV�VHFWLRQ� ZH�WDNH�D�FORVHUORRN�DW�WKLV�RSWLRQ�DQG�GHPRQVWUDWH�LWV�HIIHFW�ZLWK�DQ�H[DPSOH� 7KH�VHTXHQFH�RI)LERQDFFL�QXPEHUV LV�GHILQHG�E\�WKH�UHFXUVLRQ

Fn = Fn−1 + Fn−2 , F0 = 0, F1 = 1.

,W�LV�HDV\�WR�WUDQVODWH�WKLV�LQWR�D�0X3$' SURFHGXUH�

) � SURF�Q� 7\SH��1RQ1HJ,QW�EHJLQ

LI Q � � WKHQ QHOVH )�Q � �� � )�Q � �� HQGBLI

HQGBSURF�)�L� � L �����

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55

7KLV�ZD\�RI�FRPSXWLQJ Fn LV�KLJKO\�LQHIILFLHQW�IRU�ODUJHU�YDOXHV�RI n� 7R�VHH�ZK\�OHW�XV�WUDFH�WKH�UHFXUVLYH�FDOOV�RI ) ZKHQ�FRPSXWLQJ F4� <RX�PD\�UHJDUG�WKLV�DV�DWUHH�VWUXFWXUH� )��� FDOOV )��� DQG )���� )��� FDOOV )��� DQG )��� HWF��

F4

F3 F2

F2 F1 F1 F0

F1 F0

2QH�FDQ�VKRZ�WKDW�WKH�FDOO )�Q� OHDGV�WR�DERXW 1.45.. · (1.618..)n FDOOV�RI ) IRU�ODUJHn� 7KHVH�³FRVWV´�JURZ�GUDPDWLFDOO\�IDVW�IRU�LQFUHDVLQJ�YDOXHV�RI n�

WLPH�)������ WLPH�)������ WLPH�)������ WLPH�)�����80, 910, 10290, 113600

�����

Page 416: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

:H�UHFDOO�WKDW�WKH�IXQFWLRQ WLPH UHWXUQV�WKH�WLPH�LQ�PLOOLVHFRQGV�XVHG�WR�HYDOXDWHLWV�DUJXPHQW�

:H�VHH�WKDW�PDQ\�FDOOV��VXFK�DV� IRU�H[DPSOH� )�����DUH�H[HFXWHG�VHYHUDO�WLPHV�)RU�HYDOXDWLQJ )���� LW�LV�VXIILFLHQW�WR�H[HFXWH�RQO\�WKH�ER[HG�IXQFWLRQ�FDOOV)��������)��� LQ�WKH�DERYH�ILJXUH�DQG�WR�VWRUH�WKHVH�YDOXHV� $OO�RWKHUFRPSXWDWLRQV�RI )����)����)��� DUH�UHGXQGDQW�VLQFH�WKHLU�UHVXOWV�DUH�DOUHDG\NQRZQ� 7KLV�LV�SUHFLVHO\�ZKDW�0X3$' GRHV�ZKHQ�\RX�GHFODUH ) ZLWK�WKH RSWLRQUHPHPEHU�) � SURF�Q� 7\SH��1RQ1HJ,QW�

RSWLRQ UHPHPEHU�EHJLQ

LI Q � � WKHQ QHOVH )�Q � �� � )�Q � �� HQGBLI

HQGBSURF�

7KH�V\VWHP�LQWHUQDOO\�FUHDWHV�D�UHPHPEHU�WDEOH�IRU�WKH�SURFHGXUH )��ZKLFKLQLWLDOO\�LV�HPSW\� $W�HDFK�FDOO�WR )��0X3$' FKHFNV�ZKHWKHU�WKHUH�LV�DQ�HQWU\�IRUWKH�FXUUHQW�DUJXPHQW�VHTXHQFH�LQ�WKLV�WDEOH� ,I�WKLV�LV�WKH�FDVH� WKH�SURFHGXUH�LVQRW�H[HFXWHG�DW�DOO� DQG�WKH�UHVXOW�LV�WDNHQ�IURP�WKH�WDEOH� ,I�WKH�FXUUHQW�DUJXPHQWVGR�QRW�DSSHDU�LQ�WKH�WDEOH� WKH�V\VWHP�H[HFXWHV�WKH�SURFHGXUH�ERG\�DV�XVXDO�DQGUHWXUQV�LWV�UHVXOW� 7KHQ�LW�DSSHQGV�WKH�DUJXPHQW�VHTXHQFH�DQG�WKH�UHWXUQ�YDOXH�WRWKH�UHPHPEHU�WDEOH� 7KLV�HQVXUHV�WKDW�D�SURFHGXUH�LV�QRW�XQQHFHVVDULO\�H[HFXWHGWZLFH�ZLWK�WKH�VDPH�DUJXPHQWV�

,Q�WKH�)LERQDFFL�H[DPSOH� WKH�FDOO )�Q� QRZ�OHDGV�WR�RQO\ n+ 1 FDOOV�WR�FRPSXWH)��������)�Q�� ,Q�DGGLWLRQ� WKH�V\VWHP�VHDUFKHV�WKH�UHPHPEHU�WDEOH n− 2 WLPHV�+RZHYHU� WKLV�KDSSHQV�YHU\�TXLFNO\� ,Q�WKLV�H[DPSOH� WKH�EHQHILW�LQ�VSHHG�IURPXVLQJ RSWLRQ�UHPHPEHU LV�TXLWH�GUDPDWLF�

WLPH�)������ WLPH�)������ WLPH�)������ WLPH�)������WLPH�)������

0, 10, 0, 10, 390

7KH�UHDO�UXQQLQJ�WLPHV�DUH�VR�VPDOO�WKDW�WKH�V\VWHP�FDQQRW�PHDVXUH�WKHP�H[DFWO\�7KLV�H[SODLQV�WKH��URXQGHG��WLPHV�RI 0PLOOLVHFRQGV�IRU )���� DQG )�����

8VLQJ RSWLRQ�UHPHPEHU LQ�D�SURFHGXUH�LV�SURPLVLQJ�ZKHQHYHU�DSURFHGXUH�LV�FDOOHG�IUHTXHQWO\�ZLWK�WKH�VDPH�DUJXPHQWV�

�����

Page 417: Mupad Tutorial

Himbhgl3 ma^�K^f^f[^k�MZ[e^

2I�FRXUVH� \RX�FDQ�LPSOHPHQW�WKLV�PHWKRG�IRU�FRPSXWLQJ�WKH�)LERQDFFL�QXPEHUVGLUHFWO\�E\�FRPSXWLQJ Fn LWHUDWLYHO\�LQVWHDG�RI�UHFXUVLYHO\�DQG�VWRULQJ�DOUHDG\FRPSXWHG�YDOXHV�LQ�D�WDEOH��

) � SURF�Q� 7\SH��1RQ1HJ,QW�ORFDO L� )�

EHJLQ)>�@ � �� )>�@ � ��IRU L IURP � WR Q GR

)>L@ � )>L � �@ � )>L � �@HQGBIRU

HQGBSURF�WLPH�)������ WLPH�)������ WLPH�)������ WLPH�)������WLPH�)������

10, 0, 10, 0, 400

7KH�IXQFWLRQ QXPOLE��ILERQDFFL IURP�WKH�QXPEHU�WKHRU\�OLEUDU\�LV�\HW�IDVWHU�IRUODUJH�DUJXPHQWV� VLQFH�LW�XVHV�PRUH�HODERUDWH�DOJRULWKPV�DQG�GLUHFW�IRUPXODV�

:DUQLQJ� 7KH�UHPHPEHU�PHFKDQLVP�UHFRJQL]HV�RQO\�SUHYLRXVO\�SURFHVVHGLQSXWV� EXW�GRHV�QRW�FRQVLGHU�WKH�YDOXHV�RI�SRVVLEO\�XVHG�JOREDO�YDULDEOHV� :KHQWKH�YDOXHV�RI�WKHVH�JOREDO�YDULDEOHV�FKDQJH� WKHQ�WKH�UHPHPEHUHG�UHWXUQ�YDOXHVDUH�XVXDOO\�ZURQJ� ,Q�SDUWLFXODU� WKLV�LV�WKH�FDVH�IRU�JOREDO�HQYLURQPHQW�YDULDEOHVVXFK�DV ',*,76�IORDWH[S � SURF�[� RSWLRQ UHPHPEHU�

EHJLQ IORDW�H[S�[�� HQGBSURF�',*,76 � ��� IORDWH[S����

2.7182818284590452354

',*,76 � ��� IORDWH[S���� IORDW�H[S����2.718281828459045235360287471344923927842

2.718281828459045235360287471352662497757

+HUH� WKH�V\VWHP�RXWSXWV�WKH�UHPHPEHUHG�YDOXH�RI IORDWH[S��� ZLWK�KLJKHUSUHFLVLRQ�DIWHU�VZLWFKLQJ�IURP 20 WR 40 ',*,76��1HYHUWKHOHVV� WKLV�LV�VWLOO�WKH�YDOXHFRPSXWHG�ZLWK ',*,76 ��� WKH�RXWSXW�RQO\�VKRZV�DOO�GLJLWV�WKDW�ZHUH�XVHG

�7KLV�SURFHGXUH�LV�QRW�SURSHUO\�LPSOHPHQWHG� ZKDW�KDSSHQV�ZKHQ�\RX�FDOO )���"

�����

Page 418: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

LQWHUQDOO\� LQ�WKLV�FRPSXWDWLRQ� 7KH�ODVW�RI�WKH�WKUHH�QXPEHUV�LV�WKH�WUXH�YDOXH�RIH[S��� FRPSXWHG�ZLWK 40 GLJLWV� ,W�GLIIHUV�IURP�WKH�ZURQJO\�UHPHPEHUHG�YDOXH�DWWKH 30�WK�GHFLPDO�GLJLW�

<RX�FDQ�H[SOLFLWO\�DGG�QHZ�HQWULHV�WR�WKH�UHPHPEHU�WDEOH�RI�D�SURFHGXUH� ,Q�WKHIROORZLQJ�H[DPSOH� f LV�WKH�IXQFWLRQ x #→ VLQ(x)/x� ZKLFK�KDV�D�UHPRYDEOHVLQJXODULW\�DW x = 0� 7KH�OLPLW�LV f(0) := OLPx→0 VLQ(x)/x = 1�

I � SURF�[� EHJLQ VLQ�[��[ HQGBSURF� I���(UURU� 'LYLVLRQ E\ ]HUR�GXULQJ HYDOXDWLRQ RI I

<RX�FDQ�HDVLO\�DGG�WKH�YDOXH�IRU x = 0�

I��� � �� I���1

7KH�DVVLJQPHQW I���� � FUHDWHV�D�UHPHPEHU�WDEOH�IRU I� VR�WKDW�D�ODWHU�FDOO�RII��� GRHV�QRW�WU\�WR�HYDOXDWH�WKH�YDOXH�RI VLQ(x)/x IRU x = 0� 1RZ�\RX�FDQ�XVH IZLWKRXW�UXQQLQJ�LQWR�GDQJHU�DW x = 0�

:DUQLQJ� 7KH�FDOO�

GHOHWH I� I�[� � [A��

GRHV QRW JHQHUDWH�WKH�IXQFWLRQ f : x #→ x2� EXW�UDWKHU�FUHDWHV�D�UHPHPEHU�WDEOH�IRUWKH�LGHQWLILHU I ZLWK�WKH�HQWU\ [A� RQO\�IRU�WKH�V\PEROLF�LGHQWLILHU [� $Q\�RWKHU�FDOOWR I UHWXUQV�D�V\PEROLF�IXQFWLRQ�FDOO�

I�[�� I�\�� I���x2, f(y) , f(1)

�,QWHUQDOO\� 0X3$' XVHV�D�FHUWDLQ�QXPEHU�RI�DGGLWLRQDO�³JXDUG�GLJLWV´�H[FHHGLQJ�WKH�QXPEHU�RIGLJLWV�UHTXHVWHG�YLD ',*,76��+RZHYHU� IRU�WKH�RXWSXW� WKH�V\VWHP�WUXQFDWHV�WKH�LQWHUQDOO\�FRPSXWHG�YDOXHWR�WKH�UHTXHVWHG�QXPEHU�RI�GLJLWV�

�����

Page 419: Mupad Tutorial

Bginm�IZkZf^m^kl

Afhml�HYjYe]l]jk

7KH�GHFODUHG�IRUPDO�DUJXPHQWV�RI�D�SURFHGXUH�FDQ�EH�XVHG�OLNH�DGGLWLRQDO�ORFDOYDULDEOHV�

I � SURF�D� E� EHJLQ D � �� D � E HQGBSURF�D � �� I�D� ��� D

2

0RGLI\LQJ D ZLWKLQ�WKLV�SURFHGXUH�GRHV�QRW�DIIHFW�WKH�LGHQWLILHU D WKDW�LV�XVHGRXWVLGH�WKH�SURFHGXUH� <RX�VKRXOG�EH�FDXWLRXV�ZKHQ�\RX�DFFHVV�WKH�FDOOLQJDUJXPHQWV�YLD DUJV �SDJH �������LQ�D�SURFHGXUH�DIWHU�FKDQJLQJ�VRPH�LQSXWSDUDPHWHU� $VVLJQPHQW�WR�D�IRUPDO�SDUDPHWHU�FKDQJHV�WKH�UHWXUQ�YDOXH�RI DUJV�

I � SURF�D� EHJLQ D � �� D� DUJV��� HQGBSURF� I���1, 1

,Q�SULQFLSOH� DUELWUDU\�0X3$'� REMHFWV�PD\�EH�LQSXW�SDUDPHWHUV�RI�D�SURFHGXUH�7KXV�\RX�FDQ�XVH�VHWV� OLVWV� H[SUHVVLRQV� RU�HYHQ�SURFHGXUHV�DQG�IXQFWLRQV�

S � SURF�I� EHJLQ>I���� I���� I���@

HQGBSURF�S�J�

[g(1) , g(2) , g(3)]

S�SURF�[� EHJLQ [A� HQGBSURF�[1, 4, 9]

S�[ �! [A��[1, 8, 27]

�����

Page 420: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

,Q�JHQHUDO� XVHU�GHILQHG�SURFHGXUHV�HYDOXDWH�WKHLU�DUJXPHQWV��³FDOO�E\�YDOXH´��ZKHQ�\RX�FDOO I�[�� WKH�SURFHGXUH I NQRZV�RQO\�WKH�YDOXH�RI�WKH�LGHQWLILHU [� ,I�\RXGHFODUH�D�SURFHGXUH�ZLWK�WKH RSWLRQ�KROG� WKHQ�WKH�³FDOO�E\�QDPH´�VFKHPH�LV�XVHGLQVWHDG� WKH�H[SUHVVLRQ�RU�WKH�QDPH�RI�WKH�LGHQWLILHU�SDVVHG�DV�WKH�DFWXDODUJXPHQW�LV�VHHQ�E\�WKH�SURFHGXUH� ,Q�WKLV�FDVH� \RX�FDQ�XVH FRQWH[W WR�REWDLQ�WKHYDOXH�RI�WKH�H[SUHVVLRQ�

I � SURF�[� RSWLRQ KROG�EHJLQ [ FRQWH[W�[� HQGBSURF�

[ � ��I�[�� I�VLQ�[��� I�VLQ�3,��x = 2, VLQ(x) = VLQ(2) , VLQ(π) = 0

�����

Page 421: Mupad Tutorial

>oZenZmbhg�Pbmabg�Ikh\^]nk^l

=nYdmYlagf�Oal`af�Hjg[]\mj]k

,Q�&KDSWHU �� ZH�KDYH�GLVFXVVHG�0X3$'�¶V�HYDOXDWLRQ�VWUDWHJ\� FRPSOHWHHYDOXDWLRQ�DW�LQWHUDFWLYH�OHYHO� 7KXV�DOO�LGHQWLILHUV�DUH�UHSODFHG�E\�WKHLU�YDOXHVUHFXUVLYHO\ XQWLO�RQO\�V\PEROLF�LGHQWLILHUV�ZLWKRXW�D�YDOXH�UHPDLQ��RU�WKHHYDOXDWLRQ�GHSWK�JLYHQ�E\�WKH�HQYLURQPHQW�YDULDEOH /(9(/ LV�UHDFKHG��

GHOHWH D� E� F� [ � D � E� D � E � �� E � F�[

2 c+ 1

,Q�FRQWUDVW� ZLWKLQ�SURFHGXUHV� WKH�V\VWHP�SHUIRUPV�HYDOXDWLRQ�QRW�FRPSOHWHO\�EXWRQO\�ZLWK�HYDOXDWLRQ�GHSWK 1� 7KLV�LV�VLPLODU�WR�LQWHUQDOO\�UHSODFLQJ�HDFK�LGHQWLILHUE\ OHYHO�LGHQWLILHU� ��� HYHU\�LGHQWLILHU�LV�UHSODFHG�E\�LWV�YDOXH� EXW QRWUHFXUVLYHO\� :H�UHFDOO�IURP�SDJH ��� WKH�GLVWLQFWLRQ�EHWZHHQ�WKH YDOXH RI�DQLGHQWLILHU��WKH�HYDOXDWLRQ�DW�WKH�WLPH�RI�DVVLJQPHQW��DQG�LWV HYDOXDWLRQ �WKH³FXUUHQW�YDOXH�´ ZKHUH�V\PEROLF�LGHQWLILHUV�WKDW�KDYH�EHHQ�DVVLJQHG�D�YDOXH�LQ�WKHPHDQWLPH�DUH�UHSODFHG�E\�WKHLU�YDOXHV�DV�ZHOO�� ,Q�LQWHUDFWLYH�PRGH� FDOOLQJ�DQREMHFW�\LHOGV�LWV�FRPSOHWH�HYDOXDWLRQ� ZKLOH�LQ�SURFHGXUHV�RQO\�WKH�REMHFW¶V�YDOXH�LVUHWXUQHG� 7KLV�H[SODLQV�WKH�GLIIHUHQFH�EHWZHHQ�WKH�LQWHUDFWLYH�UHVXOW�DERYH�DQGWKH�IROORZLQJ�UHVXOW�

I � SURF�� EHJLQGHOHWH D� E� F�[ � D � E� D � E � �� E � F�[

HQGBSURF�I��a+ b

7KH�UHDVRQ�ZK\�WZR�GLIIHUHQW�EHKDYLRUV�DUH�LPSOHPHQWHG�LV�WKDW�WKH�VWUDWHJ\�RILQFRPSOHWH�HYDOXDWLRQ�PDNHV�WKH�HYDOXDWLRQ�RI�SURFHGXUHV�IDVWHU�DQG�LQFUHDVHVWKH�HIILFLHQF\�RI�0X3$' SURFHGXUHV�FRQVLGHUDEO\� )RU�D�EHJLQQHU�LQSURJUDPPLQJ�0X3$' SURFHGXUHV� WKLV�HYDOXDWLRQ�FRQFHSW�KDV�LWV�SLWIDOOV�+RZHYHU� DIWHU�VRPH�SUDFWLFH�\RX�DFTXLUH�DQ�DSSURSULDWH�SURJUDPPLQJ�VW\OH�VRWKDW�\RX�FDQ�ZRUN�ZLWK�WKH�UHVWULFWHG�HYDOXDWLRQ�GHSWK�ZLWKRXW�SUREOHPV�

�����

Page 422: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

:DUQLQJ� ,I�\RX�GR�QRW�ZRUN�LQWHUDFWLYHO\�ZLWK�0X3$'��EXW�XVH�DQ�HGLWRU�WRZULWH�\RXU�0X3$' FRPPDQGV�LQWR�D�WH[W�ILOH�ZKLFK�LV�UHDG�LQWR�D�0X3$' VHVVLRQYLD UHDG �SDJH ������ WKHVH�FRPPDQGV�DUH�H[HFXWHG�ZLWKLQ�D�SURFHGXUH��QDPHO\�UHDG�� &RQVHTXHQWO\� WKH�HYDOXDWLRQ�GHSWK�LV 1�

<RX�FDQ�XVH�WKH�V\VWHP�IXQFWLRQ OHYHO �SDJH �����WR�FRQWURO�WKH�HYDOXDWLRQ�GHSWKDQG�WR�HQIRUFH�FRPSOHWH�HYDOXDWLRQ�LI�QHFHVVDU\�

I � SURF�� EHJLQGHOHWH D� E� F�[ � D � E� D � E � �� E � F�OHYHO�[�

HQGBSURF�I��

2 c+ 1

:DUQLQJ� OHYHO GRHV�QRW�HYDOXDWH�ORFDO�YDULDEOHV�

0RUHRYHU� \RX�FDQQRW�XVH�ORFDO�YDULDEOHV�DV�V\PEROLF�LGHQWLILHUV� \RX�PXVWLQLWLDOL]H�WKHP�EHIRUH�XVH� 7KH�IROORZLQJ�SURFHGXUH� LQ�ZKLFK�D�V\PEROLF�LGHQWLILHU[ LV�SDVVHG�WR�WKH�LQWHJUDWLRQ�IXQFWLRQ� LV�LQYDOLG�

I � SURF�Q� ORFDO [� EHJLQ LQW�H[S�[AQ�� [� HQGBSURF�

<RX�FDQ�SDVV�WKH�QDPH�RI�WKH�LQWHJUDWLRQ�YDULDEOH�DV�DGGLWLRQDO�DUJXPHQW�WR�WKHSURFHGXUH� 7KXV� WKH�IROORZLQJ�YDULDQWV�DUH�YDOLG�

I � SURF�Q� [� EHJLQ LQW�H[S�[AQ�� [� HQGBSURF�I � SURF�Q� [� ORFDO \� EHJLQ

\ � [� LQW�H[S�\AQ�� \� HQGBSURF�

,I�\RX�QHHG�V\PEROLF�LGHQWLILHUV�IRU�LQWHUPHGLDWH�UHVXOWV� \RX�FDQ�JHQHUDWH�DQLGHQWLILHU�ZLWKRXW�D�YDOXH�YLD JHQLGHQW�� �SDJH �����DQG�DVVLJQ�LW�WR�D�ORFDOYDULDEOH�

�����

Page 423: Mupad Tutorial

?ng\mbhg�>gobkhgf^gml

>mf[lagf�=fnajgfe]flk

0X3$'� SURYLGHV�D�YDULHW\�RI�WRROV�IRU�KDQGOLQJ�EXLOW�LQ�PDWKHPDWLFDO�VWDQGDUGIXQFWLRQV�VXFK�DV VLQ� FRV� H[S� 7KHVH�WRROV�LPSOHPHQW�WKH�PDWKHPDWLFDOSURSHUWLHV�RI�WKHVH�IXQFWLRQV� 7\SLFDO�H[DPSOHV�DUH�WKH IORDW FRQYHUVLRQ�URXWLQH�WKH�GLIIHUHQWLDWLRQ�IXQFWLRQ GLII� RU�WKH�IXQFWLRQ H[SDQG� ZKLFK�\RX�XVH�WRPDQLSXODWH�H[SUHVVLRQV�

IORDW�VLQ����� GLII�VLQ�[�� [� [� [��H[SDQG�VLQ�[ � ���

0.8414709848, − FRV(x) , FRV(1) VLQ(x) + VLQ(1) FRV(x)

,Q�WKLV�VHQVH� WKH�PDWKHPDWLFDO�NQRZOHGJH�DERXW�WKH�VWDQGDUG�IXQFWLRQV�LVGLVWULEXWHG�RYHU�VHYHUDO�V\VWHP�IXQFWLRQV� WKH�IXQFWLRQ IORDW KDV�WR�NQRZ�KRZ�WRFRPSXWH�QXPHULFDO�DSSUR[LPDWLRQV�RI�WKH�VLQH�IXQFWLRQ� GLIIPXVW�NQRZ�LWVGHULYDWLYH� DQG H[SDQG KDV�WR�NQRZ�WKH�DGGLWLRQ�WKHRUHPV�RI�WKH�WULJRQRPHWULFIXQFWLRQV�

<RX�FDQ�LQYHQW�DUELWUDU\�QHZ�IXQFWLRQV�DV�V\PEROLF�QDPHV�RU�LPSOHPHQW�WKHP�LQSURFHGXUHV� +RZ�FDQ�\RX�SDVV�WKH�NQRZOHGJH�DERXW�WKH�PDWKHPDWLFDO�PHDQLQJDQG�WKH�UXOHV�RI�PDQLSXODWLRQ�IRU�WKH�QHZ�IXQFWLRQV�WR�WKH�RWKHU�V\VWHP�IXQFWLRQV")RU�H[DPSOH� KRZ�FDQ�\RX�WHOO�WKH�GLIIHUHQWLDWLRQ�URXWLQH GLII ZKDW�WKH�GHULYDWLYHRI�\RXU�QHZO\�FUHDWHG�IXQFWLRQ�LV" ,I�WKH�IXQFWLRQ�LV�FRPSRVHG�RI�VWDQGDUGIXQFWLRQV�NQRZQ�WR�0X3$'��VXFK�DV� IRU�H[DPSOH� f : x #→ x VLQ(x)� WKHQ�WKLV�LV�QRSUREOHP� 7KH�FDOO

I � [ �! �[ VLQ�[��� GLII�I�[�� [�VLQ(x) + x FRV(x)

LPPHGLDWHO\�\LHOGV�WKH�GHVLUHG�DQVZHU� +RZHYHU� RIWHQ�WKHUH�DUH�VLWXDWLRQV�ZKHUHWKH�QHZO\�LPSOHPHQWHG�IXQFWLRQ�FDQQRW�EH�FRPSRVHG�IURP�VWDQGDUG�REMHFWV�7KXV� RXU�JRDO�LV�WR�KDQG�WKH�UXOHV�RI�PDQLSXODWLRQ��IORDWLQJ�SRLQW�DSSUR[LPDWLRQ�GLIIHUHQWLDWLRQ�HWF���IRU V\PEROLF IXQFWLRQ�FDOOV�WR�WKH�0X3$' IXQFWLRQV IORDW�GLII HWF� 7KLV�LV�WKH�DFWXDO�FKDOOHQJH�ZKHQ�\RX�³LPSOHPHQW�D�QHZ�PDWKHPDWLFDOIXQFWLRQ�LQ�0X3$'�´�WR�GLVWULEXWH�WKH�NQRZOHGJH�DERXW�WKH�PDWKHPDWLFDOPHDQLQJ�RI�WKH�V\PERO�WR�0X3$'¶V�VWDQGDUG�WRROV� ,QGHHG� WKLV�LV�D�QHFHVVDU\WDVN� IRU�H[DPSOH� LI�\RX�ZDQW�WR�GLIIHUHQWLDWH�D�PRUH�FRPSOH[�H[SUHVVLRQFRQWDLQLQJ�ERWK�WKH�QHZ�IXQFWLRQ�DQG�VRPH�VWDQGDUG�IXQFWLRQV� WKHQ�WKLV�LV�RQO\

�����

Page 424: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

SRVVLEOH�YLD�WKH�V\VWHP¶V�GLIIHUHQWLDWLRQ�URXWLQH� 7KXV� WKH�ODWWHU�KDV�WR�OHDUQ�KRZWR�KDQGOH�WKH�QHZ�V\PEROV�

)RU�WKDW�SXUSRVH� 0X3$' SURYLGHV�WKH�GRPDLQ�W\SH '20B)81&B(19 �VKRUW�IRU�IXQFWLRQ�HQYLURQPHQW�� ,QGHHG� DOO�EXLOW�LQ�PDWKHPDWLFDO�VWDQGDUG�IXQFWLRQV�DUHRI�WKLV�W\SH�LQ�RUGHU�WR�HQDEOH IORDW� GLII� H[SDQG HWF��WR�KDQGOH�WKHP�

GRPW\SH�VLQ�'20B)81&B(19

<RX�FDQ�FDOO�D�IXQFWLRQ�HQYLURQPHQW�OLNH�DQ\�³QRUPDO´�IXQFWLRQ�RU�SURFHGXUH�

VLQ�����0.9916648105

$ IXQFWLRQ�HQYLURQPHQW�FRQVLVWV�RI�WKUHH�RSHUDQGV� 7KH�ILUVW�RSHUDQG�LV�DSURFHGXUH�WKDW�FRPSXWHV�WKH�UHWXUQ�YDOXH�RI�D�IXQFWLRQ�FDOO� 7KH�VHFRQG�RSHUDQGLV�D�SURFHGXUH�IRU�SULQWLQJ�D�V\PEROLF�IXQFWLRQ�FDOO�RQ�WKH�VFUHHQ� 7KH�WKLUGRSHUDQG�LV�D�WDEOH�FRQWDLQLQJ�LQIRUPDWLRQ�KRZ�WKH�V\VWHP�IXQFWLRQV IORDW� GLII�H[SDQG� HWF��VKRXOG�KDQGOH�V\PEROLF�IXQFWLRQ�FDOOV�

<RX�FDQ�ORRN�DW�WKH�SURFHGXUH�IRU�HYDOXDWLQJ�D�IXQFWLRQ�FDOO�ZLWK�WKH�IXQFWLRQH[SRVH� MXVW�OLNH�IRU�QRUPDO�IXQFWLRQV�

H[SRVH�VLQ�SURF�[�

QDPH VLQ�ORFDO I� \�RSWLRQ QR'HEXJ�

EHJLQLI DUJV��� � WKHQ

HUURU��QR DUJXPHQWV JLYHQ��HOVH���

HQGBSURF

7R�NHHS�WKH�H[DPSOH�PDQDJHDEOH� ZH�ZLOO�FKRRVH�WZR�FORVHO\�UHODWHG�IXQFWLRQV�DQGDFW�DV�LI�WKH\�ZHUH�QRW�LPSOHPHQWHG�LQ�0X3$' \HW� 2XU�H[DPSOH�IXQFWLRQV�DUH�WKHFRPSOHWH�HOOLSWLF�LQWHJUDO�IXQFWLRQV�RI�WKH�ILUVW�DQG�VHFRQG�NLQG� K(z) DQG E(z)�6LQFH�0X3$'�DOUHDG\�KDV�D�SUHGHILQHG�LGHQWLILHU (��ZH�ZLOO�LPSOHPHQW�WKHVHIXQFWLRQV�DV HOOLS( DQG� IRU�FRQVLVWHQF\� HOOLS.���7KH�PRUH�REYLRXV�QDPHV

�����

Page 425: Mupad Tutorial

?ng\mbhg�>gobkhgf^gml

HOOLSWLF( DQG HOOLSWLF. DUH�DOUHDG\�XVHG�E\�WKH�0X3$'�YHUVLRQV�RI�WKH�VDPHIXQFWLRQV�� 7KHVH�IXQFWLRQV�DSSHDU�LQ�VXFK�GLIIHUHQW�FRQWH[WV�DV�FDOFXODWLQJ�WKHSHULPHWHU�RI�DQ�HOOLSVLV� WKH�JUDYLWDWLRQDO�RU�HOHFWURVWDWLF�SRWHQWLDO�RI�D�XQLIRUPULQJ�RU�WKH�SUREDELOLW\�WKDW�D�UDQGRP�ZDON�LQ�WKUHH�GLPHQVLRQV�HYHU�JRHV�WKURXJKWKH�RULJLQ� )RU�WKLV�SUHVHQWDWLRQ� OHW�XV�FRQFHQWUDWH�RQ�WKH�IROORZLQJ�SURSHUWLHV�RIWKH�IXQFWLRQV E DQGK�

E′(z) =E(z)−K(z)

2 z,

K ′(z) =E(z)− (1− z)K(z)

2 (1− z)z ,

E(0) = K(0) =π

2, E(1) = 1,

K(1

2

)=

8π3/2

Γ(−1

4

)2 , K(−1) =Γ(

14

)2

4√

2π.

7KDW�LV� ZH�DUH�JRLQJ�WR�LPSOHPHQW�WKH�GHULYDWLYHV�RI E DQGK ZLWK�WKH�DERYHUHODWLRQV�DQG�PDNH�WKH�IXQFWLRQV�HYDOXDWH�DW�VSHFLDO�SRLQWV� $GGLWLRQDOO\� ZH�ZLOOLPSOHPHQW�WKH�IXQFWLRQV�LQ�VXFK�D�ZD\�WKDW�WKH\�DUH�ZULWWHQ�DV E DQGK LQ�WKHRXWSXW�

7KH�EDVLF�IXQFWLRQV�DUH�HDV\�WR�ZULWH�

HOOLS( � SURF�[� EHJLQ

LI [ � WKHQ 3,��HOLI [ � WKHQ �HOVH SURFQDPH�[� HQGBLI

HQGBSURF�HOOLS. � SURF�[� EHJLQ

LI [ � WKHQ 3,��HOLI [ ��� WKHQ � 3,A������JDPPD������A�HOLI [ �� WKHQ JDPPD�����A����VTUW�� 3,�HOVH SURFQDPH�[� HQGBLI

HQGBSURF�

�����

Page 426: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

6LQFH�WKH�YDOXHV�RI�WKH�IXQFWLRQV�DUH�NQRZQ�RQO\�DW�VSHFLILF�SODFHV� ZH�XVHSURFQDPH �SDJH ������WR�UHWXUQ�WKH�V\PEROLF�H[SUHVVLRQV HOOLS(�[� DQGHOOLS.�[�� UHVSHFWLYHO\� IRU�DOO�DUJXPHQWV�ZKHUH�WKH�IXQFWLRQ�YDOXHV�DUH�QRWNQRZQ� 7KLV�\LHOGV�

HOOLS(���� HOOLS(������HOOLS.�������� HOOLS.�[A����π

2, HOOLS(

(12

), HOOLS.

(1217

), HOOLS.

(x2 + 1

)

<RX�JHQHUDWH�D�QHZ�IXQFWLRQ�HQYLURQPHQW�E\�PHDQV�RI IXQFHQY�

RXWSXWB( � I �! KROG�(��RS�I���HOOLS( � IXQFHQY�HOOLS(� RXWSXWB(��RXWSXWB. � I �! KROG�.��RS�I���HOOLS. � IXQFHQY�HOOLS.� RXWSXWB.��

7KHVH�FRPPDQGV�FRQYHUW�WKH�SURFHGXUHV HOOLS( DQG HOOLS. WR�IXQFWLRQHQYLURQPHQWV� 7KH�ILUVW�DUJXPHQWV�DUH�WKH�SURFHGXUHV�IRU�HYDOXDWLRQ� 7KH�RSWLRQDO��VHFRQG�DUJXPHQW�RI IXQFHQY LV�WKH�SURFHGXUH�IRU�VFUHHQ�RXWSXW� :HZDQW�D�V\PEROLF�H[SUHVVLRQ HOOLS(�[� GLVSOD\HG�DV (�[� DQG�OLNHZLVH HOOLS.�[�VKDOO�EH�GLVSOD\HG�DV .�[�� 7KLV�LV�DFKLHYHG�E\�WKH�VHFRQG�DUJXPHQW�RI IXQFHQY�ZKLFK�\RX�VKRXOG�LQWHUSUHW�DV�D�FRQYHUVLRQ�URXWLQH� 2Q�LQSXW HOOLS(�[�� LWUHWXUQV�WKH�0X3$' REMHFW�WR�SULQW�RQ�WKH�VFUHHQ�LQVWHDG�RI HOOLS(�[�� 7KHDUJXPHQW I� ZKLFK�UHSUHVHQWV HOOLS(�[�� LV�FRQYHUWHG�WR�WKH�XQHYDOXDWHGIXQFWLRQ�FDOO (�[� �QRWH�WKDW [ RS�I� IRU I HOOLS(�[� DQG�WKDW�\RX�QHHGKROG�(� WR�SUHYHQW�HYDOXDWLRQ�RI�WKH�LGHQWLILHU (���7KH�V\VWHP�RXWSXWV�WKLVH[SUHVVLRQ�LQVWHDG�RI HOOLS(�[� RQ�WKH�VFUHHQ��

HOOLS(���� HOOLS(������HOOLS.�������� HOOLS.�[A����π

2, E

(12

), K

(1217

), K

(x2 + 1

)

7KH��RSWLRQDO��WKLUG�DUJXPHQW�WR IXQFHQY LV�D�WDEOH�RI IXQFWLRQ�DWWULEXWHV� ,W�WHOOVWKH�V\VWHP�IXQFWLRQV IORDW� GLII� H[SDQG HWF��KRZ�WR�KDQGOH�V\PEROLF�FDOOV�RI�WKH

�1RWH�WKDW�\RX�VKRXOG�DYRLG�UHWXUQLQJ�VWULQJV�IURP�VXFK�SURFHGXUHV� 8VLQJ�VWULQJV�EUHDNV�ERWKSUHWW\�SULQWLQJ�DQG�W\SHVHWWLQJ�RI�WKH�RXWSXW�

�����

Page 427: Mupad Tutorial

?ng\mbhg�>gobkhgf^gml

IRUP HOOLS.�[� DQG HOOLS(�[�� ,Q�WKH�H[DPSOH�DERYH� ZH�GLG�QRW�SURYLGH�DQ\VXFK�IXQFWLRQ�DWWULEXWHV� +HQFH� WKH�V\VWHP�IXQFWLRQV�GR�QRW�\HW�NQRZ�KRZ�WRSURFHHG�DQG� E\�GHIDXOW� UHWXUQ�WKH�H[SUHVVLRQ�RU�WKHPVHOYHV�V\PEROLFDOO\�

IORDW�HOOLS(������� H[SDQG�HOOLS(�[ � \���GLII�HOOLS(�[�� [�� GLII�HOOLS.�[�� [�

E

(13

), E(x+ y) ,

∂xE(x) ,

∂xK(x)

%\�DVVLJQLQJ�WR�WKH �GLII� VORW�RI�RXU�IXQFWLRQ�HQYLURQPHQWV� ZH�VHW�WKH�DWWULEXWHVIRU�WKH�GLIIHUHQWLDWLRQ�URXWLQH GLII�HOOLS(��GLII � SURF�I�[�ORFDO ]�

EHJLQ] � RS�I���HOOLS(�]� � HOOLS.�]����� ]� GLII�]� [�

HQGBSURF�

HOOLS.��GLII � SURF�I�[�ORFDO ]�

EHJLQ] � RS�I���HOOLS(�]� � ���]� HOOLS.�]���

�� ���]� ]� GLII�]� [�HQGBSURF�

7KHVH�FRPPDQGV�WHOO GLII WKDW GLII�I�[� ZLWK�D�V\PEROLF�IXQFWLRQ�FDOOI HOOLS(�]�� ZKHUH ] GHSHQGV�RQ [� VKRXOG�DSSO\�WKH�SURFHGXUH�DVVLJQHG�WRHOOLS(��GLII� 7KH�ZHOO�NQRZQ�FKDLQ�UXOH�\LHOGV

GGxE(z) = E′(z)

GzGx.

7KH�VSHFLILHG�SURFHGXUH�LPSOHPHQWV�WKLV�UXOH� ZKHUH�WKH�LQQHU�IXQFWLRQ�LQ�WKHH[SUHVVLRQ I HOOLS(�]� LV�JLYHQ�E\ ] RS�I��

�����

Page 428: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

1RZ�0X3$' NQRZV�WKH�GHULYDWLYHV�RI�WKH�IXQFWLRQV�UHSUHVHQWHG�E\�WKHLGHQWLILHUV HOOLS( DQG HOOLS.�:H�KDYH�DOUHDG\�LPSOHPHQWHG�WKH�VFUHHQ�RXWSXWV�

GLII�HOOLS(�]�� ]�� GLII�HOOLS(�\�[��� [��GLII�HOOLS(�[ VLQ�[��� [�

E(z)−K(z)2 z

,(E(y(x))−K(y(x))) ∂∂xy(x)

2 y(x)

(E(x VLQ(x))−K(x VLQ(x))) (VLQ(x) + x FRV(x))2x VLQ(x)

$V�IDU�DV GLII LV�FRQFHUQHG� WKH�LPSOHPHQWDWLRQ�RI�RXU�WZR�HOOLSWLF�LQWHJUDOV�LV�QRZFRPSOHWH�

GLII�HOOLS(�[�� [� [�E(x)−K(x)

2 x + E(x)+K(x) (x−1)2 x (x−1)

2x− E(x)−K(x)

2x2

QRUPDO�GLII�HOOLS.�� [ � ��� [� [� [��� ��� (�� [ � �� � �� .�� [ � �� � ��� [ (�� [ � �� �

���� [ .�� [ � �� � �� [ (�� [ � �� �

� ���� [ .�� [ � �� � �� [ .�� [ � ��� �

� � � � ���� [ � ��� [ � ��� [ � ���� [ � ���� [ �

��� [ � ����

$V�DQ�DSSOLFDWLRQ� ZH�QRZ�ZDQW�0X3$' WR�FRPSXWH�WKH�ILUVW�WHUPV�RI�WKH�7D\ORUH[SDQVLRQ�RI�WKH�FRPSOHWH�HOOLSWLF�LQWHJUDO�RI�WKH�ILUVW�NLQG�DURXQG x = 0� :H�FDQXVH�WKH�IXQFWLRQ WD\ORU VLQFH�LW�FDOOV GLII LQWHUQDOO\�

WD\ORU�HOOLS.�[�� [ �� ��π

2+π x

8+

9π x2

128+

25π x3

512+

1225π x4

32768+

3969π x5

131072+O

(x6)

�����

Page 429: Mupad Tutorial

?ng\mbhg�>gobkhgf^gml

%XW�WKHUH�LV�PRUH� :LWK�IXQFWLRQV�VXFK�DV�WKH�HOOLSWLF�LQWHJUDOV� ZKLFK�DSSHDU�LQWKHLU�RZQ�GHULYDWLYHV� WKH�LQWHJUDWLRQ�URXWLQH�KDV�D�JRRG�FKDQFH�RI�ILQGLQJV\PEROLF�LQWHJUDOV�RQFH�WKH GLII DWWULEXWHV�KDYH�EHHQ�LPSOHPHQWHG�

LQW�HOOLS(�[�� [�2E(x)

3− 2K(x)

3+ x

(2E(x)

3+

2K(x)3

)

([HUFLVH������ ([WHQG�WKH�GHILQLWLRQV�RI HOOLS( DQG HOOLS. E\�D �IORDW� VORW�8VH K\SHUJHRP��IORDW DQG�WKH�IROORZLQJ�HTXLYDOHQFHV�

E(z) =π

2K\SHUJHRP

([−1

2,12

], [1], z

)

K(z) =π

2K\SHUJHRP

([12,12

], [1], z

)

$OVR� H[WHQG�WKH�GHILQLWLRQV�RI�WKH�IXQFWLRQV�VXFK�WKDW�\RXU�QHZ�IORDW�HYDOXDWLRQ�LVDXWRPDWLFDOO\�XVHG�ZKHQ�D�IORDWLQJ�SRLQW�YDOXH�LV�JLYHQ�DV�LQSXW�

([HUFLVH������ ,PSOHPHQW�DQ�DEVROXWH�YDOXH�IXQFWLRQ $EV DV�D�IXQFWLRQHQYLURQPHQW� 7KH�FDOO $EV�[� VKRXOG�UHWXUQ�WKH�DEVROXWH�YDOXH�IRU�UHDO�QXPEHUV [RI�GRPDLQ�W\SH '20B,17� '20B5$7��RU '20B)/2$7��)RU�DOO�RWKHU�DUJXPHQWV� WKHV\PEROLF�RXWSXW _[_ VKRXOG�DSSHDU�RQ�WKH�VFUHHQ� 7KH�DEVROXWH�YDOXH�LVGLIIHUHQWLDEOH�RQ R\{0}� ,WV�GHULYDWLYH�LV

G |y|Gx

=|y|y

GyGx.

6HW�WKH �GLII� DWWULEXWH�DFFRUGLQJO\�DQG�FRPSXWH�WKH�GHULYDWLYH�RI $EV�[A���&RPSDUH�\RXU�UHVXOW�WR�WKH�FRUUHVSRQGLQJ�GHULYDWLYH�RI�WKH�V\VWHP�IXQFWLRQ DEV�

�����

Page 430: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

9 Hjg_jYeeaf_�=pYehd]2 <a^^]j]flaYlagf

,Q�WKLV�VHFWLRQ� ZH�GLVFXVV�DQ�H[DPSOH�GHPRQVWUDWLQJ�WKH�W\SLFDO�PRGH�RIRSHUDWLRQ�RI�D�V\PEROLF�0X3$'� SURFHGXUH� :H�LPSOHPHQW�D�V\PEROLFGLIIHUHQWLDWLRQ�URXWLQH�WKDW�FRPSXWHV�WKH�GHULYDWLYHV�RI�DOJHEUDLF�H[SUHVVLRQVFRPSRVHG�RI�DGGLWLRQV� PXOWLSOLFDWLRQV� H[SRQHQWLDWLRQV� VRPH�PDWKHPDWLFDOIXQFWLRQV��H[S� OQ� VLQ� FRV� «�� FRQVWDQWV� DQG�V\PEROLF�LGHQWLILHUV�

7KLV�H[DPSOH�LV�RQO\�IRU�LOOXVWUDWLRQ�SXUSRVHV� VLQFH�0X3$' DOUHDG\�SURYLGHV�VXFKD�IXQFWLRQ� WKH GLII URXWLQH� 7KLV�IXQFWLRQ�LV�LPSOHPHQWHG�LQ�WKH�0X3$' NHUQHODQG�WKHUHIRUH�YHU\�IDVW� $ XVHU�GHILQHG�IXQFWLRQ�WKDW�LV�ZULWWHQ�LQ�WKH�0X3$'SURJUDPPLQJ�ODQJXDJH�FDQ�KDUGO\�DFKLHYH�WKH�HIILFLHQF\�RI GLII�

7KH�IROORZLQJ�DOJHEUDLF�GLIIHUHQWLDWLRQ�UXOHV�DUH�YDOLG�IRU�WKH�FODVV�RI�H[SUHVVLRQVWKDW�ZH�FRQVLGHU�

���GfGx

= 0� LI f GRHV�QRW�GHSHQG�RQ x�

���GxGx

= 1�

���G(f + g)Gx

=GfGx

+GgGx

�OLQHDULW\��

���Ga bGx

=GaGxb+ a

GbGx

�SURGXFW�UXOH��

���G ab

Gx=

GGxeb OQ(a) = eb OQ(a)

GGx

(b OQ(a))

= ab OQ(a)GbGx

+ ab−1 bGaGx,

���GGxF (y(x)) = F ′(y)

GyGx

�FKDLQ�UXOH��

0RUHRYHU� IRU�VRPH�IXQFWLRQV F � WKH�GHULYDWLYH�LV�NQRZQ� DQG�ZH�ZDQW�WR�WDNH�WKLVLQWR�DFFRXQW�LQ�RXU�LPSOHPHQWDWLRQ� )RU�DQ�XQNQRZQ�IXQFWLRQ F � ZH�UHWXUQ�WKHV\PEROLF�IXQFWLRQ�FDOO�RI�WKH�GLIIHUHQWLDWLRQ�URXWLQH�

�����

Page 431: Mupad Tutorial

: Ikh`kZffbg`�>qZfie^3 =b__^k^gmbZmbhg

7KH�SURFHGXUH 'LII LQ�7DEOH ���� LPSOHPHQWV�WKH�DERYH�SURSHUWLHV�LQ�WKH�VWDWHGRUGHU� ,WV�FDOOLQJ�V\QWD[�LV 'LII�H[SUHVVLRQ�LGHQWLILHU��

'LII � SURF�I� [ � '20B,'(17� �� ���ORFDO D� E� )� \� EHJLQ

LI QRW KDV�I� [� WKHQ UHWXUQ��� HQGBLI� �� ���LI I [ WKHQ UHWXUQ��� HQGBLI� �� ���LI W\SH�I� �BSOXV� WKHQ

UHWXUQ�PDS�I� 'LII� [�� HQGBLI� �� ���LI W\SH�I� �BPXOW� WKHQ

D � RS�I� ��� E � VXEVRS�I� � ���UHWXUQ�'LII�D� [� E � D 'LII�E� [�� �� ���

HQGBLI�LI W\SH�I� �BSRZHU� WKHQ

D � RS�I� ��� E � RS�I� ���UHWXUQ�I OQ�D� 'LII�E� [�

� DA�E � �� E 'LII�D� [�� �� ���HQGBLI�LI RS�I� �� �! )$,/ WKHQ

) � RS�I� ��� \ � RS�I� ��� �� ���LI ) KROG�H[S� WKHQ

UHWXUQ� H[S�\� 'LII�\� [�� HQGBLI� �� ���LI ) KROG�OQ� WKHQ

UHWXUQ� ��\ 'LII�\� [�� HQGBLI� �� ���LI ) KROG�VLQ� WKHQ

UHWXUQ� FRV�\� 'LII�\� [�� HQGBLI� �� ���LI ) KROG�FRV� WKHQ

UHWXUQ��VLQ�\� 'LII�\� [�� HQGBLI� �� ���� VSHFLI\ IXUWKHU NQRZQ IXQFWLRQV KHUH �

HQGBLI�SURFQDPH�DUJV��� �� ���

HQGBSURF�

7DEOH ����� $ V\PEROLF�GLIIHUHQWLDWLRQ�URXWLQH

,Q����� ZH�LPSOHPHQWHG�DQ�DXWRPDWLF�W\SH�FKHFN�RI�WKH�VHFRQG�DUJXPHQW� ZKLFKPXVW�EH�D�V\PEROLF�LGHQWLILHU�RI�GRPDLQ�W\SH '20B,'(17�

,Q����� WKH�0X3$' IXQFWLRQ KDV FKHFNV�ZKHWKHU�WKH�H[SUHVVLRQ f WR�EHGLIIHUHQWLDWHG�GHSHQGV�RQ x�

/LQHDULW\�RI�GLIIHUHQWLDWLRQ�LV�LPSOHPHQWHG�LQ�����E\�PHDQV�RI�WKH�0X3$'

�����

Page 432: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

IXQFWLRQ PDS�PDS�I��[� � I��[� � I��[�� 'LII� [�

'LII(I�(x) , x) +'LII(I�(x) , x) +'LII(I�(x) , x)

,Q����� ZH�KDQGOH�D�SURGXFW�H[SUHVVLRQ f = f1 · f2 · . . . � WKH�FRPPDQG D� RS�I���GHWHUPLQHV�WKH�ILUVW�IDFWRU a = f1� WKHQ VXEVRS�I�� �� UHSODFHV�WKLV�IDFWRU�E\ 1�VXFK�WKDW b DVVXPHV�WKH�YDOXH f2 · f3 · . . . � 7KHQ� ZH�FDOO 'LII�D�[� DQG 'LII�E�[��,I b = f2 · f3 · . . . LV�LWVHOI�D�SURGXFW� WKLV�OHDGV�WR�DQRWKHU�H[HFXWLRQ�RI�VWHS�����DWWKH�QH[W�UHFXUVLRQ�OHYHO� ,Q�WKLV�ZD\� ����KDQGOHV�SURGXFWV�RI�DUELWUDU\�OHQJWK�

6WHS�����GLIIHUHQWLDWHV�SRZHUV� )RU f = ab� WKH�FDOO RS�I��� UHWXUQV�WKH�EDVH a DQGRS�I��� WKH�H[SRQHQW b� ,Q�SDUWLFXODU� WKLV�FRYHUV�DOO�PRQRPLDO�H[SUHVVLRQV�RI�WKHIRUP f = xn IRU�FRQVWDQW n� 7KH�UHFXUVLYH�FDOOV�WR 'LII IRU a = x DQG b = n WKHQ\LHOG 'LII�D�[� � DQG 'LII�E�[� �� UHVSHFWLYHO\� DQG�WKH�H[SUHVVLRQ�UHWXUQHGLQ�����VLPSOLILHV�WR�WKH�FRUUHFW�UHVXOW nxn−1�

,I�WKH�H[SUHVVLRQ f LV�D�V\PEROLF�IXQFWLRQ�FDOO�RI�WKH�IRUP f = F (y)� ZH�H[WUDFW�WKH³RXWHU´�IXQFWLRQ F LQ�����YLD )� RS�I��� �RWKHUZLVH� ) JHWV�WKH�YDOXH )$,/���1H[W�ZH�KDQGOH�WKH�FDVH�ZKHUH F LV�D�IXQFWLRQ�ZLWK�RQH�DUJXPHQW y DQG�H[WUDFW�WKH³LQQHU´�IXQFWLRQ�E\ \� RS�I���� ,I F LV�WKH�QDPH�RI�D�IXQFWLRQ�ZLWK�NQRZQGHULYDWLYH��VXFK�DV F = H[S, OQ, VLQ, FRV�� WKHQ�ZH�DSSO\�WKH�FKDLQ�UXOH� ,W�LV�HDV\�WRH[WHQG�WKLV�OLVW�RI�IXQFWLRQV F ZLWK�NQRZQ�GHULYDWLYHV� ,Q�SDUWLFXODU� \RX�FDQ�DGG�DIRUPXOD�IRU�GLIIHUHQWLDWLQJ�V\PEROLF�H[SUHVVLRQV�RI�WKH�IRUP LQW�·� ·�� ([WHQVLRQVWR�IXQFWLRQV F ZLWK�PRUH�WKDQ�RQH�DUJXPHQW�DUH�DOVR�SRVVLEOH�

)LQDOO\� VWHS�����UHWXUQV 'LII�I�[� V\PEROLFDOO\�LI�QR�VLPSOLILFDWLRQV�RI�WKHH[SUHVVLRQ f KDSSHQ�LQ�VWHSV�����WKURXJK�����

'LII¶V�PRGH�RI�RSHUDWLRQ�LV�DGRSWHG�IURP�WKH�V\VWHP�IXQFWLRQ GLII� &RPSDUH�WKHIROORZLQJ�UHVXOWV�WR�WKRVH�UHWXUQHG�E\ GLII�

'LII�[ OQ�[ � ��[�� [�

OQ

(x+

1x

)−x(

1x2 − 1

)

x+ 1x

'LII�I�[� VLQ�[A��� [�VLQ(x2)'LII(f(x) , x) + 2x FRV

(x2)f(x)

�����

Page 433: Mupad Tutorial

Ikh`kZffbg`�>q^k\bl^l

Hjg_jYeeaf_�=p]j[ak]k

([HUFLVH������ :ULWH�D�VKRUW�SURFHGXUH GDWH WKDW�WDNHV�WKUHH�LQWHJHUV PRQWK� GD\�\HDU DV�LQSXW�DQG�SULQWV�WKH�GDWH�LQ�WKH�XVXDO�ZD\� )RU�H[DPSOH� WKH�FDOOGDWH���������� VKRXOG�\LHOG�WKH�VFUHHQ�RXWSXW ���������

([HUFLVH������ :H�GHILQH�WKH�IXQFWLRQ f : N→ N E\

f(x) =

{3x+ 1 IRU�RGG x ,

x/2 IRU�HYHQ x .

7KH�³(3x+ 1) SUREOHP´�DVNV�ZKHWKHU�IRU�DQ�DUELWUDU\�LQLWLDO�YDOXH x0 ∈ N� WKHVHTXHQFH�UHFXUVLYHO\�GHILQHG�E\ xi+1 := f(xi) FRQWDLQV�WKH�YDOXH 1� :ULWH�DSURJUDP�WKDW�RQ�LQSXW x0 UHWXUQV�WKH�VPDOOHVW�LQGH[ i ZLWK xi = 1�

([HUFLVH������ ,PSOHPHQW�D�IXQFWLRQ *FG WR�FRPSXWH�WKH�JUHDWHVW�FRPPRQGLYLVRU�RI�WZR�SRVLWLYH�LQWHJHUV� 2I�FRXUVH� \RX�VKRXOG�QRW�XVH�WKH�V\VWHPIXQFWLRQV JFG DQG LJFG� +LQW� WKH�(XFOLGHDQ�$OJRULWKP IRU�FRPSXWLQJ�WKH�JFG�LVEDVHG�RQ�WKH�REVHUYDWLRQ

JFG(a, b) = JFG(a mod b, b) = JFG(b, a mod b)

DQG�WKH�IDFWV JFG(0, b) = JFG(b, 0) = b�

([HUFLVH������ ,PSOHPHQW�D�IXQFWLRQ 4XDGUDWXUH� )RU�D�IXQFWLRQ I DQG�D0X3$'� OLVW ; RI�QXPHULFDO�YDOXHV

x0 < x1 < · · · < xn,

WKH�FDOO 4XDGUDWXUH�I�;� VKRXOG�FRPSXWH�D�QXPHULFDO�DSSUR[LPDWLRQ�RI�WKHLQWHJUDO

∫ xn

x0

f(x) dx ≈n−1∑

i=0

(xi+1 − xi) f(xi).

�����

Page 434: Mupad Tutorial

*0 FnI:=� Ikh\^]nk^l

([HUFLVH������ 1HZWRQ¶V�PHWKRG�IRU�ILQGLQJ D�QXPHULFDO�URRW�RI�D�IXQFWLRQf : R #→ R HPSOR\V�WKH�LWHUDWLRQ xi+1 = F (xi)� ZKHUH F (x) = x− f(x)/f ′(x)�:ULWH�D�SURFHGXUH 1HZWRQ� 7KH�FDOO 1HZWRQ�I�[��Q�� ZLWK�DQ�H[SUHVVLRQ I�VKRXOG�UHWXUQ�WKH�ILUVW n+ 1 HOHPHQWV x0, . . . , xn RI�WKH�1HZWRQ�VHTXHQFH�

)RU�ERQXV�SRLQWV� H[WHQG�WKH�H[DPSOH�RQ�SDJH ����� WR�GLVSOD\�WKH�VHTXHQFH�MXVWIRXQG�

([HUFLVH������ 7KH 6LHUSLQVNL�WULDQJOH LV�D�ZHOO�NQRZQ�IUDFWDO� :H�GHILQH�DYDULDQW�RI�LW�DV�IROORZV� 7KH�6LHUSLQVNL�WULDQJOH�LV�WKH�VHW�RI�DOO�SRLQWV (x, y) ∈ N×NZLWK�WKH�IROORZLQJ�SURSHUW\� WKHUH�H[LVWV�DW�OHDVW�RQH�SRVLWLRQ�LQ�WKH�ELQDU\H[SDQVLRQV�RI x DQG y VXFK�WKDW�ERWK�KDYH�D 1�ELW�DW�WKLV�SRVLWLRQ� :ULWH�D�SURJUDP6LHUSLQVNL WKDW�RQ�LQSXW [PD[�\PD[ SORWV�WKH�VHW�RI�DOO�VXFK�SRLQWV�ZLWK�LQWHJHUFRRUGLQDWHV�LQ�WKH�UDQJH 1 ≤ x ≤ [PD[� 1 ≤ y ≤ \PD[� +LQW� WKH�IXQFWLRQQXPOLE��JBDGLF FRPSXWHV�WKH�ELQDU\�H[SDQVLRQ�RI�DQ�LQWHJHU� 7KH�JUDSKLFDOSULPLWLYH SORW��3RLQW/LVW�G FDQ�EH�XVHG�WR�FUHDWH�D�SORW�RI�WKH�SRLQWV�

([HUFLVH������ $ ORJLFDO�IRUPXOD LV�FRPSRVHG�RI�LGHQWLILHUV�DQG�WKH�RSHUDWRUVDQG� RU� DQG QRW� )RU�H[DPSOH�

IRUPXOD � �[ DQG \� RU��\ RU ]� DQG �QRW [� DQG \ DQG ]�

6XFK�D�IRUPXOD�LV�FDOOHG VDWLVILDEOH LI�LW�LV�SRVVLEOH�WR�DVVLJQ�WKH�YDOXHV 758( DQG)$/6( WR�DOO�LGHQWLILHUV�LQ�VXFK�D�ZD\�WKDW�WKH�IRUPXOD�FDQ�EH�HYDOXDWHG�WR 758(�:ULWH�D�SURJUDP�WKDW�FKHFNV�ZKHWKHU�DQ�DUELWUDU\�ORJLFDO�IRUPXOD�LV�VDWLVILDEOH�

�����

Page 435: Mupad Tutorial

9

6ROXWLRQV�WR�([HUFLVHV

([HUFLVH ���� 7KH�KHOS�SDJH "GLII WHOOV�\RX�KRZ�WR�FRPSXWH�KLJKHU�RUGHUGHULYDWLYHV�

GLII�VLQ�[A��� [� [� [� [� [�32x5 FRV

(x2)− 120x FRV

(x2)

+ 160x3 VLQ(x2)

<RX�FDQ�DOVR�XVH�WKH�ORQJHU�FRPPDQG GLII�GLII�����[��[��

([HUFLVH ���� 7KH�H[DFW�UHSUHVHQWDWLRQV�DUH�

VTUW���� � � VTUW���� FRV�3,���√

3,

√√2 + 22

7KH�QXPHULFDO�DSSUR[LPDWLRQV�DUH�

',*,76 � ��IORDW�VTUW���� � � VTUW����� IORDW�FRV�3,����

1.7321, 0.92388

7KH\�DUH�FRUUHFW�WR�ZLWKLQ 5 GLJLWV�

([HUFLVH ����

H[SDQG��[A� � \�A��x10 + 5x8 y + 10x6 y2 + 10x4 y3 + 5x2 y4 + y5

Page 436: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����

QRUPDO��[A� � ����[ � ���x− 1

([HUFLVH ���� <RX�FDQ�SORW�WKH�VLQJXODU�IXQFWLRQ f(x) = 1/ VLQ(x) RQ�WKH�LQWHUYDO[1, 10] ZLWKRXW�DQ\�SUREOHPV�

SORW���VLQ�[�� [ ������

([HUFLVH ���� 0X3$'� LPPHGLDWHO\�UHWXUQV�WKH�FODLPHG�OLPLWV�

OLPLW�VLQ�[��[� [ ���OLPLW��� � FRV�[���[� [ ���OLPLW�OQ�[�� [ �� 5LJKW�

1, 0, −∞

OLPLW�[AVLQ�[�� [ ���OLPLW��� � ��[�A[� [ LQILQLW\��OLPLW�OQ�[��H[S�[�� [ LQILQLW\�

1, H, 0

$��

Page 437: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

OLPLW�[AOQ�[�� [ �� 5LJKW��OLPLW��� � 3,�[�A[� [ LQILQLW\��OLPLW����� � H[S����[��� [ �� /HIW�∞, Hπ, 0

7KH�UHVXOW XQGHILQHG GHQRWHV�D�QRQ�H[LVWLQJ�OLPLW�

OLPLW�H[S�FRW�[��� [ ��XQGHILQHG

([HUFLVH ���� <RX�REWDLQ�WKH�ILUVW�UHVXOW�LQ�WKH�GHVLUHG�IRUP�E\�IDFWRULQJ�

VXP�NA� � N � � � N ���Q�� � IDFWRU���n3

3+ n2 +

5n3

=(

13

)· n ·

(n2 + 3n+ 5

)

VXP��� N � �����N � �� �N � �� �N � ����N ���LQILQLW\�

−14

VXP�N��N � ��A���N � ��A�� N ���LQILQLW\�516

([HUFLVH ����

$ � PDWUL[�>>�� �� �@� >�� �� �@� >�� �� �@@��% � PDWUL[�>>�� �� �@� >�� �� �@� >�� �� �@@��� �$ � %�� $ %⎛

⎜⎝4 6 68 10 1414 18 0

⎟⎠ ,

⎜⎝1 4 24 10 57 7 8

⎟⎠

$��

Page 438: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

�$ � %�A����⎛

⎜⎝−5

232 −5

752 −3

267

−12

12 −2

7

⎟⎠

([HUFLVH ���� D��7KH�IXQFWLRQ QXPOLE��PHUVHQQH UHWXUQV�D�OLVW�RI�YDOXHV�IRU p\LHOGLQJ�WKH 44 FXUUHQWO\�NQRZQ�0HUVHQQH�SULPHV�WKDW�KDYH�EHHQ�IRXQG�RQVXSHUFRPSXWHUV� 7KH�DFWXDO�FRPSXWDWLRQ�IRU 1 < p ≤ 1000 FDQ�EH�HDVLO\SHUIRUPHG�LQ�0X3$'�

VHOHFW�>� �������@� LVSULPH��VHOHFW��� S �! �LVSULPH��AS � ����

$IWHU�VRPH�WLPH�\RX�REWDLQ�WKH�GHVLUHG�OLVW�RI�YDOXHV�RI p�

VHOHFW��� S �! �LVSULPH��AS � ����[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607]

7KH�FRUUHVSRQGLQJ�0HUVHQQH�SULPHV�DUH�

PDS��� S �! ��AS����>�� �� ��� ���� ����� ������� ������� �����������

�������������������� ����������������������������

���������������������������������� ��� @

E��'HSHQGLQJ�RQ�\RXU�FRPSXWHU¶V�VSHHG� \RX�FDQ�WHVW�RQO\�WKH�ILUVW 13 RU 14 )HUPDWQXPEHUV�LQ�D�UHDVRQDEOH�DPRXQW�RI�WLPH� 1RWH�WKDW�WKH 12�WK�)HUPDW�QXPEHUDOUHDG\�KDV 1234 GHFLPDO�GLJLWV�

)HUPDW � Q �! �A��AQ� � �� LVSULPH�)HUPDW�����)$/6(

7KH�RQO\�NQRZQ�)HUPDW�SULPHV�DUH�WKH�ILUVW�ILYH�)HUPDW�QXPEHUV��LQFOXGLQJ)HUPDW����� ,QGHHG� LI�0X3$' WHVWV�WKH�ILUVW 12 )HUPDW�QXPEHUV� WKHQ�DIWHU�VRPHWLPH�LW�UHWXUQV�WKH�IROORZLQJ�ILYH�YDOXHV�

$��

Page 439: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

VHOHFW�>)HUPDW�L� � L �����@� LVSULPH�[3, 5, 17, 257, 65537]

([HUFLVH ���� 7KH�ILUVW�RSHUDQG�RI�D�SRZHU�LV�WKH�EDVH� WKH�VHFRQG�LV�WKHH[SRQHQW� 7KH�ILUVW�DQG�VHFRQG�RSHUDQG�RI�DQ�HTXDWLRQ�LV�WKH�OHIW�DQG�WKHULJKW�KDQG�VLGH� UHVSHFWLYHO\� 7KH�RSHUDQGV�RI�D�IXQFWLRQ�FDOO�DUH�LWV�DUJXPHQWV�

RS�DAE� ��� RS�DAE� ��a, b

RS�D E� ��� RS�D E� ��a, b

RS�I�D� E�� ��� RS�I�D� E�� ��a, b

([HUFLVH ���� 7KH�OLVW�ZLWK�WKH�WZR�HTXDWLRQV�LV RS�VHW���� ,WV�VHFRQG�RSHUDQGLV�WKH�HTXDWLRQ \ ���� ZKRVH�VHFRQG�RSHUDQG�LV�WKH�ULJKW�KDQG�VLGH�

VHW � VROYH�^[�VLQ��� \ H[S�D��\�VLQ��� \ H[S��D�`� ^[�\`�

{[x =

Ha VLQ(3)− Ha + VLQ(3)Ha

VLQ(3)− 1, y = − 1

Ha (VLQ(3)− 1)

]}

\ � RS�VHW� >�� �� �@�

− 1Ha (VLQ(3)− 1)

8VH DVVLJQ�RS�VHW�� WR�SHUIRUP�DVVLJQPHQWV�RI�ERWK�XQNQRZQV [ DQG \VLPXOWDQHRXVO\�

([HUFLVH ���� ,I�DW�OHDVW�RQH�QXPEHU�LQ�D�QXPHULFDO�H[SUHVVLRQ�LV�D�IORDWLQJ�SRLQWQXPEHU� WKHQ�WKH�UHVXOW�LV�D�IORDWLQJ�SRLQW�QXPEHU�

��� � ��� � ���� ����� � ��� � ���1, 1.0

$��

Page 440: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ���� <RX�REWDLQ�WKH�GHVLUHG�IORDWLQJ�SRLQW�QXPEHUV�LPPHGLDWHO\�E\DSSO\LQJ IORDW�

IORDW�3,A�3,A3,��� IORDW�H[S�3, VTUW���������1.340164183 · 1018, 640320.0

1RWH�WKDW�RQO\�WKH�ILUVW 10 GLJLWV�RI�WKHVH�YDOXHV�DUH�UHOLDEOH�VLQFH�WKLV�LV�WKH�GHIDXOWSUHFLVLRQ� ,QGHHG� IRU�ODUJHU�YDOXHV�RI ',*,76��\RX�ILQG�

',*,76 � ����IORDW�3,A�3,A3,��� IORDW�H[S�3, VTUW���������

�����������������������������������������������������?������������������������������������������������

�����������������������������������������������������?������������������������������������������������

:H�FRPSXWH 235 GHFLPDO�GLJLWV�RI 3, WR�REWDLQ�WKH�FRUUHFW 234�WK�GLJLW�DIWHU�WKHGHFLPDO�SRLQW� $IWHU�VHWWLQJ ',*,76� ���� WKH�UHVXOW�LV�WKH�ODVW�VKRZQ�GLJLW�RIIORDW�3,���$ PRUH�HOHJDQW�ZD\�LV�WR�PXOWLSO\�E\ 10234� 7KHQ�WKH�GHVLUHG�GLJLW�LVWKH�ILUVW�GLJLW�EHIRUH�WKH�GHFLPDO�SRLQW� DQG�ZH�REWDLQ�LW�E\�WUXQFDWLQJ�WKH�GLJLWVDIWHU�WKH�GHFLPDO�SRLQW�

',*,76 � ���� WUXQF���A��� 3,� � �� WUXQF���A��� 3,�6

([HUFLVH ���� D��,QWHUQDOO\� 0X3$' FRPSXWHV�ZLWK�VRPH�DGGLWLRQDO�GLJLWV�QRWVKRZQ�LQ�WKH�RXWSXW�

',*,76 � ��� [ � ��A������� IORRU�[�3.333333333 · 1049

33333333333333333307484730568084080731669827420160

$��

Page 441: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

E��$IWHU�LQFUHDVLQJ ',*,76��0X3$' GLVSOD\V�WKH�DGGLWLRQDO�GLJLWV� +RZHYHU� QRWDOO�RI�WKHP�DUH�FRUUHFW�

',*,76 � ��� [3.333333333333333330748473056808408073167 · 1049

5HVWDUW�WKH�FRPSXWDWLRQ�ZLWK�WKH�LQFUHDVHG�YDOXH�RI ',*,76 WR�REWDLQ�WKH�GHVLUHGSUHFLVLRQ�

',*,76 � ��� [ � ��A������3.333333333333333333333333333333333333333 · 1049

([HUFLVH ���� 7KH�QDPHV FDXWLRQ���� [�\� DQG -DFN-LOO DUH�LQYDOLG�VLQFH�WKH\FRQWDLQ�WKH�VSHFLDO�FKDUDFWHUV �� �� DQG � UHVSHFWLYHO\� 6LQFH�DQ�LGHQWLILHU¶V�QDPHPXVW�QRW�VWDUW�ZLWK�D�QXPEHU� �[ LV�QRW�YDOLG�HLWKHU� 7KH�QDPHV GLII DQG H[S DUHYDOLG�QDPHV�RI�LGHQWLILHUV� +RZHYHU� \RX�FDQQRW�DVVLJQ�YDOXHV�WR�WKHP�VLQFH�WKH\DUH�SURWHFWHG�QDPHV�RI�0X3$' IXQFWLRQV� 7KH�QDPH �� LV�D�YDOLG�QDPH�RILGHQWLILHU� +RZHYHU� \RX�FDQQRW�DVVLJQ�YDOXHV�WR�LW�EHFDXVH� VWDUWLQJ�ZLWK�D�KDVKPDUN� LW�FDQQRW�EH�DVVLJQHG�WR�

([HUFLVH ���� :H�XVH�WKH�VHTXHQFH�RSHUDWRU � �SDJH ������WR�JHQHUDWH�WKH�VHW�RIHTXDWLRQV�DQG�WKH�VHW�RI�XQNQRZQV� 7KHQ�D�FDOO�WR VROYH UHWXUQV�D�VHW�RI�VLPSOHUHTXDWLRQV�

HTXDWLRQV � ^�[�L � [��L��� �� � L ������[�� 3,`�

XQNQRZQV � ^[�L � L �����`�VROXWLRQV � VROYH�HTXDWLRQV� XQNQRZQV�

^>[� � � 3,� [�� 3,� [�� � � 3,� [�� 3,�

[�� � � 3,� [�� 3,� [�� � � 3,� [�� 3,�

[�� � � 3,� [�� 3,� [�� � � 3,� [� 3,�

[�� 3,� [� � � 3,� [� 3,� [� � � 3,�

[� 3,� [� � � 3,� [� 3,� [� � � 3,@`

$��

Page 442: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

:H�XVH�WKH�IXQFWLRQ DVVLJQ WR�DVVLJQ�WKH�FRPSXWHG�YDOXHV�WR�WKH�LGHQWLILHUV�

DVVLJQ�RS�VROXWLRQV� ���� [�� [�� [�� [�� [�� [�1− π, π, 1− π, π, 1− π, π

([HUFLVH ���� 0X3$' VWRUHV�WKH�H[SUHVVLRQ DAE�VLQ�D�E� LQ�WKH�IRUPVLQ�D EA����� �����DAE� ,WV�H[SUHVVLRQ�WUHH�LV�

VLQ

D A

E ��

��

A

D E

([HUFLVH ���� :H�REVHUYH�WKDW�

RS������ RS�[���2, 3

x,13

7KH�UHDVRQ�LV�WKDW ��� LV�RI�GRPDLQ�W\SH '20B5$7��ZKRVH�RSHUDQGV�DUH�WKHQXPHUDWRU�DQG�WKH�GHQRPLQDWRU� 7KH�GRPDLQ�W\SH�RI�WKH�V\PEROLF�H[SUHVVLRQ [��LV '20B(;35 DQG�LWV�LQWHUQDO�UHSUHVHQWDWLRQ�LV [ ������ 7KH�VLWXDWLRQ�LV�VLPLODU�IRU��� , DQG [�� ,�RS�� � � ,�� RS�[ � � ,�

1, 2

x, 2 L

$��

Page 443: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

7KH�ILUVW�REMHFW�LV�RI�GRPDLQ�W\SH '20B&203/(;��,WV�RSHUDQGV�DUH�WKH�UHDO�DQG�WKHLPDJLQDU\�SDUW� 7KH�RSHUDQGV�RI�WKH�V\PEROLF�H[SUHVVLRQ [�� , DUH�WKH�ILUVW�DQGWKH�VHFRQG�WHUP�RI�WKH�VXP�

([HUFLVH ����� 7KH�H[SUHVVLRQ�WUHH�RI FRQGLWLRQ �QRW�D��DQG��E�RU�F� LV�

DQG

QRW

D

RU

E F

7KXV RS�FRQGLWLRQ��� QRW�D DQG RS�FRQGLWLRQ��� E�RU�F� :H�REWDLQ�WKHDWRPV D� E� F DV�IROORZV�RS�FRQGLWLRQ� >�� �@�� RS�FRQGLWLRQ� >�� �@��RS�FRQGLWLRQ� >�� �@�a, b, c

([HUFLVH ����� <RX�FDQ�XVH�ERWK�WKH�DVVLJQPHQW�IXQFWLRQ BDVVLJQ SUHVHQWHG�RQSDJH ��� DQG�WKH�DVVLJQPHQW�RSHUDWRU � �

IRU L IURP � WR ��� GR BDVVLJQ�[�L� L�� HQG�IRU L IURP � WR ��� GR [�L � L� HQG�

<RX�FDQ�DOVR�SDVV�D�VHW�RI�DVVLJQPHQW�HTXDWLRQV�WR�WKH DVVLJQ IXQFWLRQ�

DVVLJQ�^[�L L � L ������`��

([HUFLVH ����� 6LQFH�D�VHTXHQFH�LV�D�YDOLG�DUJXPHQW�RI�WKH�VHTXHQFH�RSHUDWRU�\RX�PD\�DFKLHYH�WKH�GHVLUHG�UHVXOW�DV�IROORZV�

�[�L � L� � L �����[�� [�� [�� [�� [�� [�� [�� [�� [�� [�� ���

$��

Page 444: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� :H�XVH�WKH�DGGLWLRQ�IXQFWLRQ BSOXV DQG�JHQHUDWH�LWV�DUJXPHQWVHTXHQFH�YLD ��

BSOXV���L�M�A���� � M ��� L� � L ������1464232069232792560

([HUFLVH �����

/� � >D� E� F� G@� /� � >�� �� �� �@�/��/�� ]LS�/�� /�� BPXOW�

[a, b, c, d, 1, 2, 3, 4] , [a, 2 b, 3 c, 4 d]

([HUFLVH ����� 7KH�IXQFWLRQ BPXOWPXOWLSOLHV�LWV�DUJXPHQWV�

PDS�>�� [� �@� BPXOW� PXOWLSOLHU�[PXOWLSOLHU,PXOWLSOLHU x, 2PXOWLSOLHU]

:H�XVH PDS WR�DSSO\�WKH�IXQFWLRQ VXEOLVW �!PDS�VXEOLVW�BPXOW��� WR�WKHVXEOLVWV�LQ�D�QHVWHG�OLVW�

/ � >>�� [� �@� >3,@� >���� �@@�PDS�/� PDS� BPXOW� ��[[2, 2x, 4] , [2π] ,

[43, 2]]

([HUFLVH ����� )RU

; � >[�� [�� [�@� < � >\�� \�� \�@�

WKH�SURGXFWV�DUH�JLYHQ�LPPHGLDWHO\�E\

BSOXV�RS�]LS�;� <� BPXOW���[� \�+ [� \�+ [� \�

$���

Page 445: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

7KH�IROORZLQJ�IXQFWLRQ IPXOWLSOLHV�HDFK�HOHPHQW�RI�WKH�OLVW < E\�LWV�LQSXWSDUDPHWHU [ DQG�UHWXUQV�WKH�UHVXOWLQJ�OLVW�

I � [ �! �PDS�<� BPXOW� [���

7KH�QH[W�FRPPDQG�UHSODFHV�HDFK�HOHPHQW�RI ; E\�WKH�OLVW�UHWXUQHG�E\ I�

PDS�;� I�[[[� \�, [� \�, [� \�] , [[� \�, [� \�, [� \�] , [[� \�, [� \�, [� \�]]

([HUFLVH ����� )RU�HDFKm� ZH�XVH�WKH�VHTXHQFH�JHQHUDWRU � WR�FUHDWH�D�OLVW�RI�DOOLQWHJHUV�WR�EH�FKHFNHG� 7KHQ�ZH�H[WUDFW�DOO�SULPHV�IURP�WKH�OLVW�YLDVHOHFW�·�LVSULPH�� 7KH�QXPEHU�RI�SULPHV�LV�MXVW QRSV RI�WKH�UHVXOWLQJ�OLVW� :HFRPSXWH�WKLV�YDOXH�IRU�DOOm EHWZHHQ 0 DQG 41�

QRSV�VHOHFW�>�QA� � Q � P� � Q ������@� LVSULPH��� P ������� ��� �� ��� �� ��� �� ��� �� ��� �� ��� �� ��� ��

��� �� ��� �� ��� �� ��� �� ��� �� ��� �� ��� ��

��� �� ��� �� ��� �� ��� �� ��� �� ��� �� ��

7KHUH�LV�D�VLPSOH�H[SODQDWLRQ�IRU�WKH�]HUR�YDOXHV�IRU�HYHQ m > 0� 6LQFHn2 + n = n (n+ 1) LV�DOZD\V�HYHQ� n2 + n+m LV�DQ�HYHQ�LQWHJHU�JUHDWHU�WKDQ 2 DQGKHQFH�QRW�D�SULPH�

([HUFLVH ����� :H�VWRUH�WKH�FKLOGUHQ�LQ�D�OLVW & DQG�UHPRYH�WKH�RQH�WKDW�ZDVFRXQWHG�RXW�DW�WKH�HQG�RI�HDFK�URXQG� :H�UHSUHVHQW�WKH�SRVLWLRQV 1, 2, . . . , n RI nFKLOGUHQ�E\�D�OLVW�ZLWK�WKH�FRUUHVSRQGLQJ�LQWHJHUV� /HW RXW ∈ {1, . . . , n} EH�WKHSRVLWLRQ�RI�WKH�ODVW�FKLOG�WKDW�ZDV�FRXQWHG�RXW� $IWHU�GHOHWLQJ�WKH RXW�WK�HOHPHQW�WKH�QH[W�URXQG�EHJLQV�DW�SRVLWLRQ RXW LQ�WKH�VKRUWHQHG�OLVW� $W�WKH�HQG�RI�WKHURXQG� DIWHUm ZRUGV� WKH�FKLOG�DW�SRVLWLRQ RXW�P�� LQ�WKH�FXUUHQW�OLVW�LV�FRXQWHGRXW� 6LQFH�ZH�DUH�FRXQWLQJ�F\FOLFDOO\� ZH�WDNH�WKLV�YDOXH�PRGXOR�WKH�QXPEHU�RIUHPDLQLQJ�FKLOGUHQ� 1RWH� KRZHYHU� WKDW D�PRG�E SURGXFHV�QXPEHUV�LQ�WKH�UDQJH0, 1, . . . , b− 1 UDWKHU�WKDQ 1, . . . , b� 7KLV�LV�RYHUFRPH�E\�XVLQJ ��D����PRG�E���LQVWHDG�RI D�PRG�E�P � �� Q � ��� & � >� ���Q@� RXW � ��

$���

Page 446: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

RXW � ��RXW � P � �� PRG QRSV�&�� � ��&>RXW@

8

GHOHWH &>RXW@�RXW � ��RXW � P � �� PRG QRSV�&�� � ��&>RXW@

4

HWF� ,W�LV�XVHIXO�WR�LPSOHPHQW�WKH�FRPSOHWH�FRXQWLQJ�E\�D�ORRS��&KDSWHU ����

P � �� Q � ��� & � >� ���Q@� RXW � ��UHSHDW

RXW � ��RXW � P � �� PRG QRSV�&�� � ��SULQW�&>RXW@��GHOHWH &>RXW@�

XQWLO QRSV�&� � HQGBUHSHDW��

��

���

([HUFLVH �����

VHW � ^RS�OLVW�`� OLVW � >RS�VHW�@�

1RWH�WKDW�WKH�WZR�FRQYHUVLRQV list #→ set #→ list LQ�JHQHUDO�FKDQJH�WKH�RUGHU�RI�WKHOLVW�HOHPHQWV�

$���

Page 447: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH �����

$ � ^D� E� F`� % � ^E� F� G`� & � ^E� F� H`�$ XQLRQ % XQLRQ &� $ LQWHUVHFW % LQWHUVHFW &�$ PLQXV �% XQLRQ &�{a, b, c, d, e} , {b, c} , {a}

([HUFLVH ����� <RX�REWDLQ�WKH�XQLRQ�YLD BXQLRQ�0 � ^^�� �`� ^�� �`� ^�� �`� ^�� �`� ^�� �� �� �``�BXQLRQ�RS�0��{1, 2, 3, 4, 5, 7}

DQG�WKH�LQWHUVHFWLRQ�YLD BLQWHUVHFW�BLQWHUVHFW�RS�0��{3}

([HUFLVH �����

WHOHSKRQH'LUHFWRU\ � WDEOH�)RUG �����5HDJDQ ����� %XVK ����� &OLQWRQ ������

$Q�LQGH[HG�FDOO�UHWXUQV�)RUG¶V�QXPEHU�

WHOHSKRQH'LUHFWRU\>)RUG@1815

<RX�FDQ�XVH VHOHFW WR�H[WUDFW�DOO�WDEOH�HQWULHV�FRQWDLQLQJ�WKH�QXPEHU������

VHOHFW�WHOHSKRQH'LUHFWRU\� KDV� �����

&OLQWRQ 5678

([HUFLVH ����� 7KH�FRPPDQG >RS�7DEOH�@ UHWXUQV�D�OLVW�RI�DOO�DVVLJQPHQWHTXDWLRQV� 7KH�FDOO PDS�·�RS�L� �i = 1, 2��H[WUDFWV�WKH�OHIW�DQG�WKH�ULJKW�KDQGVLGHV� UHVSHFWLYHO\� RI�WKH�HTXDWLRQV�

7 � WDEOH�D �� E ��� � VLQ�[� �GHULYDWLYH RI [ � FRV�[�� ��

$���

Page 448: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

LQGLFHV � PDS�>RS�7�@� RS� ��[a, b, 1− VLQ(x)]

YDOXHV � PDS�>RS�7�@� RS� ��[1, 2, ´GHULYDWLYH�RI�[���FRV�[�´]

([HUFLVH ����� 7KH�IROORZLQJ�WLPLQJV��LQ�PLOOLVHFRQGV��VKRZ�WKDW�JHQHUDWLQJ�DWDEOH�LV�PRUH�WLPH�FRQVXPLQJ�

Q � �������WLPH��7 � WDEOH��L L� � L ���Q����WLPH��/ � >L � L ���Q@��

422, 165

+RZHYHU� ZRUNLQJ�ZLWK�WDEOHV�LV�QRWDEO\�IDVWHU� 7KH�IROORZLQJ�DVVLJQPHQWV�FUHDWHDQ�DGGLWLRQDO�WDEOH�HQWU\�DQG�H[WHQG�WKH�OLVW�E\�RQH�HOHPHQW� UHVSHFWLYHO\�

WLPH��7>Q � �@ � 1HZ��� WLPH��/ � /�>1HZ@��0, 84

([HUFLVH ����� :H�XVH�WKH�VHTXHQFH�JHQHUDWRU � WR�FUHDWH�D�QHVWHG�OLVW�DQG�SDVVLW�WR DUUD\�Q � ���DUUD\����Q� ���Q�

>>���L � M � �� � M ���Q@ � L ���Q@��

([HUFLVH �����

758( DQG �)$/6( RU QRW �)$/6( RU QRW )$/6(��)$/6(

([HUFLVH ����� :H�XVH�WKH�IXQFWLRQ ]LS WR�JHQHUDWH�D�OLVW�RI�FRPSDULVRQV� :HSDVV�WKH�V\VWHP�IXQFWLRQ BOHVV DV�WKLUG�DUJXPHQW� ZKLFK�JHQHUDWHV�LQHTXDOLWLHV�RIWKH�IRUP D�E� 7KHQ�ZH�H[WUDFW�WKH�VHTXHQFH�RI�LQHTXDOLWLHV�YLD RS DQG�SDVV�LW�WRBDQG�

$���

Page 449: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

/� � >�� LA� � LA� � L �����@�/� � >LA� � �� L � L �����@�BDQG�RS�]LS�/�� /�� BOHVV���

� � �� DQG �� � �� DQG �� � �� DQG �� � ��� DQG

��� � ��� DQG ��� � ��� DQG ��� � ��� DQG

��� � ��� DQG �� � ��� DQG � � ����

)LQDOO\� HYDOXDWLQJ�WKLV�H[SUHVVLRQ�ZLWK ERRO DQVZHUV�WKH�TXHVWLRQ�

ERRO���758(

([HUFLVH ����� 7KH�IXQFWLRQ VRUW GRHV�QRW�VRUW�WKH�LGHQWLILHUV�DOSKDEHWLFDOO\�E\WKHLU�QDPHV�EXW�DFFRUGLQJ�WR�DQ�LQWHUQDO�RUGHU��SDJH ������ 7KXV� ZH�FRQYHUWWKHP�WR�VWULQJV�YLD H[SU�WH[W EHIRUH�VRUWLQJ�

VRUW�PDS�DQDPHV�$OO�� H[SU�WH[W��>�$[�� �$[LRP�� �$[LRP&RQVWUXFWRU�� �&B�� �&DW��

�&DWHJRU\�� ��� � �]HWD�� �]LS�@

([HUFLVH ����� :H�FRPSXWH�WKH�UHIOHFWLRQ�RI�WKH�SDOLQGURPH

WH[W � �1HYHU RGG RU HYHQ��

E\�SDVVLQJ�WKH�UHIOHFWHG�VHTXHQFH�RI�LQGLYLGXDO�FKDUDFWHUV�WR�WKH�IXQFWLRQ BFRQFDW�ZKLFK�FRQYHUWV�LW�WR�D�VWULQJ�DJDLQ�

Q � OHQJWK�WH[W��BFRQFDW�WH[W>Q � L � �@ � L ���Q�

´QHYH�UR�GGR�UHYH1´

7KLV�FDQ�DOVR�EH�DFKLHYHG�ZLWK�WKH�FDOO UHYHUW�WH[W��

([HUFLVH �����

I � [ �! �[A��� J � [ �! �VTUW�[���

$���

Page 450: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

�I#I#J����� �I##�����[�4, x1267650600228229401496703205376

([HUFLVH ����� 7KH�IROORZLQJ�IXQFWLRQ�GRHV�WKH�MRE�

I � / �! >/>QRSV�/� � � � L@ � L ���QRSV�/�@/ �! >/>�QRSV�/� � �� � L@ � L ���QRSV�/�@

I�>D� E� F@�[c, b, a]

+RZHYHU� WKH�VLPSOHVW�VROXWLRQ�LV�WR�XVH I� UHYHUW�

([HUFLVH ����� <RX�FDQ�XVH�WKH�IXQFWLRQ ODVW �&KDSWHU ������WR�JHQHUDWH�WKH&KHE\VKHY�SRO\QRPLDOV�DV�H[SUHVVLRQV�

7� � �� 7� � [�7� � � [ � � ��� 7�� � [ � � ��� 7�� � [ � � ��

2x2 − 1

2x(2x2 − 1

)− x

1− 2x2 − 2x(x− 2x

(2x2 − 1

))

$ PXFK�PRUH�HOHJDQW�ZD\�LV�WR�WUDQVODWH�WKH�UHFXUVLYH�GHILQLWLRQ�LQWR�D�0X3$'IXQFWLRQ�WKDW�ZRUNV�UHFXUVLYHO\�

7 � �N� [� �!�LI N � �

WKHQ [ANHOVH � [ 7�N � �� [� � 7�N � �� [�

HQGBLI��

7KHQ�ZH�REWDLQ�

7�L� ���� � L ����

−79, −23

27,

1781,

241243

$���

Page 451: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

7�L� ����� � L ����−0.7822, −0.846252, 0.22367368, 0.9938766288

7�L� [� � L ����� �

� [ � �� � [ �� [ � �� � [�

� �� � � [ � � [ �[ � � [ �� [ � ���� [ �

� �� [ �� [ � �� � � [ �� [ �[ � � [ �� [ � ��� �

�� [ � ��

<RX�FDQ�REWDLQ�H[SDQGHG�UHSUHVHQWDWLRQV�RI�WKH�SRO\QRPLDOV�E\�LQVHUWLQJ�D�FDOO�WRH[SDQG �SDJH �����LQ�WKH�IXQFWLRQ�GHILQLWLRQ�

7KH�&KHE\VKHY�SRO\QRPLDOV�DUH�DOUHDG\�LPSOHPHQWHG�LQ RUWKSRO\� WKH�OLEUDU\�IRURUWKRJRQDO�SRO\QRPLDOV� 7KH i�WK�&KHE\VKHY�SRO\QRPLDO�LV�UHWXUQHG�E\�WKH�FDOORUWKSRO\��FKHE\VKHY��L� [��

([HUFLVH ����� ,Q�SULQFLSOH� \RX�FDQ�FRPSXWH�WKH�GHULYDWLYHV�RI f LQ�0X3$' DQGVXEVWLWXWH x = 0� %XW�LW�LV�VLPSOHU�WR�DSSUR[LPDWH�WKH�IXQFWLRQ�E\�D�7D\ORU�VHULHVZKRVH�OHDGLQJ�WHUPV�GHVFULEH�WKH�EHKDYLRU�LQ�WKH�QHLJKERUKRRG�RI x = 0�

WD\ORU�WDQ�VLQ�[�� � VLQ�WDQ�[��� [ ��x7

30+

29x9

756+

1913x11

75600+O

(x13)

7KXV� f(x) = x7/30 · (1 +O(x2))� DQG�KHQFH f KDV�D�URRW�RI�RUGHU 7 DW x = 0�

$���

Page 452: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� 7KH�UHDVRQ�IRU�WKH�GLIIHUHQFH�EHWZHHQ�WKH�UHVXOWV

WD\ORU�GLII����� � [�� [�� [��GLII�WD\ORU����� � [�� [�� [�

1 + 2x+ 3x2 + 4x3 + 5x4 + 6x5 +O(x6)

1 + 2x+ 3x2 + 4x3 + 5x4 +O(x5)

LV�WKH�WUXQFDWLRQ�GHWHUPLQHG�E\�WKH�HQYLURQPHQW�YDULDEOH 25'(5 ZLWK�WKH�GHIDXOWYDOXH 6� %RWK WD\ORU FDOOV�FRPSXWH�WKH�FRUUHVSRQGLQJ�VHULHV�XS�WR O(x6)�

WD\ORU����� � [�� [�1 + x+ x2 + x3 + x4 + x5 +O

(x6)

7KH�RUGHU�WHUP 2�[A�� DSSHDUV�ZKHQ�WKH�WHUP 2�[A�� LV�GLIIHUHQWLDWHG�

GLII��� [�1 + 2x+ 3x2 + 4x3 + 5x4 +O

(x5)

([HUFLVH ����� $Q�DV\PSWRWLF�H[SDQVLRQ�\LHOGV�

I � VTUW�[ � �� � VTUW�[ � ���J � VHULHV�I� [ LQILQLW\�

1√x

+1

8x 52

+7

128x 92

+O(

1x

112

)

7KXV

f ≈ 1√x

(1 +

18x2

+7

128x4+ · · ·

),

DQG�KHQFH f(x) ≈ 1/√x IRU�DOO�UHDO x≫ 1� 7KH�QH[W�EHWWHU�DSSUR[LPDWLRQ�LV

f(x) ≈ 1√x

(1 +

18x2

)�

$���

Page 453: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� 7KH�FRPPDQG "UHYHUW UHTXHVWV�WKH�FRUUHVSRQGLQJ�KHOS�SDJH�

I � WD\ORU�VLQ�[ � [A��� [�� J � UHYHUW���

x+5x3

6− 59x5

120+O

(x7)

x− 5x3

6+

103x5

40+O

(x7)

7R�FKHFN�WKLV�UHVXOW� ZH�FRQVLGHU�WKH�FRPSRVLWLRQ�RI f DQG g� ZKRVH�VHULHVH[SDQVLRQ�LV�WKDW�RI�WKH�LGHQWLW\�IXQFWLRQ x #→ x�

J#Ix+O

(x7)

([HUFLVH ����� :H�SHUIRUP�WKH�FRPSXWDWLRQ�RYHU�WKH�VWDQGDUG�FRHIILFLHQW�ULQJ�SDJH ������ ZKLFK�FRPSULVHV�ERWK�UDWLRQDO�QXPEHUV�DQG�IORDWLQJ�SRLQW�QXPEHUV�

Q � ���+ � PDWUL[�Q� Q� �L� M� �! ��L � M ���A�������H � PDWUL[�Q� �� ��� E � + H�

:H�ILUVW�FRPSXWH�WKH�VROXWLRQ�RI�WKH�V\VWHP�RI�HTXDWLRQV H x⃗ = b⃗ ZLWK�H[DFWDULWKPHWLF�RYHU�WKH�UDWLRQDO�QXPEHUV� 7KHQ�ZH�FRQYHUW�DOO�HQWULHV�RI H DQG b⃗ WRIORDWLQJ�SRLQW�QXPEHUV�DQG�VROYH�WKH�V\VWHP�QXPHULFDOO\��DQG�LQ�WKH�XQVWDEOH�DQGVORZ�ZD\� XVLQJ�DQ�H[SOLFLW�LQYHUVH�LQVWHDG�RI�FDOOLQJ QXPHULF��PDWOLQVROYH��

H[DFW +A���� E� QXPHULFDO IORDW�+�A���� IORDW�E�

H[DFW =

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

11. . .

111

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, QXPHULFDO =

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0.9999999931.000000164. . .

−11.8754.0361328120.3975830078

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

$���

Page 454: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

7KH�HUURUV�LQ�WKH�QXPHULFDO�VROXWLRQ�RULJLQDWH�IURP�URXQGLQJ�HUURUV� 7RGHPRQVWUDWH�WKLV� ZH�UHSHDW�WKH�QXPHULFDO�FRPSXWDWLRQ�ZLWK�KLJKHU�SUHFLVLRQ�

',*,76 � ���QXPHULFDO IORDW�+�A���� IORDW�E�

QXPHULFDO =

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1.00000005050041473190.99999992752642441474

. . .

0.999998897314071655271.00000032782554626460.9999999580904841423

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

([HUFLVH ����� :H�ORRN�IRU�YDOXHV�ZKHUH�WKH�GHWHUPLQDQW�RI�WKH�PDWUL[�YDQLVKHV�

PDWUL[�>>�� D� E@� >�� �� F@� >�� �� �@@��IDFWRU�OLQDOJ��GHW����

(c− 1) · (a− 1)

7KH�PDWUL[�LV�LQYHUWLEOH�XQOHVV a = 1 RU c = 1�

([HUFLVH ����� :H�ILUVW�VWRUH�WKH�PDWUL[�GDWD�LQ�DUUD\V� 7KHVH�DUUD\V�DUH�XVHGODWHU�WR�JHQHUDWH�PDWULFHV�RYHU�GLIIHUHQW�FRHIILFLHQW�ULQJV�

D � DUUD\������ ����� >> �� �� �@�>��� �� �@�> �� �� �@@��

E � DUUD\������ ����� >>�� ��@� >�� �@� >�� �@@��

1RZ�ZH�GHILQH�WKH�FRQVWUXFWRU 04 IRU�PDWULFHV�RYHU�WKH�UDWLRQDO�QXPEHUV�DQGFRQYHUW�WKH�DUUD\V�WR�FRUUHVSRQGLQJ�PDWULFHV�

04 � 'RP��0DWUL[�'RP��5DWLRQDO�� $ � 04�D�� % � 04�E��

$���

Page 455: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

7KH�PHWKRG �WUDQVSRVH� RI�WKH�FRQVWUXFWRU�FRPSXWHV�WKH�WUDQVSRVH�RI�D�PDWUL[ %YLD 04��WUDQVSRVH�%���� $ � % 04��WUDQVSRVH�%��A����⎛

⎜⎝

341885

71508 − 153

754011

3770 − 313016

89315080

− 473770

2013016 − 731

15080

⎟⎠

&RPSXWLQJ�RYHU�WKH�UHVLGXH�FODVV�ULQJ�PRGXOR 7 ZH�ILQG�

0PRG� � 'RP��0DWUL[�'RP��,QWHJHU0RG�����$ � 0PRG��D�� % � 0PRG��E��& � �� $ � % 0PRG���WUDQVSRVH�%��� &A����⎛

⎜⎝3PRG 7 0PRG 7 1PRG 71PRG 7 3PRG 7 2PRG 74PRG 7 2PRG 7 2PRG 7

⎟⎠

:H�FKHFN�WKLV�E\�PXOWLSO\LQJ�WKH�LQYHUVH�E\�WKH�RULJLQDO�PDWUL[� ZKLFK�\LHOGV�WKHLGHQWLW\�PDWUL[�RYHU�WKH�FRHIILFLHQW�ULQJ�

� &⎛

⎜⎝1PRG 7 0PRG 7 0PRG 70PRG 7 1PRG 7 0PRG 70PRG 7 0PRG 7 1PRG 7

⎟⎠

([HUFLVH ����� :H�FRPSXWH�RYHU�WKH�FRHIILFLHQW�ULQJ�RI�UDWLRQDO�QXPEHUV�

04 � 'RP��0DWUL[�'RP��5DWLRQDO��

:H�FRQVLGHU 3× 3PDWULFHV� 7R�GHILQH�WKH�PDWUL[� ZH�SDVV�D�IXQFWLRQ�WR�WKHFRQVWUXFWRU�WKDW�PDSV�WKH�LQGLFHV�WR�WKH�FRUUHVSRQGLQJ�PDWUL[�HQWULHV�

$���

Page 456: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

$ � 04��� �� �L� M� �! �LI L M WKHQ � HOVH � HQGBLI��⎛

⎜⎝0 1 11 0 11 1 0

⎟⎠

7KH�GHWHUPLQDQW�RI A LV

OLQDOJ��GHW�$�2

7KH�HLJHQYDOXHV�DUH�WKH�URRWV�RI�WKH�FKDUDFWHULVWLF�SRO\QRPLDO�

S � OLQDOJ��FKDUSRO\�$� [�x3 − 3x− 2

VROYH�S� [�{−1, 2}

$OWHUQDWLYHO\� WKH OLQDOJ SDFNDJH�SURYLGHV�D�IXQFWLRQ�IRU�FRPSXWLQJ�HLJHQYDOXHV�

OLQDOJ��HLJHQYDOXHV�$�{−1, 2}

/HW ,G GHQRWH�WKH 3× 3 LGHQWLW\�PDWUL[� 7KH�HLJHQVSDFH�IRU�WKH�HLJHQYDOXHλ ∈ {−1, 2} LV�WKH�VROXWLRQ�VSDFH�RI�WKH�V\VWHP�RI�OLQHDU�HTXDWLRQV(A− λ · ,G) x⃗ = 0⃗� 7KH�VROXWLRQ�YHFWRUV�VSDQ�WKH�QXOOVSDFH��WKH�³NHUQHO´��RI�WKHPDWUL[ A− λ · ,G� 7KH�IXQFWLRQ OLQDOJ��QXOOVSDFH FRPSXWHV�D�EDVLV�IRU�WKHNHUQHO�RI�D�PDWUL[�

,G � 04��LGHQWLW\����ODPEGD � ��� OLQDOJ��QXOOVSDFH�$ � ODPEGD ,G�⎡

⎢⎣

⎜⎝−110

⎟⎠ ,

⎜⎝−101

⎟⎠

⎥⎦

7KHUH�DUH�WZR�OLQHDUO\�LQGHSHQGHQW�EDVLV�YHFWRUV� +HQFH�WKH�HLJHQVSDFH�IRU�WKHHLJHQYDOXH λ = −1 LV�WZR�GLPHQVLRQDO� 7KH�RWKHU�HLJHQYDOXH�LV�VLPSOH�

$���

Page 457: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

ODPEGD � �� OLQDOJ��QXOOVSDFH�$ � ODPEGD ,G�⎡

⎢⎣

⎜⎝111

⎟⎠

⎥⎦

$OWHUQDWLYHO\� OLQDOJ��HLJHQYHFWRUV FRPSXWHV�DOO�HLJHQVSDFHV�VLPXOWDQHRXVO\�

OLQDOJ��HLJHQYHFWRUV�$�⎡

⎢⎣

⎢⎣−1, 2,

⎢⎣

⎜⎝−110

⎟⎠ ,

⎜⎝−101

⎟⎠

⎥⎦

⎥⎦ ,

⎢⎣2, 1,

⎢⎣

⎜⎝111

⎟⎠

⎥⎦

⎥⎦

⎥⎦

7KH�UHWXUQ�YDOXH�LV�D�QHVWHG�OLVW� )RU�HDFK�HLJHQYDOXH λ� LW�FRQWDLQV�D�OLVW�RI�WKH�IRUP

[ λ , PXOWLSOLFLW\�RI λ , HLJHQVSDFH�EDVLV ].

([HUFLVH �����

S � SRO\�[A� � [A� � [A� � ��� T � SRO\�[A� � ���S � TA�

SRO\(x7 − x6 − x4 + 3x3 − 2, [x]

)

7KH�SRO\QRPLDO p LV�D�PXOWLSOH�RI q�

S�TSRO\

(x4 + 1, [x]

)

7KLV�LV�FRQILUPHG�E\�D�IDFWRUL]DWLRQ�

IDFWRU�S�SRO\ (x− 1, [x]) · SRO\

(x2 + x+ 1, [x]

)· SRO\

(x4 + 1, [x]

)

IDFWRU�T�SRO\ (x− 1, [x]) · SRO\

(x2 + x+ 1, [x]

)

$���

Page 458: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� :H�XVH�WKH�LGHQWLILHU 5 WR�DEEUHYLDWH�WKH�OHQJWK\�W\SH�QDPH'RP��,QWHJHU0RG���� :LWK DOLDV� 0X3$'�DOVR�XVHV 5 DV�DQ�DOLDV�LQ�WKH�RXWSXW�RIWKH�IROORZLQJ�SRO\QRPLDOV�

S � �� DOLDV�5 'RP��,QWHJHU0RG�S���

:H�RQO\�QHHG�WR�WU\�WKH�SRVVLEOH�UHPDLQGHUV 0, 1, 2PRGXOR 3 IRU�WKH�FRHIILFLHQWVa, b, c LQ a x2 + b x+ c� :H�JHQHUDWH�D�OLVW�RI�DOO 18 TXDGUDWLF�SRO\QRPLDOV�ZLWK a ̸= 0DV�IROORZV�

>��SRO\�D [A� � E [ � F� >[@� 5� � D ���S���� E ���S��� � F ���S��@�

7KH�FRPPDQG VHOHFW�·�LUUHGXFLEOH� H[WUDFWV�WKH 6 LUUHGXFLEOH�SRO\QRPLDOV�

VHOHFW��� LUUHGXFLEOH�� �

>SRO\�[ � �� >[@� 5�� SRO\�� [ � [ � �� >[@� 5��

�SRO\�� [ � � [ � �� >[@� 5��

� �SRO\�� [ � �� >[@� 5�� SRO\�[ � [ � �� >[@� 5��

�SRO\�[ � � [ � �� >[@� 5�@

([HUFLVH ���� 7KH�YDOXH�RI [ LV�WKH�LGHQWLILHU D�� 7KH�HYDOXDWLRQ�RI [ \LHOGV�WKHLGHQWLILHU F�� 7KH�YDOXH�RI \ LV�WKH�LGHQWLILHU E�� 7KH�HYDOXDWLRQ�RI \ \LHOGV�WKHLGHQWLILHU F�� 7KH�YDOXH�RI ] LV�WKH�LGHQWLILHU D�� 7KH�HYDOXDWLRQ�RI ] \LHOGV ���

7KH�HYDOXDWLRQ�RI X� OHDGV�WR�DQ�LQILQLWH�UHFXUVLRQ� ZKLFK�0X3$' DERUWV�ZLWK�DQHUURU�PHVVDJH� 7KH�HYDOXDWLRQ�RI X� \LHOGV�WKH�H[SUHVVLRQ Y�A����

([HUFLVH ���� 7KH�UHVXOW�RI VXEVRS�E�D�� F� LV E�F DQG�QRW F�D� DV�\RX�PLJKWKDYH�H[SHFWHG� 7KH�UHDVRQ�LV�WKDW VXEVRS HYDOXDWHV�LWV�DUJXPHQWV� 7KH�V\VWHPUHRUGHUV�WKH�VXP�LQWHUQDOO\�ZKHQ�HYDOXDWLQJ�LW� DQG�WKXV VXEVRS SURFHVVHV D�ELQVWHDG�RI E�D� 8SRQ�UHWXUQ� WKH�UHVXOW F�E LV�UHRUGHUHG�DJDLQ�

$���

Page 459: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ���� 7KH�KLJKHVW�GHULYDWLYH�RFFXUULQJ�LQ J LV�WKH 6�WK�GHULYDWLYHGLII�I�[��[���� :H�SDVV�WKH�VHTXHQFH�RI�UHSODFHPHQW�HTXDWLRQV�

GLII�I�[�� [������ �I�� GLII�I�[�� [������ �I�� ����GLII�I�[�� [�� �I�� I�[�� �I�

WR�0X3$'¶V�VXEVWLWXWLRQ�IXQFWLRQ� 1RWH�WKDW� LQ�DFFRUGDQFH�ZLWK�WKH�XVXDOPDWKHPDWLFDO�QRWDWLRQ� GLII UHWXUQV�WKH�IXQFWLRQ�LWVHOI�DV�WKH 0�WK�GHULYDWLYH�GLII�I�[��[��� GLII�I�[�� I�[��GHOHWH I� J � GLII�I�[��GLII�I�[�� [�� [ � ���VXEV�J� �GLII�I�[�� [ � � � L� I����L�� � L �����

� � � ��� I� � I� �� I� ��� I� I� ��� I� I������� � ���� � ������ � ���������� � ���������� �

� I� � � �I� I� I� I�

I� I� �� I� I� �� I� I� I� �� I� I� I������ � �������� � ����������� � ����������� �

� � � �I� I� I� I�

� � ��� I� I� I� ��� I� I� I� �� I� I� I������������� � ������������� � ������������

� � �I� I� I�

([HUFLVH ���� 7KH�IROORZLQJ�FRPPDQGV�\LHOG�WKH�GHVLUHG�HYDOXDWLRQ�RI�WKHIXQFWLRQ�

I � VLQ�[��[� [ � ����� I0.7662510585

+RZHYHU� [ QRZ�KDV�D�YDOXH� 7KH�IROORZLQJ�FDOO GLII�I�[� ZRXOG�LQWHUQDOO\�OHDGWR�WKH�FRPPDQG GLII�������������������� VLQFH GLII HYDOXDWHV�LWV�DUJXPHQWV�

$���

Page 460: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

<RX�FDQ�FLUFXPYHQW�WKLV�SUREOHP�E\�SUHYHQWLQJ�D�FRPSOHWH�HYDOXDWLRQ�RI�WKHDUJXPHQWV�YLD OHYHO RU KROG �SDJH �����

J � GLII�OHYHO�I� ��� KROG�[��� JFRV(x)x− VLQ(x)x2

−0.3512303507

+HUH�WKH�HYDOXDWLRQ�RI KROG�[� LV�WKH�LGHQWLILHU [ DQG�QRW�LWV�YDOXH� :ULWLQJKROG�I� LQVWHDG�RI OHYHO�I��� ZRXOG�\LHOG�WKH�ZURQJ�UHVXOWGLII�KROG�I��KROG�[�� �� VLQFH KROG�I� GRHV�QRW�FRQWDLQ KROG�[�� 8VLQJOHYHO�I��� UHSODFHV I E\�LWV�YDOXH VLQ�[��[ �SDJH ����� 7KH�QH[W�FDOO�RI J UHWXUQVWKH�HYDOXDWLRQ�RI J� QDPHO\�WKH�YDOXH�RI�WKH�GHULYDWLYH�DW [ ����� $OWHUQDWLYHO\\RX�FDQ�GHOHWH�WKH�YDOXH�RI [�

GHOHWH [� GLII�I� [� _ [ ����−0.3512303507

([HUFLVH ���� 7KH�ILUVW�WKUHH�GHULYDWLYHV�RI�WKH�QXPHUDWRU�DQG�WKH�GHQRPLQDWRUYDQLVK�DW�WKH�SRLQW x = 0�

= � [ �! �[A� VLQ�[��� 1 � [ �! ��� � FRV�[��A���=���� 1���� =���� 1���� =���� 1����=���� 1���

0, 0, 0, 0, 0, 0, 0, 0

)RU�WKH�IRXUWK�GHULYDWLYHV� ZH�KDYH�

=���� 1���24, 6

7KXV�WKH�OLPLW�LV Z ′′′′(0)/N ′′′′(0) = 4� DFFRUGLQJ�WR�GH�O¶+RVSLWDO¶V�UXOH� 7KHIXQFWLRQ OLPLW FRPSXWHV�WKH�VDPH�UHVXOW�

OLPLW�=�[��1�[�� [ ��4

$���

Page 461: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ���� 7KH�ILUVW�RUGHU�SDUWLDO�GHULYDWLYHV�RI f1 DUH�

I� � VLQ�[� [��� GLII�I�� [��� GLII�I�� [��[� FRV([� [�) , [� FRV([� [�)

7KH�VHFRQG�RUGHU�GHULYDWLYHV�DUH�

GLII�I�� [�� [��� GLII�I�� [�� [���GLII�I�� [�� [��� GLII�I�� [�� [��

�� [� VLQ�[� [��� FRV�[� [�� � [� [� VLQ�[� [���

�FRV�[� [�� � [� [� VLQ�[� [��� � [� VLQ�[� [��

7KH�WRWDO�GHULYDWLYH�RI f2 ZLWK�UHVSHFW�WR t LV�

I� � [A� \A�� [ � VLQ�W�� \ � FRV�W�� GLII�I�� W�

2 FRV(t)3 VLQ(t)− 2 FRV(t) VLQ(t)3

([HUFLVH ����

LQW�VLQ�[� FRV�[�� [ ���3,����LQW����VTUW�� � [A���� [ ������LQW�[ DUFWDQ�[�� [ ������LQW���[� [ �������

12,π

2,π

4− 1

2, − OQ(2)

([HUFLVH ����

LQW�[��� D [ � [A��A������ [�x

a√

2 a x− x2

LQW�VTUW�[A� � DA��� [�

x√x2 − a22

−a2 OQ

(x+√x2 − a2

)

2

$���

Page 462: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

LQW����[ VTUW�� � [A���� [�

− DUFVLQK(√

1x2

)

([HUFLVH ���� 7KH�IXQFWLRQ LQWOLE��FKDQJHYDU RQO\�SHUIRUPV�D�FKDQJH�RIYDULDEOHV�ZLWKRXW�LQYRNLQJ�WKH�LQWHJUDWLRQ�

LQWOLE��FKDQJHYDU�KROG�LQW��VLQ�[� VTUW�� � VLQ�[���[ �3,����3,���� W VLQ�[��

∫ 1

−1

t√t+ 1√

1− t2Gt

$ IXUWKHU�HYDOXDWLRQ�DFWLYDWHV�WKH�LQWHJUDWLRQ�URXWLQH�

HYDO���� � IORDW���2√

23

= 0.9428090416

1XPHULFDO�TXDGUDWXUH�UHWXUQV�WKH�VDPH�UHVXOW�

QXPHULF��LQW�VLQ�[� VTUW�� � VLQ�[���[ �3,����3,���

0.9428090416

([HUFLVH ���� 7KH�HTXDWLRQ�VROYHU�UHWXUQV�WKH�JHQHUDO�VROXWLRQ�

HTXDWLRQV � ^D � E � F � G � H ��D � � E � � F � � G � � H ��D � � E � � F � � G � � H ��D � E � F � G � H �`�

VROYH�HTXDWLRQV� ^D� E� F� G� H`�{[a = 2, b = z + 2 ]�− 3, c = 2− 3 ]�− 2 z, d = z, e = ]�]}

7KH�IUHH�SDUDPHWHUV�DUH�RQ�WKH�ULJKW�KDQG�VLGHV�RI�WKH�VROYHG�HTXDWLRQV� <RX�FDQGHWHUPLQH�WKHP�LQ�0X3$' E\�H[WUDFWLQJ�WKH�ULJKW�KDQG�VLGHV�DQG�XVLQJ LQGHWV WRILQG�WKH�LGHQWLILHUV�FRQWDLQHG�WKHUHLQ�

$���

Page 463: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

PDS��� PDS� RS� ��� LQGHWV���{[2, z + 2 ]�− 3, 2− 3 ]�− 2 z, z, ]�]}

{z, ]�}

([HUFLVH ���� 7KH�V\PEROLF�VROXWLRQ�LV�

VROXWLRQ � VROYH�RGH�^\�[� \�[� � � ]�[�� ]�[� \�[�`� ^\�[�� ]�[�`��{[z(x) =

&� H2 x

2− &�Hx, y(x) =

&�Hx

+ &� H2 x]}

ZLWK�IUHH�FRQVWDQWV &��&�� :H�UHPRYH�WKH�RXWHU�FXUO\�EUDFHV�YLD RS�

VROXWLRQ � RS�VROXWLRQ�[z(x) =

&� H2 x

2− &�Hx, y(x) =

&�Hx

+ &� H2 x]

1RZ� ZH�VHW x = 0 DQG�VXEVWLWXWH \��� DQG ]���� UHVSHFWLYHO\� IRU�WKH�LQLWLDOFRQGLWLRQV� 7KHQ� ZH�VROYH�WKH�UHVXOWLQJ�OLQHDU�V\VWHP�RI�HTXDWLRQV�IRU &� DQG &��

VROYH��VROXWLRQ _ [ �� _ >\��� �� ]��� �@�^&�� &�`�

{[&� = −1

3,&� =

43

]}

$JDLQ� ZH�UHPRYH�WKH�RXWHU�FXUO\�EUDFHV�YLD RS DQG�DVVLJQ�WKH�VROXWLRQ�YDOXHV�WR &�DQG &� E\�PHDQV�RI DVVLJQ�

DVVLJQ�RS�����

7KXV� WKH�YDOXH�DW x = 1 RI�WKH�V\PEROLF�VROXWLRQ�IRU�WKH�DERYH�LQLWLDO�FRQGLWLRQV�LV�

[ � �� VROXWLRQ[z(1) =

13 H

+2 H2

3, y(1) =

4 H2

3− 1

3 H

]

$���

Page 464: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

)LQDOO\� ZH�DSSO\ IORDW�

IORDW���[z(1) = 5.04866388, y(1) = 9.729448318]

([HUFLVH ����

VROYH�RGH�\�[��\�[�A� ��[� \�[���{− 1&�+ OQ(x)

}

VROYH�RGH�^\�[� � VLQ�[� \�[� �� \��� �`� \�[���{

HFRV(1)

HFRV(x) VLQ(1)

}

VROYH�RGH�^� \�[� � \�[��[ �� \��� 3,`� \�[���{− 2π√x

}

$���

Page 465: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

VROYH�RGH�^GLII�[�W��W� \�W� ]�W��GLII�\�W��W� [�W� ]�W��GLII�]�W��W� W ]�W�`�

^[�W��\�W��]�W�`��^ ��^ _^ _^ _^ _ [�W� ^ ��

� ��� ��� ��� ?_ � &�� ��3,� HUI�� W ��� ,� _

&�� H[S_ � ����������������������������������� _? � �

������������������������������������������������ ��

� ��� ��� ��� ?_ � &�� ��3,� HUI�� W ��� ,� _

&�� H[S_ ����������������������������������� _? � �

������������������������������������������������

\�W�

� ��� ��� ��� ?_ � &�� ��3,� HUI�� W ��� ,� _

&�� H[S_ � ����������������������������������� _? � �

������������������������������������������������ ��

� ��� ��� ��� ?_ � &�� ��3,� HUI�� W ��� ,� _

&�� H[S_ ����������������������������������� _? � �

������������������������������������������������

�� `_ `

� � ? _ `_ W _ _ `

]�W� &�� H[S_ �� _ _ `? � � �� `

$���

Page 466: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ���� 7KH�IXQFWLRQ VROYH GLUHFWO\�\LHOGV�WKH�VROXWLRQ�RI�WKH�UHFXUUHQFH�

VROYH�UHF�)�Q� )�Q��� � )�Q���� )�Q��^)��� �� )��� �`��

⎧⎨

√5(√

52 + 1

2

)n

5−

√5(

12 −

√5

2

)n

5

⎫⎬

([HUFLVH ���� <RX�REWDLQ�WKH�DQVZHU�LPPHGLDWHO\�IURP VLPSOLI\�

VLPSOLI\�FRV�[�A� � VLQ�[� FRV�[��FRV(2x)

2+VLQ(2x)

2+

12

<RX�JHW�WKH�VDPH�UHVXOW�E\�DSSO\LQJ FRPELQH WR�UHZULWH�SURGXFWV�RI�WULJRQRPHWULFIXQFWLRQV�DV�VXPV�

FRPELQH�FRV�[�A� � VLQ�[� FRV�[�� VLQFRV�FRV(2x)

2+VLQ(2x)

2+

12

([HUFLVH ����

H[SDQG�FRV�� [���VLQ�� [� FRV�[�A���

FRV(x)2

2 VLQ(x)− 5 VLQ(x) +

5 VLQ(x)3

2 FRV(x)2

I � �VLQ�[�A� � H[S�� [�� ��VLQ�[�A� � � VLQ�[� H[S�[� � H[S�� [���

QRUPDO�H[SDQG�I��

−Hx − VLQ(x)Hx + VLQ(x)

I � �VLQ�� [� � � VLQ�[� FRV�[�� ��VLQ�[� �� � WDQ�[�A����

$���

Page 467: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

FRPELQH�QRUPDO�H[SDQG�I��� VLQFRV�

− 3 FRV(x)WDQ(x)2 + 1

I � VTUW��� � � VTUW�� �� VTUW�� � �� VTUW�� � � VTUW��������

VLPSOLI\�I� VTUW�√

2 + 3

([HUFLVH ���� $V�D�ILUVW�VWHS� ZH�SHUIRUP�D�QRUPDOL]DWLRQ�

LQW�VTUW�VLQ�[� � ��� [�� QRUPDO�GLII��� [��

3 FRV(x)2 VLQ(x) + FRV(x)2 + 2 VLQ(x)3 − 2 VLQ(x)FRV(x)2

√VLQ(x) + 1

7KHQ�ZH�HOLPLQDWH�WKH�FRVLQH�WHUPV�

UHZULWH��� VLQ�

2 VLQ(x) + VLQ(x)2 − 2 VLQ(x)3 + 3 VLQ(x)(VLQ(x)2 − 1

)− 1

√VLQ(x) + 1

(VLQ(x)2 − 1

)

7KH�ILQDO�QRUPDOL]DWLRQ�VWHS�DFKLHYHV�WKH�GHVLUHG�VLPSOLILFDWLRQ�

QRUPDO���√VLQ(x) + 1

2Q�WKH�RWKHU�KDQG� FDOOLQJ 6LPSOLI\ LV�PXFK�HDVLHU�

LQW�VTUW�VLQ�[� � ��� [�� 6LPSOLI\�GLII��� [��√VLQ(x) + 1

$���

Page 468: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ���� 7KH�SUREOHP�REYLRXVO\�LV�WKDW�WKH�YDOXDWLRQ�IXQFWLRQ�GRHV�QRW�ORRNFDUHIXOO\�HQRXJK�DW�WKH�WHUP�WR�DQDO\]H� 7R�UHPHG\�WKLV� ZH�XVH�WKH�IXQFWLRQOHQJWK ZKLFK�UHWXUQV�D�JHQHUDO� TXLFNO\�FRPSXWHG�³FRPSOH[LW\�´

QR7DQJHQW � [ �! LI KDV�[� >KROG�WDQ�� KROG�FRW�@�WKHQ �������HOVH OHQJWK�[� HQGBLI�

6LPSOLI\�WDQ�[� � FRW�[�� 9DOXDWLRQ QR7DQJHQW�

−2 FRV(2x)VLQ(2x)

([HUFLVH ���� 7KH�RSHUDWRU DVVXPLQJ �SDJH ������DVVLJQV�SURSHUWLHV�WRLGHQWLILHUV� 7KHVH�DUH�WDNHQ�LQWR�DFFRXQW�E\ OLPLW�

OLPLW�(L�D [�� [ LQILQLW\�OLPx→∞

(L(a x)

OLPLW�(L�D [�� [ LQILQLW\� DVVXPLQJ D ! �∞

OLPLW�(L�D [�� [ LQILQLW\� DVVXPLQJ D � �0

([HUFLVH ����� ,Q�DQDORJ\�WR�WKH�SUHYLRXV�JFG�H[DPSOH� ZH�REWDLQ�WKH�IROORZLQJH[SHULPHQW�

GLH � UDQGRP�������H[SHULPHQW � >>GLH��� GLH��� GLH��@ � L ������@�GLFH6FRUHV � PDS�H[SHULPHQW�

[ �! �[>�@ � [>�@ � [>�@���IUHTXHQFLHV � 'RP��0XOWLVHW�RS�GLFH6FRUHV���VRUWLQJ2UGHU � �[� \� �! �[>�@ � \>�@��

$���

Page 469: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

VRUW�>RS�IUHTXHQFLHV�@� VRUWLQJ2UGHU�>>�� �@� >�� �@� >�� �@� >�� �@� >�� ��@� >�� ��@�

>��� ��@� >��� ��@� >��� ��@� >��� ��@� >��� ��@�

>��� ��@� >��� �@� >��� �@� >��� �@@

,Q�WKLV�H[SHULPHQW� WKH�VFRUH 3 GLG�QRW�RFFXU�

([HUFLVH ����� D��7KH�FRPPDQG IUDQGRP LV�D�JHQHUDWRU�IRU�UDQGRPIORDWLQJ�SRLQW�QXPEHUV�LQ�WKH�LQWHUYDO [0, 1]� 7KH�FDOO

Q � �����DEV9DOXHV � >VTUW�IUDQGRP��A��IUDQGRP��A�� � L ���Q@�

UHWXUQV�D�OLVW�ZLWK�WKH�DEVROXWH�YDOXHV�RI n UDQGRP�YHFWRUV�LQ�WKH�UHFWDQJOHQ = [0, 1]× [0, 1]� 7KH�QXPEHU�RI�YDOXHV ≤ 1 LV�WKH�QXPEHU�RI�UDQGRP�SRLQWV�LQ�WKHULJKW�XSSHU�TXDGUDQW�RI�WKH�XQLW�FLUFOH�

P � QRSV�VHOHFW�DEV9DOXHV� ] �! �] � ����821

6LQFH P�Q DSSUR[LPDWHV�WKH�DUHD π/4 RI�WKH�ULJKW�XSSHU�TXDGUDQW�RI�WKH�XQLW�FLUFOH�ZH�REWDLQ�WKH�IROORZLQJ�DSSUR[LPDWLRQ�WR π�

IORDW�� P�Q�3.284

E��)LUVW�ZH�GHWHUPLQH�WKH�PD[LPXP�RI f � 7KH�IROORZLQJ�IXQFWLRQ�SORW�VKRZV�WKDW fLV�PRQRWRQLFDOO\�LQFUHDVLQJ�RQ�WKH�LQWHUYDO [0, 1]�

$���

Page 470: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

I � [ �! [ VLQ�[� � FRV�[� H[S�[��SORWIXQF�G�I�[�� [ ������

7KXV f(x) DVVXPHV�LWV�PD[LPDO�YDOXH�DW�WKH�ULJKW�HQG�RI�WKH�LQWHUYDO� 7KHUHIRUH�M = f(1) LV�DQ�XSSHU�ERXQG�IRU�WKH�IXQFWLRQ�

0 � I�����2.310164925

:H�XVH�WKH�UDQGRP�QXPEHU�JHQHUDWRU�GHILQHG�DERYH�WR�JHQHUDWH�UDQGRP�SRLQWV�LQWKH�UHFWDQJOH [0, 1]× [0,M ]�

Q � �����SRLQWOLVW � >>IUDQGRP��� 0 IUDQGRP��@ � L ���Q@�

:H�VHOHFW�WKRVH�SRLQWV p = [x, y] IRU�ZKLFK 0 ≤ y ≤ f(x) KROGV�

VHOHFW�SRLQWOLVW� S �! �S>�@ � I�S>�@����P � QRSV���

742

7KXV�WKH�IROORZLQJ�LV�DQ�DSSUR[LPDWLRQ�RI�WKH�LQWHJUDO�

P�Q 01.714142374

$���

Page 471: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

7KH�H[DFW�YDOXH�LV�

IORDW�LQW�I�[�� [ ������1.679193292

([HUFLVH ����� :LWK�WKH�IROORZLQJ�GHILQLWLRQ�RI�WKH SRVW2XWSXWPHWKRG�RI 3UHI�WKH�V\VWHP�SULQWV�DQ�DGGLWLRQDO�VWDWXV�OLQH�

3UHI��SRVW2XWSXW�SURF��EHJLQ

�E\WHV� � �H[SU�WH[W�RS�E\WHV��� ��� � � �ORJLFDO� � � �H[SU�WH[W�RS�E\WHV��� ��� � � �SK\VLFDO��

HQGBSURF��IORDW�VXP���L�� L ��������

2.718281828

E\WHV� ������ �ORJLFDO� � ������� �SK\VLFDO�

([HUFLVH ����� :H�ILUVW�JHQHUDWH�WKH�VHW S�

I � L �! �LA������LA��LA�������� ��LA������LA��� LA������� L�LA���������

6 � ^I�L� � L ���������` XQLRQ ^I�L� � L �������`�

7KHQ�ZH�DSSO\ GRPW\SH WR�DOO�HOHPHQWV�RI�WKH�VHW�WR�GHWHUPLQH�WKHLU�GRPDLQ�W\SHV�

PDS�6� GRPW\SH�{'20B5$7,'20B,17,'20B(;35}

<RX�VHH�WKH�H[SODQDWLRQ�IRU�WKLV�UHVXOW�E\�ORRNLQJ�DW�VRPH�HOHPHQWV�

I����� I���� I���� I���� I���� I���

(−2)52 + 3−

√2 L

2 (−2)32 + (−2)

52 + 1 +

√2 L, −1, 0,

3√

2 + 39√

2 + 9,

8√

3 + 816√

3 + 16,

35

7KH�IXQFWLRQ QRUPDO VLPSOLILHV�WKH�H[SUHVVLRQV�FRQWDLQLQJ�VTXDUH�URRWV�

$���

Page 472: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

PDS��� QRUPDO�

3, −1, 0,13,

12,

35

1RZ�ZH�DSSO\ QRUPDO WR�DOO�HOHPHQWV�RI�WKH�VHW�EHIRUH�TXHU\LQJ�WKHLU�GDWD�W\SH�

PDS�6� GRPW\SH#QRUPDO�{'20B5$7,'20B,17}

7KXV� DOO�QXPEHUV�LQ S DUH�LQGHHG�UDWLRQDO��LQ�SDUWLFXODU�WKHUH�DUH�WZR�LQWHJHUYDOXHV I��� �� DQG I��� ��� 7KH�UHDVRQ�LV�WKDW I�L� FDQ�EH�VLPSOLILHG�WR�L�����L����QRUPDO�I�L��i− 1i+ 1

([HUFLVH ����� :H�DSSO\ WHVWW\SH�·��VLQ�� WR�HDFK�HOHPHQW�RI�WKH�OLVW�

OLVW � >VLQ�L 3,����� � L ������@�

WR�ILQG�RXW�ZKHWKHU�LW�LV�UHWXUQHG�LQ�WKH�IRUP VLQ�·�� 7KH�IROORZLQJ VSOLWFRPPDQG��SDJH ������GHFRPSRVHV�WKH�OLVW�DFFRUGLQJO\�

GHFRPSRVLWLRQ � VSOLW�OLVW� WHVWW\SH� �VLQ���

0X3$'�VLPSOLILHG 9 RI�WKH 101 FDOOV�

$���

Page 473: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

PDS�GHFRPSRVLWLRQ� QRSV�� GHFRPSRVLWLRQ>�@>��� �� �@

�� ��� ��� ��� ��� ��� ���_ � �� � � � � �� � � �_ �� ���� � ���� �������������� ��������������������� � � �

��� ��� ��� ���� � �� � ������� ���� � ���� ��������������

� � �

��� ��� ��� ��� �� � �� _������������������� � _

� ��

([HUFLVH ����� <RX�FDQ�XVH VHOHFW �SDJH ������WR�H[WUDFW�WKRVH�HOHPHQWV�WKDWWHVWW\SH LGHQWLILHV�DV�SRVLWLYH�LQWHJHUV� )RU�H[DPSOH�

VHW � ^��� ���� �� [� ���� �`�VHOHFW�VHW� WHVWW\SH� 7\SH��3RV,QW�{2, 4}

1RWH�WKDW�WKLV�VHOHFWV�RQO\�WKRVH�REMHFWV�WKDW DUH SRVLWLYH�LQWHJHUV� EXW�QRW�WKRVHWKDW�PLJKW UHSUHVHQW SRVLWLYH�LQWHJHUV� VXFK�DV�WKH�LGHQWLILHU [ LQ�WKH�DERYHH[DPSOH� 7KLV�LV�QRW�SRVVLEOH�ZLWK WHVWW\SH� ,QVWHDG� \RX�FDQ�XVH DVVXPH WR�VHWWKLV�SURSHUW\�DQG�TXHU\�LW�YLD LV�

DVVXPH�[� 7\SH��3RV,QW��VHOHFW�VHW� LV� 7\SH��3RV,QW�{2, 4, x}

$���

Page 474: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� :H�FRQVWUXFW�WKH�GHVLUHG�W\SH�VSHFLILHU�DQG�HPSOR\�LW�DV�IROORZV�

7 � 7\SH��/LVW2I�7\SH��/LVW2I�7\SH��$Q\7\SH� �� ��� �� ��

7\SH::/LVW2I(7\SH::/LVW2I(7\SH::$Q\7\SH, 3, 3) , 2, 2)

WHVWW\SH�>>D� E� F@� >�� �� �@@� 7��WHVWW\SH�>>D� E� F@� >�� �@@� 7�

758(, )$/6(

([HUFLVH ����� &RQVLGHU�WKH�FRQGLWLRQV [�!��DQG�$ DQG [ ��RU�$��UHVSHFWLYHO\�$IWHU�HQWHULQJ�

[ � ��

LW�LV�QRW�SRVVLEOH�WR�HYDOXDWH�WKHP�GXH�WR�WKH�VLQJXODULW\�LQ x/(x− 1)�

[ �! � DQG $(UURU� 'LYLVLRQ E\ ]HUR >BSRZHU@

[ � RU $(UURU� 'LYLVLRQ E\ ]HUR >BSRZHU@

+RZHYHU� WKLV�LV�QRW�D�SUREOHP�ZLWKLQ�DQ LI VWDWHPHQW�VLQFH�WKH�%RROHDQHYDOXDWLRQ�RI [�!� DQG [ � DOUHDG\�WHOOV�XV�WKDW [�!��DQG�$ HYDOXDWHV�WR )$/6(DQG [ ��RU�$ WR 758(��UHVSHFWLYHO\�

�LI [ �! � DQG $ WKHQ ULJKW HOVH ZURQJ HQGBLI���LI [ � RU $ WKHQ ULJKW HOVH ZURQJ HQGBLI�

ZURQJ, ULJKW

2Q�WKH�RWKHU�KDQG� HYDOXDWLRQ�RI�WKH�IROORZLQJ LI VWDWHPHQW�VWLOO�SURGXFHV�DQHUURU� VLQFH�LW�LV�QHFHVVDU\�WR�HYDOXDWH $ LQ�RUGHU�WR�GHWHUPLQH�WKH�WUXWK�YDOXH�RI[ ��DQG�$�LI [ � DQG $ WKHQ ULJKW HOVH ZURQJ HQGBLI

(UURU� 'LYLVLRQ E\ ]HUR >BSRZHU@

$���

Page 475: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� :LWK�WKH�IRUPXODV�JLYHQ� LPSOHPHQWDWLRQ�RI�WKH �IORDW� VORWV�LVVWUDLJKWIRUZDUG�

HOOLS(��IORDW � ] �! IORDW�3,��� K\SHUJHRP��IORDW�>����� ���@� >�@� ]��

HOOLS.��IORDW � ] �! IORDW�3,��� K\SHUJHRP��IORDW�>���� ���@� >�@� ]��

7KLV�LV�DOPRVW�VXIILFLHQW�

HOOLS(�����

E

(13

)

IORDW���1.430315257

+RZHYHU� RQH�WKLQJ�LV�PLVVLQJ� ZH�ZRXOG�OLNH�WR�KDYH�FDOOV�ZLWK�IORDWLQJ�SRLQWDUJXPHQWV�HYDOXDWHG�LPPHGLDWHO\�

HOOLS(�����E(0.1)

7R�WKLV�HQG� ZH�PXVW�H[WHQG�WKH�GHILQLWLRQV�RI HOOLS( DQG HOOLS.�HOOLS( � SURF�[� EHJLQ

LI GRPW\SH�[� '20B)/2$7RU GRPW\SH�[� '20B&203/(;

DQG �GRPW\SH�5H�[�� '20B)/2$7RU GRPW\SH�,P�[�� '20B)/2$7�

WKHQUHWXUQ�HOOLS(��IORDW�[���

HQGBLI�LI [ � WKHQ 3,��HOLI [ � WKHQ �HOVH SURFQDPH�[� HQGBLI

HQGBSURF�

([WHQG HOOLS. DQDORJRXVO\� WKHQ�PDNH�WKHVH�QHZ�IXQFWLRQV�LQWR�IXQFWLRQHQYLURQPHQWV��WKH�DERYH�GHILQLWLRQ�UHSODFHG�WKH�ROG�RQH�FRPSOHWHO\��DQG�DGG�WKHIXQFWLRQ�VORWV�ZH�XVHG�IRU�WKH�RULJLQDO�GHILQLWLRQV�DQG�WKH�QHZ�IORDW�VORWV� DQG�\RXJHW�

$���

Page 476: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

HOOLS(�����1.530757637

([HUFLVH ����� 7KH�IROORZLQJ�SURFHGXUH�HYDOXDWHV�WKH $EV IXQFWLRQ�

$EV � SURF�[�EHJLQ

LI GRPW\SH�[� '20B,17 RU GRPW\SH�[� '20B5$7RU GRPW\SH�[� '20B)/2$7WKHQ LI [ ! � WKHQ [ HOVH �[ HQGBLI�HOVH SURFQDPH�[��

HQGBLIHQGBSURF�

:H�FRQYHUW $EV WR�D�IXQFWLRQ�HQYLURQPHQW�DQG�VXSSO\�D�IXQFWLRQ�SURGXFLQJ�WKHGHVLUHG�VFUHHQ�RXWSXW�

$EV � IXQFHQY�$EV�SURF�I� EHJLQ

�_� � H[SU�WH[W�RS�I�� � �_�HQGBSURF�1,/��

7KHQ�ZH�VHW�WKH�IXQFWLRQ�DWWULEXWH�IRU�GLIIHUHQWLDWLRQ�

$EV��GLII � SURF�I�[� EHJLQI�RS�I� GLII�RS�I�� [�

HQGBSURF�

1RZ�ZH�KDYH�WKH�IROORZLQJ�EHKDYLRU�

$EV�������� $EV�[�� $EV�[A� � \ � ]������ _[_� _[A� � \ � ]_

7KH GLII DWWULEXWH�RI�WKH�V\VWHP�IXQFWLRQ DEV \LHOGV�D�VOLJKWO\�GLIIHUHQW�EXWHTXLYDOHQW�UHVXOW�

GLII�$EV�[A��� [�� GLII�DEV�[A��� [�� _[A�_ ��������� � _[_ VLJQ�[�

[

$���

Page 477: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� :H�XVH H[SU�WH[W �SDJH ������WR�FRQYHUW�WKH�LQWHJHUV�SDVVHG�DVDUJXPHQWV�WR�VWULQJV� 7KHQ�ZH�FRPELQH�WKHP� WRJHWKHU�ZLWK�VRPH�VODVKHV� YLD�WKHFRQFDWHQDWLRQ�RSHUDWRU�³�´�

GDWH � SURF�PRQWK� GD\� \HDU� EHJLQSULQW�8QTXRWHG� H[SU�WH[W�PRQWK� � ��� �

H[SU�WH[W�GD\� � ��� �H[SU�WH[W�\HDU��

HQGBSURF�

([HUFLVH ����� :H�SUHVHQW�D�VROXWLRQ�XVLQJ�D ZKLOH ORRS� 7KH�FRQGLWLRQ [�PRG� � FKHFNV�ZKHWKHU [ LV�HYHQ�I � SURF�[� ORFDO L�EHJLQ

L � ��XVHULQIR��� �WHUP � � H[SU�WH[W�L� � �� � �

H[SU�WH[W�[���ZKLOH [ �! � GR

LI [ PRG � � WKHQ [ � [��HOVH [ � � [�� HQGBLI�L � L � ��XVHULQIR��� �WHUP � � H[SU�WH[W�L� � �� � �

H[SU�WH[W�[��HQGBZKLOH�L

HQGBSURF�I���� I������� I�������� I�����������

2, 132, 60, 177

,I�ZH�VHW VHWXVHULQIR�I��� �SDJH ������ WKHQ�WKH XVHULQIR FRPPDQG�RXWSXWV�DOOWHUPV�RI�WKH�VHTXHQFH�XQWLO�WKH�SURFHGXUH�WHUPLQDWHV�

VHWXVHULQIR�I� ��� I���,QIR� WHUP �� �,QIR� WHUP �� �,QIR� WHUP �� �

2

$���

Page 478: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

,I�\RX�GR�QRW�EHOLHYH�LQ�WKH 3x+ 1 FRQMHFWXUH� \RX�VKRXOG�LQVHUW�D�VWRSSLQJFRQGLWLRQ�IRU�WKH�LQGH[ L WR�HQVXUH�WHUPLQDWLRQ�

([HUFLVH ����� $ UHFXUVLYH�LPSOHPHQWDWLRQ�EDVHG�RQ�WKH�UHODWLRQJFG(a, b) = JFG(a PRG b, b) OHDGV�WR�DQ�LQILQLWH�UHFXUVLRQ� ZH�KDYHa PRG b ∈ {0, 1, . . . , b− 1}� DQG�KHQFH

(a PRG b) PRG b = a PRG b

LQ�WKH�QH[W�VWHS� 7KXV�WKH�IXQFWLRQ JFG ZRXOG�DOZD\V�FDOO�LWVHOI�UHFXUVLYHO\�ZLWK�WKHVDPH�DUJXPHQWV� +RZHYHU� D�UHFXUVLYH�FDOO�RI�WKH�IRUP JFG(a, b) = JFG(b, a PRG b)PDNH�VHQVH� 6LQFH a PRG b < b� WKH�IXQFWLRQ�FDOOV�LWVHOI�UHFXUVLYHO\�IRU�GHFUHDVLQJYDOXHV�RI�WKH�VHFRQG�DUJXPHQW� ZKLFK�ILQDOO\�EHFRPHV�]HUR�

*FG � SURF�D� E� EHJLQ � UHFXUVLYH YDULDQW �LI E �

WKHQ DHOVH *FG�E� D PRG E�

HQGBLIHQGBSURF�

)RU�ODUJH�YDOXHV�RI a DQG b� \RX�PD\�QHHG�WR�LQFUHDVH�WKH�YDOXH�RI�WKH�HQYLURQPHQWYDULDEOH 0$;'(37+ LI *FG H[KDXVWV�WKH�YDOLG�UHFXUVLRQ�GHSWK� 7KH�IROORZLQJLWHUDWLYH�YDULDQW�DYRLGV�WKLV�SUREOHP�

*&' � SURF�D� E�EHJLQ

ZKLOH E �! � GR>D� E@ � >E� D PRG E@�

HQGBZKLOH�D

HQGBSURF�

7KHVH�LPSOHPHQWDWLRQV�\LHOG�WKH�VDPH�UHVXOWV�DV�WKH�IXQFWLRQV LJFG DQG JFGSURYLGHG�E\�WKH�V\VWHP�

D � ������� E � �������*FG�D� E�� *&'�D� E�� LJFG�D� E�� JFG�D� E�

20576, 20576, 20576, 20576

$���

Page 479: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� ,Q�WKH�IROORZLQJ�LPSOHPHQWDWLRQ� ZH�JHQHUDWH�D�VKRUWHQHG�FRS\Y = [x1, . . . , xn] RIX = [x0, . . . , xn] DQG�FRPSXWH�WKH�OLVW�RI�GLIIHUHQFHV[x1 − x0, . . . , xn − xn−1] YLD ]LS DQG BVXEWUDFW �BVXEWUDFW�\�[� \�[�� 7KHQ�ZHPXOWLSO\�HDFK�HOHPHQW�RI�WKLV�OLVW�ZLWK�WKH�FRUUHVSRQGLQJ�QXPHULFDO�YDOXH�LQ�WKHOLVW [f(x0), f(x1), . . . ]� )LQDOO\� WKH�IXQFWLRQ BSOXV DGGV�DOO�HOHPHQWV�RI�WKHUHVXOWLQJ�OLVW�

[(x1 − x0) f(x0), . . . , (xn − xn−1) f(xn−1)] :

4XDGUDWXUH � SURF�I� ;�ORFDO <� GLVWDQFHV� QXPHULFDO9DOXHV� SURGXFWV�EHJLQ

< � ;� GHOHWH <>�@�GLVWDQFHV � ]LS�<� ;� BVXEWUDFW��QXPHULFDO9DOXHV � PDS�;� IORDW#I��SURGXFWV � ]LS�GLVWDQFHV� QXPHULFDO9DOXHV� BPXOW��BSOXV�RS�SURGXFWV��

HQGBSURF�

,Q�WKH�IROORZLQJ�H[DPSOH� ZH�XVH n = 1000 HTXLGLVWDQW�VDPSOH�SRLQWV�LQ�WKHLQWHUYDO [0, 1]�

I � [ �! �[ H[S�[��� Q � �����4XDGUDWXUH�I� >L�Q � L ���Q@�

0.9986412288

7KLV�LV�D��FUXGH��QXPHULFDO�DSSUR[LPDWLRQ�RI∫ 10 x e

x dx (= 1)�

$���

Page 480: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� 7KH�VSHFLILFDWLRQ�RI 1HZWRQ UHTXLUHV�WKDW�WKH�ILUVW�DUJXPHQW I EHDQ H[SUHVVLRQ DQG�QRW�D�0X3$' IXQFWLRQ� 7KXV� WR�FRPSXWH�WKH�GHULYDWLYH� ZHILUVW�XVH LQGHWV WR�GHWHUPLQH�WKH�XQNQRZQ�LQ I� :H�VXEVWLWXWH�D�QXPHULFDO�YDOXHIRU�WKH�XQNQRZQ�WR�HYDOXDWH�WKH�LWHUDWLRQ�IXQFWLRQ F (x) = x− f(x)/f ′(x) DW�DSRLQW�

1HZWRQ � SURF�I� [�� Q�ORFDO YDUV� [� )� VHTXHQFH� L�EHJLQ

YDUV � LQGHWV�IORDW�I���LI QRSV�YDUV� �! �

WKHQ HUURU��WKH IXQFWLRQ PXVW FRQWDLQ H[DFWO\ RQH XQNQRZQ�

�HOVH [ � RS�YDUV�

HQGBLI�) � [ � I�GLII�I�[�� VHTXHQFH � [��IRU L IURP � WR Q GR

[� � IORDW�VXEV�)� [ [����VHTXHQFH � VHTXHQFH� [�

HQGBIRU�UHWXUQ�VHTXHQFH�

HQGBSURF�

,Q�WKH�IROORZLQJ�H[DPSOH� 1HZWRQ FRPSXWHV�WKH�ILUVW�WHUPV�RI�D�VHTXHQFH�UDSLGO\FRQYHUJLQJ�WR�WKH�VROXWLRQ

√2�

1HZWRQ�[A� � �� �� ���� ���� ������������ ������������ ������������

������������ �����������

$���

Page 481: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

7KH�IROORZLQJ�SURFHGXUH�XVHV 1HZWRQ WR�SORW�WKH�DSSUR[LPDWLRQ�SURFHGXUH�

SORW1HZWRQ � SURF�I� [�� Q�

ORFDO [� [YDOV� [PLQ� [PD[� L� [L�EHJLQ

[YDOV � 1HZWRQ�I� [�� Q��[PLQ � PLQ�[YDOV��[PD[ � PD[�[YDOV��>[PLQ� [PD[@ �

>[PLQ � �[PD[ � [PLQ���� [PD[ � �[PD[ � [PLQ���@�[ � RS�LQGHWV�I���SORW�SORW��)XQFWLRQ�G�I� [ [PLQ��[PD[��

SORW��/LQH�G�>[YDOV>L@� I _ [ [YDOV>L@@�>[YDOV>L��@� �@� � L ���QRSV�[YDOV����

SORW��/LQH�G�>[YDOV>L@� �@� >[YDOV>L@� I _ [ [YDOV>L@@�/LQH6W\OH 'DVKHG� � L ���QRSV�[YDOV��

SORW��/LQH�G��/LQH&RORU 5*%��5HG�HQGBSURF�SORW1HZWRQ�[A� � �� �� ��

$���

Page 482: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

SORW1HZWRQ�VLQ�[��� VLQ�[���� �� ��

([HUFLVH ����� 7KH�FDOO QXPOLE��JBDGLF������ \LHOGV�WKH�ELQDU\�H[SDQVLRQ�RI�DQLQWHJHU�DV�D�OLVW�RI�ELWV�

QXPOLE��JBDGLF��� ��� QXPOLE��JBDGLF���� ��[1, 1, 1] , [0, 0, 0, 0, 1]

,QVWHDG�RI�FDOOLQJ QXPOLE��JBDGLF GLUHFWO\� RXU�VROXWLRQ�XVHV�D�VXESURFHGXUHELQDU\ IXUQLVKHG�ZLWK�WKH RSWLRQ�UHPHPEHU� 7KLV�DFFHOHUDWHV�WKH�FRPSXWDWLRQQRWDEO\�VLQFH QXPOLE��JBDGLF LV�FDOOHG�IUHTXHQWO\�ZLWK�WKH�VDPH�DUJXPHQWV� 7KHFDOO LV63RLQW�>[�\@� UHWXUQV 758( ZKHQ�WKH�SRLQW�VSHFLILHG�E\�WKH�OLVW >[�\@ LV�D6LHUSLQVNL�SRLQW� 7R�FKHFN�WKLV� WKH�IXQFWLRQ�PXOWLSOLHV�WKH�OLVWV�ZLWK�WKH�ELWV�RI�WKHWZR�FRRUGLQDWHV� $W�WKRVH�SRVLWLRQV�ZKHUH�ERWK [ DQG \ KDYH�D 1 ELW� PXOWLSOLFDWLRQ\LHOGV�D 1� ,Q�DOO�RWKHU�FDVHV� 0 · 0� 1 · 0� 0 · 1� SURGXFH�WKH�UHVXOW 0� ,I�WKH�OLVW�RISURGXFWV�FRQWDLQV�DW�OHDVW�RQH 1� WKH�SRLQW�LV�D�6LHUSLQVNL�SRLQW�

$���

Page 483: Mupad Tutorial

Lhenmbhgl�mh�>q^k\bl^l

:H�XVH VHOHFW �SDJH ������WR�H[WUDFW�WKH�6LHUSLQVNL�SRLQWV�IURP�DOO�SRLQWVFRQVLGHUHG� )LQDOO\� ZH�FUHDWH�D SORW��3RLQW/LVW�G ZLWK�WKHVH�SRLQWV�DQG�FDOOSORW �SDJH ������

6LHUSLQVNL � SURF�[PD[� \PD[�ORFDO ELQDU\� LV63RLQW� DOO3RLQWV� L� M� 63RLQWV�EHJLQ

ELQDU\ � SURF�[� RSWLRQ UHPHPEHU� EHJLQQXPOLE��JBDGLF�[� ��

HQGBSURF�LV63RLQW � SURF�3RLQW� ORFDO [� \� EHJLQ

[ � ELQDU\�3RLQW>�@��\ � ELQDU\�3RLQW>�@��KDV�]LS�[� \� BPXOW�� ��

HQGBSURF�DOO3RLQWV � >�>L� M@ � L ���[PD[� � M ���\PD[@�63RLQWV � VHOHFW�DOO3RLQWV� LV63RLQW��SORW�SORW��3RLQW/LVW�G�63RLQWV� &RORU 5*%��%ODFN���

HQGBSURF�

)RU [PD[ \PD[ ���� VD\� \RX�REWDLQ�D�TXLWH�DSSHDOLQJ�SLFWXUH�

6LHUSLQVNL����� ����

$���

Page 484: Mupad Tutorial

: Lhenmbhgl�mh�>q^k\bl^l

([HUFLVH ����� :H�SUHVHQW�D�UHFXUVLYH�VROXWLRQ� *LYHQ�DQ�H[SUHVVLRQIRUPXOD�[��[������ ZLWK�WKH�LGHQWLILHUV [��[������ ZH�FROOHFW�WKH�LGHQWLILHUV�LQWKH�VHW [ ^[��[�����` E\�PHDQV�RI LQGHWV� 7KHQ�ZH�VXEVWLWXWH 758( DQG )$/6(�UHVSHFWLYHO\� IRU [� � RS�[����� DQG�FDOO�WKH�SURFHGXUH�UHFXUVLYHO\�ZLWK�WKHDUJXPHQWV IRUPXOD�758(�[��[������ DQG IRUPXOD�)$/6(�[��[�������UHVSHFWLYHO\� ,Q�WKLV�ZD\� ZH�WHVW�DOO�SRVVLEOH�FRPELQDWLRQV�RI 758( DQG )$/6( IRUWKH�LGHQWLILHUV�XQWLO�WKH�H[SUHVVLRQ�FDQ�ILQDOO\�EH�VLPSOLILHG�WR 758( RU )$/6( DW�WKHERWWRP�RI�WKH�UHFXUVLRQ� 7KLV�YDOXH�LV�UHWXUQHG�WR�WKH�FDOOLQJ�SURFHGXUH� ,I�DW�OHDVWRQH�RI�WKH 758(�)$/6( FRPELQDWLRQV�\LHOGV 758(��WKHQ�WKH�SURFHGXUH�UHWXUQV 758(�LQGLFDWLQJ�WKDW�WKH�IRUPXOD�LV�VDWLVILDEOH� DQG�RWKHUZLVH�LW�UHWXUQV )$/6(�

VDWLVILDEOH � SURF�IRUPXOD� ORFDO [�EHJLQ

[ � LQGHWV�IRUPXOD��LI [ ^` WKHQ UHWXUQ�IRUPXOD� HQGBLI�UHWXUQ�VDWLVILDEOH�VXEV�IRUPXOD� RS�[� �� 758(��

RU VDWLVILDEOH�VXEV�IRUPXOD� RS�[� �� )$/6(���HQGBSURF�

,I�WKH�QXPEHU�RI�LGHQWLILHUV�LQ�WKH�LQSXW�IRUPXOD�LV n� WKHQ�WKH�UHFXUVLRQ�GHSWK�LVDW�PRVW n DQG�WKH�WRWDO�QXPEHU�RI�UHFXUVLYH�FDOOV�RI�WKH�SURFHGXUH�LV�DW�PRVW 2n�:H�DSSO\�WKLV�SURFHGXUH�LQ�WZR�H[DPSOHV�

)� � ��[ DQG \� RU �\ RU ]�� DQG �QRW [� DQG \ DQG ]�)� � ��[ DQG \� RU �\ RU ]�� DQG �QRW \� DQG �QRW ]��VDWLVILDEOH�)��� VDWLVILDEOH�QRW )���VDWLVILDEOH�)��� VDWLVILDEOH�QRW )��

758(, 758(, )$/6(, 758(

7KH�FDOO VLPSOLI\�·�ORJLF� �SDJH ������VLPSOLILHV�ORJLFDO�IRUPXODH� )RUPXOD )�FDQ�EH�VLPSOLILHG�WR�IDOVH� QR�PDWWHU�ZKDW�WKH�YDOXHV�RI x� y� DQG z DUH�

VLPSOLI\�)�� ORJLF�� VLPSOLI\�)�� ORJLF�

¬x ∧ y ∧ z, )$/6(

$���

Page 485: Mupad Tutorial

:

'RFXPHQWDWLRQ�DQG�5HIHUHQFHV

<RX�FDQ�ILQG�D�OLVW�RI�DOO�GRFXPHQWV�WKDW�DUH�DYDLODEOH�LQ�\RXU�LQVWDOODWLRQ�E\FKRRVLQJ�³%URZVH�+HOS´�IURP�WKH�³+HOS´�PHQX�RI�DQ\�0X3$'� QRWHERRN�ZLQGRZDQG�WKHQ�FOLFNLQJ�RQ�³&RQWHQWV´�

7KH�0X3$' 4XLFN�5HIHUHQFH�OLVWV�DOO�0X3$' GDWD�W\SHV� IXQFWLRQV� DQG�OLEUDULHV�DQG�SURYLGHV�D�VXUYH\�RI�LWV�IXQFWLRQDOLW\�

<RX�DOVR�ILQG�OLQNV�WR�YDULRXV�OLEUDULHV�VXFK�DV� IRU�H[DPSOH� 'RP �WKH�OLEUDU\�IRUSUH�LQVWDOOHG�GDWD�W\SHV�� 7KH�FRUUHVSRQGLQJ�GRFXPHQWDWLRQ�FRQWDLQV�D�FRQFLVHGHVFULSWLRQ�RI�DOO�GRPDLQV�SURYLGHG�E\ 'RP� ,Q�D�0X3$' VHVVLRQ� WKH�FRPPDQG"'RP GLUHFWO\�RSHQV�WKLV�GRFXPHQW� 0RUHRYHU� \RX�FDQ�DFFHVV�WKH�GHVFULSWLRQ�RILQGLYLGXDO�GDWD�VWUXFWXUHV�IURP�WKLV�GRFXPHQW� VXFK�DV 'RP��0DWUL[� GLUHFWO\WKURXJK�WKH�FDOO "'RP��0DWUL[� $QRWKHU�H[DPSOH�LV�WKH�GRFXPHQWDWLRQ�IRU�WKHOLQDOJ SDFNDJH��OLQHDU�DOJHEUD�� ,W�FDQ�EH�UHTXHVWHG�GLUHFWO\�YLD "OLQDOJ�

$V�DQ�LQWURGXFWLRQ�WR�0X3$'��WDUJHWHG�HVSHFLDOO\�DW�:LQGRZV� XVHUV� ZHUHFRPPHQG�WKH�IROORZLQJ�ERRNV�

>0DM ��@ 0� 0DMHZVNL *HWWLQJ�6WDUWHG�ZLWK�0X3$'��6SULQJHU�+HLGHOEHUJ� �����,6%1 �������������

>0DM ��@ 0� 0DMHZVNL 0X3$'�3UR�&RPSXWLQJ�(VVHQWLDOV� 6HFRQG�(GLWLRQ�6SULQJHU�+HLGHOEHUJ� ������,6%1 �������������

,Q�DGGLWLRQ�WR�WKH�0X3$' GRFXPHQWDWLRQ� ZH�UHFRPPHQG�WKH�IROORZLQJ�ERRNVDERXW�FRPSXWHU�DOJHEUD�LQ�JHQHUDO�DQG�WKH�XQGHUO\LQJ�DOJRULWKPV�

>:HV ��@ 0��:HVWHU��HG��� &RPSXWHU�$OJHEUD�6\VWHPV��$ 3UDFWLFDO�*XLGH��:LOH\������

Page 486: Mupad Tutorial

; =h\nf^gmZmbhg�Zg]�K^_^k^g\^l

>**��@ -��YRQ�]XU�*DWKHQ�DQG�-��*HUKDUG� 0RGHUQ�&RPSXWHU�$OJHEUD�&DPEULGJH�8QLYHUVLW\�3UHVV� �����

>+HF ��@ $��+HFN� ,QWURGXFWLRQ�WR�0DSOH��6SULQJHU� �����

>'76��@ -�+��'DYHQSRUW� (��7RXUQLHU�DQG�<��6LUHW� &RPSXWHU�$OJHEUD� 6\VWHPVDQG�$OJRULWKPV�IRU�$OJHEUDLF�&RPSXWDWLRQ��$FDGHPLF�3UHVV� �����

>*&/��@ .�2��*HGGHV� 6�5��&]DSRU�DQG�*��/DEDKQ� $OJRULWKPV�IRU�&RPSXWHU$OJHEUD��.OXZHU� �����

%��

Page 487: Mupad Tutorial

;

Page 488: Mupad Tutorial

< @kZiab\l�@Zee^kr

*UDSKLFV�*DOOHU\

&��

Page 489: Mupad Tutorial

@kZiab\l�@Zee^kr

&��

Page 490: Mupad Tutorial

< @kZiab\l�@Zee^kr

&��

Page 491: Mupad Tutorial

@kZiab\l�@Zee^kr

&��

Page 492: Mupad Tutorial

< @kZiab\l�@Zee^kr

&��

Page 493: Mupad Tutorial

@kZiab\l�@Zee^kr

&��

Page 494: Mupad Tutorial

< @kZiab\l�@Zee^kr

&��

Page 495: Mupad Tutorial

@kZiab\l�@Zee^kr

&��

Page 496: Mupad Tutorial

< @kZiab\l�@Zee^kr

&���

Page 497: Mupad Tutorial

@kZiab\l�@Zee^kr

&���

Page 498: Mupad Tutorial

< @kZiab\l�@Zee^kr

&���

Page 499: Mupad Tutorial

@kZiab\l�@Zee^kr

&���

Page 500: Mupad Tutorial

< @kZiab\l�@Zee^kr

&���

Page 501: Mupad Tutorial

@kZiab\l�@Zee^kr

&���

Page 502: Mupad Tutorial

< @kZiab\l�@Zee^kr

&���

Page 503: Mupad Tutorial

@kZiab\l�@Zee^kr

&���

Page 504: Mupad Tutorial
Page 505: Mupad Tutorial

<

&RPPHQWV�RQ�WKH�*UDSKLFV*DOOHU\

2Q�WKH�FRORU�SDJHV�RI�WKLV�ERRN�\RX�ILQG�D�JDOOHU\�RI�SLFWXUHV�GHPRQVWUDWLQJ�WKHSRZHU�RI�WKH�0X3$'� JUDSKLFV� 7KHVH�SLFWXUHV�DUH�GLVFXVVHG�DW�YDULRXV�ORFDWLRQVLQ�WKLV�ERRN�DQG�WKH�RQOLQH SORW GRFXPHQWDWLRQ� UHVSHFWLYHO\� 7KHUH� IXUWKHUGHWDLOV�LQFOXGLQJ�WKH�0X3$'�FRPPDQGV�IRU�UHSURGXFLQJ�WKH�SLFWXUHV�FDQ�EHIRXQG�)LJXUH�� VKRZV�D�SORW�RI�VHYHUDO�IXQFWLRQV� 6LQJXODULWLHV�DUH�KLJKOLJKWHG�E\³YHUWLFDO�DV\PSWRWHV�´ 6HH�SDJH ����� )LJXUH�� VKRZV�D�IXQFWLRQ�SORW�WRJHWKHUZLWK�D�VSOLQH�LQWHUSRODWLRQ�WKURXJK�D�VHW�RI�VDPSOH�SRLQWV� 6HH�SDJH ������

)LJXUH�� GHPRQVWUDWHV�VRPH�OD\RXW�SRVVLELOLWLHV� 6HH�WKH�H[DPSOHV�RQ�WKH�KHOSSDJH�RI�WKH�JUDSKLFDO�DWWULEXWH /D\RXW LQ�WKH�RQOLQH SORW GRFXPHQWDWLRQ�

)LJXUH�� GHPRQVWUDWHV�WKH�FRQVWUXFWLRQ�RI�F\FORLGV�YLD�SRLQWV�IL[HG�WR�D�UROOLQJZKHHO� 6HH�SDJH ������

)LJXUH�� DQG )LJXUH�� GHPRQVWUDWH�KDWFKHG�DUHDV�EHWZHHQ�IXQFWLRQV�DQG�ZLWKLQFORVHG�FXUYHV� UHVSHFWLYHO\� 6HH�WKH�H[DPSOHV�RQ�WKH�KHOS�SDJH�RI SORW��+DWFK�

)LJXUH�� VKRZV�YDULRXV�VWDWLVWLFDO�GLVWULEXWLRQ�IXQFWLRQV�

)LJXUH�� VKRZV�DQ�LPSRUWHG�ELWPDS�LQVLGH�IXQFWLRQ�SORWV� 6HH�SDJH �������)LJXUH�� VKRZV�VRPH�IUDPHV�RI�DQ�DQLPDWLRQ�RI�WKH�SHUWXUEHG�RUELW�RI�D�VPDOOSODQHW�NLFNHG�RXW�RI�D�VRODU�V\VWHP�E\�D�JLDQW�SODQHW�DIWHU�D�QHDU�FROOLVLRQ� 7KHDQLPDWLRQ�LV�JHQHUDWHG�LQ�6HFWLRQ ”$QLPDWLRQV“�”([DPSOHV“ RI�WKH�RQOLQH SORWGRFXPHQWDWLRQ�

)LJXUH��� YLVXDOL]HV�WKH�YHFWRU�ILHOG v⃗(x, y) = (VLQ(3π y), VLQ(3π x))�

Page 506: Mupad Tutorial

= <hff^gml�hg�ma^�@kZiab\l�@Zee^kr

)LJXUH��� VKRZV�WKUHH�VROXWLRQ�FXUYHV�RI�DQ�2'( LQVLGH�WKH�GLUHFWLRQDO�YHFWRUILHOG�DVVRFLDWHG�ZLWK�WKH�2'(��6HH�WKH�H[DPSOHV�RQ�WKH�KHOS�SDJH�RISORW��9HFWRU)LHOG�G�

)LJXUH��� VKRZV�VHYHUDO�URWDWHG�FRSLHV�RI�D�IXQFWLRQ�JUDSK� 6HH�WKH�H[DPSOHV�RQWKH�KHOS�SDJH�RI SORW��5RWDWH�G�

)LJXUH��� VKRZV�WKH�0DQGHOEURW�VHW�WRJHWKHU�ZLWK�WZR�EORZ�XSV�RI�UHJLRQV�RIVSHFLDO�LQWHUHVW� 6HH�WKH�H[DPSOHV�RQ�WKH�KHOS�SDJH�RI SORW��'HQVLW\�

)LJXUH��� VKRZV�D�EDU�SORW�RI�VWDWLVWLFDO�GDWD� 6HH�WKH�H[DPSOHV�RQ�WKH�KHOS�SDJHRI SORW��%DUV�G�

)LJXUH��� VKRZV�WKH�LPDJH�RI�D�UHFWDQJOH�LQ�WKH�FRPSOH[�SODQH�XQGHU�WKH�PDSz → VLQ(z2)� 6HH�WKH�H[DPSOHV�RQ�WKH�KHOS�SDJH�RI SORW��&RQIRUPDO�

)LJXUH��� VKRZV�VRPH�HOOLSWLF�FXUYHV�JHQHUDWHG�DV�D�FRQWRXU�SORW� 6HH�WKHH[DPSOHV�RQ�WKH�KHOS�SDJH�RI SORW��,PSOLFLW�G�

)LJXUH��� VKRZV�WKH�)HLJHQEDXP�GLDJUDP�RI�WKH�ORJLVWLF�PDS� 6HH�WKH�H[DPSOHVRQ�WKH�KHOS�SDJH�RI SORW��3RLQW/LVW�G�

)LJXUH��� VKRZV�%HVVHO�IXQFWLRQV Jν(√x2 + y2) ZLWK ν = 0, 4, 8� 6HH�WKH

H[DPSOHV�RQ�WKH�KHOS�SDJH�RI SORW��)XQFWLRQ�G�

)LJXUH��� VKRZV��' IXQFWLRQ�SORWV�ZLWK�FRRUGLQDWH�JULG�OLQHV� 6HH�WKH�H[DPSOHVRQ�WKH�KHOS�SDJH�RI�WKH�JUDSKLFDO�DWWULEXWH *ULG9LVLEOH�

)LJXUH��� VKRZV�³.OHLQ¶V�ERWWOH´��D�IDPRXV�WRSRORJLFDO�REMHFW�� 7KLV�VXUIDFH�GRHVQRW�KDYH�DQ�RULHQWDWLRQ� WKHUH�LV�QR�³LQVLGH´�DQG�QR�³RXWVLGH´�RI�WKLV�REMHFW� 6HHWKH�H[DPSOHV�RQ�WKH�KHOS�SDJH�RI SORW��6XUIDFH�

)LJXUH���PRGHOV�YDULRXV�VQDLOV�XVLQJ SORW��6XUIDFH �E\�0DLNH�.UDPHU�-ND��

)LJXUH��� GHPRQVWUDWHV�WKH�UH�FRQVWUXFWLRQ�RI�DQ�REMHFW�ZLWK�URWDWLRQDOV\PPHWU\�IURP�PHDVXUHPHQWV�RI�LWV�UDGLXV�DW�YDULRXV�SRLQWV� 6HH�SDJH ������

)LJXUH��� VKRZV�WKH�VROXWLRQ�VHW�RI�WKH�HTXDWLRQ z2 = VLQ(z − x2 · y2) LQ��'��6HHWKH�H[DPSOHV�RQ�WKH�KHOS�SDJH�RI SORW��,PSOLFLW�G�

)LJXUH��� VKRZV�WKH�³/RUHQ]�DWWUDFWRU�´ 6HH�SDJH �������

'��

Page 507: Mupad Tutorial

Af\]p�� � � � � � � � � � � � � � � � � � � � � � � � � � VHH IDFW� � � � � � � � � � � � � � � � � � � � � � � � VHH VWULQJV� � � � � � � � � � � � � � � � � � � � � � � � ����� ���� � � � � � � � � � � � � � � � � � � � ����� ����� ���µ � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � � � � VHH BPXOW� � � � � � � � � � � � � � � � � � � � � � � � � VHH BSOXV� � � � � � � � � � � VHH BQHJDWH DQG BVXEWUDFW��! � � � � � � � � � � � � � � � � � � � � � ����� �����! � � � � ����� ����� ����� ����� ����� �����

����� ����� ����� ���� ���� ���� ���������� ����� ����� ����� ������ ������$��� $���� $���� $���� $���±$����

$���� � � � � � � � � VHH BFRQFDW DQG FRQFDWHQDWLRQ�� � � � � � � � � � � VHH H[SUHVVLRQV� UDQJH ������� � � � � � � � � � � � � � � � � � � � � � � � VHH KXOO� � � � � � � � � � � � � � � � � � � � � � � VHH BGLYLGH� ���� � � � � � � � � � � � � � � VHH FRPPHQWV�� � � � � � � � � � � � � � � � � � � � VHH FRPPHQWV� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� � � � � � � � � � � � � � � � � � � � VHH DVVLJQPHQW� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���� � � � � � � � � � � � � � � � � � � � � � � � � VHH BOHVV� � � � � � � � � � � � � � � � � � � � � VHH BOHHTXDO�! � � � � � � � � � � � � � � � � � � � � VHH BXQHTXDO � � � � � � � � � � � � � � � � � � � � � � � VHH BHTXDO! � � � � � � � � � � � � � � � � � � � � � � � � VHH BOHVV! � � � � � � � � � � � � � � � � � � � � VHH BOHHTXDO" � � � � � � � � � � � � � � � � � � � � � � � � � VHH KHOS# � ����� ����� ����� ���� $���� $���� $���## � � � � � � � � � � � � � � � � � ����� ����� ����� � � � � ����±����� ����� ����� ����� �����

����� ���� ���� ����±����� ����������� $��� $��� $��±$����

$���±$���� $���� $���� $���±$���� � � � � ���� ����� ����� ����� ����� ���� ����

���� ���� ����� ����� ����� $��� $����$���� $���� $���� $���� $���±$����

$���� $���A � � � � � � � � � � � � � � � � � � � � � � � VHH BSRZHU

BDQG � � � � � � � � � � � � � � � � � � � � � � VHH DQGBDVVLJQ � � � � � � � � � � � � � � � � � � ���� $��BFRQFDW � � � ����� ����� ����� ����� $���BFRQFDW � � � � � � � � � � � � � � � � � � � � � ����BGLYLGH � � � � � � � � � � � � � ����� ����� ����BHTXDO � � � � � � � � � � � � � � ����� ����� ����BH[SUVHT � � � � � � � � � � � � � � � � � ����� ����BIFRQFDW � � � � � � � � � � � � � � � � � � � � � VHH #BIQHVW � � � � � � � � � � � � � � � � � � � � � VHH ##BIQHVW � � � � � � � � � � � � � � � � � � � � � � � ����BIRU � � � � � � � � � � � � � � � � � � � � � � � � � ����BLI � � � � � � � � � � � � � � � � � � � � � � � � � � ����BLQGH[ � � � � � � � � � � � � � � � � � � � � � � ����BLQWHUVHFW � � � � � � � � � � � � VHH LQWHUVHFWBOHHTXDO � � � � � � � � � � � � � � � � � � � � � � ����BOHVV � � � � � � � � � � ����� ����� ����� $���BPXOW � � ����� ����� ����� ����� ���� �����

$���BQHJDWH � � � � � � � � � � � � � � � � � � � � � ����BSOXV � ����� ����� ����� ����� ����� �����

����� ���� ����� ������ $���BSRZHU ����� ����� ����� ����� ����� �����

����BVHTJHQ � � � � � � � � � � � � ����� ����� ����BVXEWUDFW � � � � � � � � � � � ����� ����� $���BXQHTXDO � � � � � � � � � � � � � � � � � � � � � � ����BXQLRQ � � � � � � � � � � � � � � � � � � ����� ����

$DEEUHYLDWLRQ � � � � � � � � � � � VHH DVVLJQPHQWDEV � � � � � � � � � � � � � ����� ���� ����� $���DGGLWLRQ�WKHRUHPV � � � � � � � � � � � ����� ���DOJHEUDLF�VWUXFWXUHV � � � � � � � � � ����±����DOLDV � � � � � � � � � � � � � � � � � � � � � � � $���DQDPHV � � � � � � � � � � � � � ����� ����� $���DQG � � � � � � � � � � � � � � � � ����� ����� $���DQG � � � ����� ����� ����� ����� ������ $���

$���DQLPDWLRQV ����� ������ ������ �����±�����

H[DPSOHV � � � � � � � � � � � � �����±�����IUDPH�E\�IUDPH ∼ � � � � � � �����±�����

Page 508: Mupad Tutorial

Bg]^q

MRLQ ∼ � � � � � � � � � � � � � � � � � � � �����QXPEHU�RI�IUDPHV � � � � � � �����±�����SOD\LQJ ∼ � � � � � � � � � � � � � � � � � �����VLPSOH ∼ � � � � � � � � � � � � �����±�����VWDUW � � � � � � � � � � � � � � � � � � � � � ����V\QFKURQL]DWLRQ�RI ∼ � � � �����±�����WLPH�UDQJH�RI ∼ � � � � � � � �����±�����

DQQXOXV � � � � � � � � � � � � � � � � � � � � � � �����DSSUR[LPDWH�VROXWLRQ � � � � � � � � � VHH IORDWDUFFRV � � � � � � � � � � � � � � � � � � � � � � � ����DUFFRVK � � � � � � � � � � � � � � � � � � � � � � ����DUFVLQ � � � � � � � � � � � � � � � � � � � � � � � ����DUFVLQK � � � � � � � � � � � � � � � � � � � � � � ����DUFWDQ � � � � � � � � � � � � � � � � � � � � � � � ����DUFWDQK � � � � � � � � � � � � � � � � � � � � � � ����DUJ � � � � � � � � � � � � � � � � � � � � � � � � � ����DUJV � � � � � � � � � � � � � � ����� ������ �����DUUD\ � � � � � � � � � ����� ����� ����� $���DUUD\V � � � � � � � � � � � � � � � � � � � ����±����

0�WK�RSHUDQG � � � � � � � � � � � � � � ����GHOHWLQJ�HOHPHQWV � � � � � � � � � � � ����GLPHQVLRQ � � � � � � � � � � � � � � � � ����JHQHUDWLRQ�RI ∼ � � � � � � � � � � � � ����PDWUL[�PXOWLSOLFDWLRQ � � � � � � � � �����

DUURZ�RSHUDWRU���!� � � � � � � � � � � � � � ����DVVLJQ � � � ����� ���� $��� $��� $��� $���DVVLJQPHQW � � � � � � � � � � � � � � � � ����� ���

GHOHWH � � � � � � � � � � � � � � � � � � � � ���LQGH[HG � � � � � � � � � � � � � � � � � � ����VLPXOWDQHRXV ∼ � � � � � ���� ����� $��WR�VXEOLVW � � � � � � � � � � � � � � � � � ����

DVVXPH � ���� ����� ����� ����� ����� ����DVVXPH$OVR � � � � � � � � � � � � � � � � � � � ����DVVXPLQJ � � � � � � � � � � � � ����� ����� $���DVVXPLQJ$OVR � � � � � � � � � � � � � � � � � ����DVVXPSWLRQV�DERXW�SURSHUWLHV � � � � � � � �

� � � � � � � � � � � � � VHH DVVXPH DQG LVDV\PSWRWLF�H[SDQVLRQ �VHULHV� � � � � � ����DWRPV � � � � � � � � � � � � � � � � � � � � ���� ����DWWULEXWHV � � � � � � � VHH JUDSKLFV� DWWULEXWHV

%EDFNWLFN � � � � � � � � � � � � � � � � � � � � � � ���

EDWFK�PRGH � � � � � � � � � � � � ������±������%LWPDS � � � � � � � � � � � � � � � � � � � � � ������ERRO � � � � � � � � � � � � � � � � � � � � ����� $���%RROHDQ�H[SUHVVLRQV � � � � � � � � ����±����

HYDOXDWLRQ��ERRO� � � � � � � � � � � � ����EUDQFKLQJ��LI DQG FDVH� � � � � � � ����±����

UHWXUQ�YDOXH � � � � � � � � � � � ����� ����

&C � � � � VHH &B DQG GRPDLQ� 'RP��&RPSOH[&�&���DGGLWLRQV�WR�0X3$' � � � � � � � � ���&B � � � � � � � � � � � � � � � � � � � � � � � � � � � ���FDPHUDV�LQ��' � � � � � � � � � � � ������±������

DQLPDWHG � � � � � � � � � � � � � � � � ������FDQYDV � � � � � � � � � � � � � � � � � � � � � � � �����FDVH � � � � � � � � � � � � � � � � � � � � � � � � � ����FDVH�GLVWLQFWLRQ � � � � � � � � � VHH SLHFHZLVHFHLO � � � � � � � � � � � � � � � � � � � � � � � � � ���FKDUDFWHULVWLF�SRO\QRPLDO

�OLQDOJ��FKDUSRO\� � � � ����� $���&KHE\VKHY�SRO\QRPLDOV � � � � � � ����� $���χ2�WHVW �VWDWV��FV*2)7� � � � � � � � � � � ����FRHII � � � � � � � � � � � � � � � � � � � ����� ����FRLQ�WRVVHV � � � � � � � � � � � � � � � � � � � � ����FROOHFW � � � � � � � � � � � � � � � � � � � � � � ���FRORQ � � � � � � � � � � � � � � � � � � � � � � � � ���FRORUV � � � � � � � � � � � � � � � � � � �����±�����

ILQGLQJ ∼ E\�QDPH � � � � � � � � � � �����+69 ∼ � � � � � � � � � � � � � � �����±�����RSDFLW\ � � � � � � � � � � � � � � � � � � �����5*% ∼ � � � � � � � � � � � � � � �����±�����5*%D ∼ � � � � � � � � � � � � � � � � � � �����

FROXPQ�YHFWRUV � � � � � � � � � � � � ����� ����FRPELQH � � � � � � � � � � � � � ���� ����� $���FRPPHQWV � � � � � � � � � � � � � � � � � � � � � ����FRPSDULVRQV � � � � � � � � � � � � � � � � � � � ����FRPSRVLWLRQ�RSHUDWRU � � � � � � � � � � � � VHH #FRPSRVLWLRQ�RSHUDWRU��#� � � � � � � � � � ����FRPSXWDWLRQ

H[DFW � � � � VHH V\PEROLF FRPSXWDWLRQVK\EULG � � � � � � � � � � � � � � � � � � � � ���QXPHULFDO � � � � � � � � � � � � � � � � � �� � � � � � VHH QXPHULFDO FRPSXWDWLRQV

,QGH[��

Page 509: Mupad Tutorial

Bg]^q

V\PEROLF � VHH V\PEROLF FRPSXWDWLRQVFRPSXWHU�DOJHEUD � � � � � � � � � � � � � � � � ���FRQFDWHQDWLRQ��� DQG BFRQFDW�

RI�OLVWV � � � � � � � � � � � � � � � � � � � ����RI�PDWULFHV � � � � � � � � � � � � � � � � � ����RI�QDPHV � � � � � � � � � � � � � � � � � ����RI�VWULQJV � � � � � � � � � � � � � � � � � ����

FRQMXJDWH � � � � � � � � � � � � � � � � ����� ����FRQWDLQV � � � � � � � � ����� ����� ����� ����FRQWH[W � � � � � � � � � � � � � � � � � � � � � �����F\FORLG � � � � � � � � � � � � � � � � � � � � � � �����

'' � � � � � � � � � � � � � � � � � � � ����� ����� ���GDWD�W\SH � � � � � � � � � � � � � � � � � � � � � � ���GHEXJJHU � � � � � � � � � � � � � � � � � � � � � � ���GHFRPSRVH � � � � � � � � � � � � � VHH RSHUDQGVGHJUHH � � � � � � � � � � � � � � � � � � � � � � ����GHOD\HG�HYDOXDWLRQ��KROG� � � � � � � � � � � ���GHOHWH � � � ���� ���� ����� ���� ���� �����

����� ����� ����� ����� ����� ���������� ����� ���±���� ���� ����� ���������� ����� ����� ������ ������ ������

������ $���� $���� $���� $���GHQRP � � � � � � � � � � � � � � � � � � � � � � � � ���GHQRPLQDWRU��GHQRP� � � � � � � � � � � � � � ���GHULYDWLYH � � � � � � � � � � VHH GLIIHUHQWLDWLRQGHWHUPLQDQW��OLQDOJ��GHW� � � � � � � � � ����GLDJRQDO�PDWULFHV � � � � � � � � � � � � � � ����GLH � � � � � � � � � � � � � � � � � � � � � � � � � � ����GLII � � � ���� ����� ����� ����� ����� �����

����� ����� ����� ���� ���� ���� ���������±����� ����� ����� ����� ������������ ������ ������ ������ $����

$���� $���� $���GLIIHUHQWLDO�HTXDWLRQV��RGH� � � � � � � � � ����

LQLWLDO�DQG�ERXQGDU\�FRQGLWLRQV � � ����QXPHULFDO�VROXWLRQV � � � � � � � � � � �� � � � � � � � � � � � � � ����� ������ ������

GLIIHUHQWLDO�RSHUDWRU�� DQG '� � � � � � � � �� � � � � � � � � � � � � � � � ����� ����� ���

GLIIHUHQWLDWLRQ��GLII� � � � � � � � � � � ���±���KLJKHU�GHULYDWLYHV � � � � � � � � � � � � ���

SDUWLDO�GHULYDWLYHV � � � � � � � � � � � � ���VDPSOH�SURFHGXUH � � � � � �����±�����

',*,76 � � � ���� ���� ���� ���� ����� ������������ $��� $���

GLVFRQW � � � � � � � � � � � � � � � � � � � � � ����GLVFRQWLQXLWLHV��GLVFRQW� � � � � � � � � � ����GLVWULEXWLRQ�IXQFWLRQ � � � � � � � � � � � � ����GLY � � � � � � � � � � � � � � � � � � � � � � ���� ����GLYLGH � � � � � � � � � � � � � � � � � � � � � � ����'RGHFDKHGURQ � � � � � � � � � � � � � � � � � �����GRPDLQ

'RP��([SUHVVLRQ)LHOG � � � � � � ����'RP��)ORDW � � � � � � � � � � � � � � � ����'RP��)ORDW,9 � � � � � � � � � � � � � ����'RP��,PDJH6HW � � � � � � � � � � � � � ���'RP��,QWHJHU � � � � � � � � � � � � � ����'RP��,QWHJHU0RG � � � � � � ����� $���'RP��,QWHUYDO � � � � � � � � � � � � � ����'RP��0DWUL[ � � � � � � � � � � � � � � ����'RP��0XOWLVHW � � � � � � � ����� $���'RP��5DWLRQDO � � � � � � � � � � � � ����'RP��5HDO � � � � � � � � � � � � � � � � ����'RP��6TXDUH0DWUL[ � � � � ����� ����'20B$55$< � � � � � � � � � � � � � � � � ����'20B%22/ � � � � � � � � � � � � � � � � � ����'20B&203/(; � � � � � � � � � � � � � � � ���'20B(;35 � � � � � � � � � � � � � � � � � � ����'20B)/2$7 � � � � � � � � � � � � � � � � � ���'20B)81&B(19 � � � � � � � � � � � � � �����'20B,'(17 � � � � � � � � � � � � � � � � � ���'20B,17 � � � � � � � � � � � � � � � � � � � ���'20B,17(59$/ � � � � � � � � � � � � � � ����'20B18// � � � � � � � � � � � � � � � � � ����'20B32/< � � � � � � � � � � � � � � � � � ����'20B352& � � � � � � � � � � � � � � � � � � ����'20B5$7 � � � � � � � � � � � � � � � � � � � ���'20B6(7 � � � � � � � � � � � � � � � � � � ����'20B675,1* � � � � � � � � � � � � � � � ����'20B7$%/( � � � � � � � � � � � � � � � � ����'20B9$5 � � � � � � � � � � � � � � � � � � �����SLHFHZLVH � � � � � � � � � � � � � � � � ����UHFWIRUP � � � � � � � � � � � � � � � � � � ����6HULHV��3XLVHX[ � � � � � � � � � � � ����

,QGH[��

Page 510: Mupad Tutorial

Bg]^q

VROYHOLE��%DVLF6HW � � � � � � � � � ���7\SH � � � � � � � � � � � � � � � ����� �����

GRPDLQ�W\SH � � � � � � � � � � � � � � � � � � � ���GHILQLQJ�\RXU�RZQ ∼ � � � � � � � � � � ���GHWHUPLQLQJ�WKH ∼ � � � � � VHH GRPW\SH

GRPW\SH � ���� ���� ���� ����� ����� ���������� ����� ���� ���� ���±����� �����

����� ����� ����� ����� ����� ���������� ����� ����� ������ ������ $����

$���� $���

(( � � � � � � � � � � � � � � � � � � � � � � � � � ���� ���e � � � � � � � � � � � � � � � � � � � � � � � � � � � VHH (HLJHQYDOXHV

QXPHULFDO � � � � � � � � � � � � � � � � � �� � � � � � � VHH QXPHULF��HLJHQYDOXHV

V\PEROLF � � VHH OLQDOJ��HLJHQYDOXHVHLJHQYHFWRUV

QXPHULFDO � � � � � � � � � � � � � � � � � �� � � � � � VHH QXPHULF��HLJHQYHFWRUV

V\PEROLF � VHH OLQDOJ��HLJHQYHFWRUVHOOLSWLF�LQWHJUDOV � � � � � � � � � � � �����±�����HQYLURQPHQW�YDULDEOHV � � � � � � � � � � � � ���

',*,76 � � � � � � � � � � � � � � � � � � � � ���+,6725< � � � � � � � � � � � � � � � � � � � ����/(9(/ � � � � � � � � � � � � � � � � � � � � ���0$;'(37+ � � � � � � � � � � � � � � � � � � ����0$;/(9(/ � � � � � � � � � � � � � � � � � � ���25'(5 � � � � � � � � � � � � � � � � � � � ����35(77<35,17 � � � � � � � � � � � � � � � ����UHLQLWLDOL]DWLRQ��UHVHW��� � ���� �����7(;7:,'7+ � � � � � � � � � � � � � � � � � ����

HSV � � � � � � � � � � � � � � � � � � � � � � � � ������HTXDWLRQV � � � � � � � � � � � � � � � � � � � � � ����

DVVLJQLQJ�VROXWLRQV �DVVLJQ� � � � � ���GLIIHUHQWLDO ∼ �RGH� � � � � � � � � � � � ����JHQHUDO ∼ � � � � � � � � � � � � � � � � � � ���LQWHJHU�VROXWLRQV � � � � � � � � � � � � ���OLQHDU ∼ � � � � � � � � � � � � � � � � � � � ���QXPHULFDO�VROXWLRQV � � � � � � � � � � ���

QXPHULF��UHDOURRWV � � � � � � � ���QXPHULF��IVROYH � � � � � � � � � � ���

QXPHULF��VROYH � � � � � � � � � � � ���VHDUFK�UDQJH � � � � � � � � � � � � � ����

SDUDPHWULF ∼ � � � � � � � � � � � � � � ����SRO\QRPLDO ∼ � � � � � � � � � � � � � � � ���UDWLRQDO�VROXWLRQV � � � � � � � � � � � � ���UHDO�VROXWLRQV � � � � � � � � � � � � � � � ���UHFXUUHQFH ∼ �UHF� � � � � � � � � � � � ����VROYLQJ��VROYH� � � � � � � � � � � ���±����VXUYH\�RI�DOO�VROYHUV��"VROYHUV� � � �� � � � � � � � � � � � � � � � � � � � � ���� ���

HVFDSH � � � � � � � � � � � � � � � � � ����� �����(XFOLGHDQ�$OJRULWKP � � � � � � � � � � � � �����HYDO � � � � � � � � � � ���� ���� ���� ����� ����HYDO$W � � � � � � � � � � � � � � ����� ����� ���HYDOS � � � � � � � � � � � � � � � � � � � � � � � ����HYDOXDWLRQ � � � � � � � � � � � � � � � � � � ���±���

FRPSOHWH ∼ � � � � � � � � � � � � � ���±���GHOD\HG ∼ �KROG� � � � � � � � � � � � � ���HQIRUFLQJ ∼ �HYDO� � � � � � � � � ���� ���∼ WUHH � � � � � � � � � � � � � � � � � � � � ���LQFRPSOHWH � � � � � � � � � � � � � � � � � ���LQYRNH � � � � � � � � � � � � � � � � � � � � ���OHYHO��/(9(/� � � � � � � � � � � � � � � � ���PD[LPDO�GHSWK �0$;/(9(/� � � � � � ���RI�DUUD\V � � � � � � � � � � � � � � � � � � ���RI�PDWULFHV � � � � � � � � � � � � � � � � � ���RI�SRO\QRPLDOV � � � � � � � � � � � � � � ���RI�WDEOHV � � � � � � � � � � � � � � � � � � � ���XS�WR�D�SDUWLFXODU�OHYHO �OHYHO� � � ���ZLWKLQ�SURFHGXUHV � � � � � � � � � � � ���

HYDOXDWRU � � � � � � � � � � � � � � � � � � � � � � ���H[DFW�FDOFXODWLRQV � � � � � � � � � � � � � � � � ���H[DPSOHV

WUDQVIHU ∼ WR�QRWHERRN � � � � � � � � ���H[S � � � � � � � � � � � � � � ���� ����� ����� ���H[SDQG � � ����� ����� ����� ����� ���� ����

$���H[SRQHQWLDO�IXQFWLRQ��H[S� � � � � � � � � � ���

IRU�PDWULFHV � � � � � � ������ ����� ����H[SRQHQWLDWLRQ��A� � � � � � � � � � � VHH BSRZHUH[SRUWLQJ�JUDSKLFV � � � � � � � ������±������

EDWFK�PRGH � � � � � � � � ������±������LQWHUDFWLYH � � � � � � � � � � � � � � � ������

,QGH[��

Page 511: Mupad Tutorial

Bg]^q

H[SRVH � � � � � � � � � � � � � � ���� ���� �����([SU � � � � � � � � � � � � � � � � � � � � � � � � ����H[SU � � � � � � � � � � � ����� ����� ����� ����H[SU�WH[W � � � ����� ����� ����� ����� �����

$���� $���H[SUHVVLRQV � � � � � � � � � � � � � � � ����±����

0�WK�RSHUDQG � � � � � � � � � � � � � � � ����DWRPV � � � � � � � � � � � � � � � � � � � ����%RROHDQ ∼ � � � � � � � � � � � � ����±����HYDOXDWLRQ��ERRO� � � � � � � � � � ����

FRPSDULVRQV � � � � � � � � � � � � � � � ����HTXDWLRQV � � � � � � � � � � � � � � � � � � ����H[SUHVVLRQ�WUHHV � � � � � � � � ����±����IDFWRULDO��IDFW� � � � � � � � � � � � � � � ����JHQHUDWLRQ�YLD�RSHUDWRUV � � ����±����LQGHWHUPLQDWHV��LQGHWV� � � � � � � ����LQHTXDOLWLHV � � � � � � � � � � � � � � � � ����PDQLSXODWLRQ � � � � � � � � � � � ���±����RSHUDQGV��RS� � � � � � � � � � � ����±����TXRWLHQW�PRGXOR��GLY� � � � � � � � � ����UDQJH����� � � � � � � � � � � � � � � � � � ����UHPDLQGHU�PRGXOR��PRG� � � � � � � � ����UHGHILQLWLRQ � � � � � � � � � � � � � � ����

VHTXHQFH�JHQHUDWRU���� � � � � � � � ����VLPSOLILFDWLRQ � � � � � � � � � � � � � � � �� VHH�DOVR FRPELQH� QRUPDO� UDGVLPS�VLPSOLI\� DQG 6LPSOLI\� ����±����

WUDQVIRUPDWLRQ � � � � VHH�DOVR FROOHFW�FRPELQH� IDFWRU� QRUPDO� SDUWIUDF�

UHFWIRUP� UHZULWH� ���±����

)IDFW � � � � � � � � � � � � ����� ���� ����� ����IDFWRU � � � ����� ���� ����� ����� ���� ����

$���IDFWRULDO � � � � � � � � � � � � � � � � � � � VHH IDFW)$,/ � � � � � � � � � � � � � � � � ����� ����� ����)$/6( � � � � � � � � � � � � � � � � � � � � � � � ����)HUPDW�QXPEHUV � � � � � � � � � � � ����� $��)LERQDFFL�QXPEHUV � � � � � � � � � � � � � � �

� � � � � ���� ����� �����±������ $���ILHOG�H[WHQVLRQ

�'RP��$OJHEUDLF([WHQVLRQ� � � ����

ILHOGV � � � � � � � � � � � � � � � � � � � � ����±����ILOHV

UHDGLQJ ∼ �UHDG� � � � � � � � � � � � � ����UHDGLQJ�GDWD �LPSRUW��UHDGGDWD� �� � � � � � � � � � � � � � � � � � � � � � � � ����

ZULWLQJ ∼ �ZULWH� � � � � � � � � � � � � ����IORDW ���� ���� ���� ����� ���� ����� �����

����� ����� ���� ���� ���� ����� ���������� ������ ������ ������ $��� $����

$���� $���� $���±$���IORDWLQJ�SRLQW � VHH QXPHULFDO FRPSXWDWLRQVIORDWLQJ�SRLQW�DUUD\V � � � � � � � � ����±����IORDWLQJ�SRLQW�LQWHUYDOV � � � � � � � ����±����IORRU � � � � � � � � � � � � � � � � � � � � ���� $��IRQWV � � � � � � � � � � � � � � � � � ������±������IRU � � � � � � � � � � � � � � � � � ����� ����� $��IRUPXOD�PDQLSXODWLRQ � � � � � � � � � � � � � ���)RXULHU�H[SDQVLRQ � � � � � � � � � � � � � � � ����IUDF � � � � � � � � � � � � � � � � � � � � � � � � � ���IUDQGRP � � � � � � � � � � � � � � � � � � � � � ����IUHTXHQFLHV��'RP��0XOWLVHW� � ����� $���IXQFHQY � � � � � � � � � � � � � � � � ������ $���IXQFWLRQ�HQYLURQPHQWV � � � � � � �����±�����

IXQFWLRQ�DWWULEXWHV � � � � � � � � � � �����JHQHUDWLRQ � � � � � � � � � � � VHH IXQFHQYRSHUDQGV � � � � � � � � � � � � � � � � � �����VRXUFH�FRGH��H[SRVH� � � � � � � � � � ���

IXQFWLRQV � � � � � � � � � � � � � ����� ����±����DV�SURFHGXUHV � � � � � � � � � � ����±�����FRPSRVLWLRQ � � � � � � � � � � � � � � � VHH #FRQVWDQW ∼ � � � � � � � � � � � � � � � � ����FRQYHUWLQJ�H[SUHVVLRQV�WR ∼ ���!� �� � � � � � � � � � � � � � � � � � � � � � � � ����

H[WUHPD � � � � � � � � � � � � � � � � � � ����IXQFWLRQDO�H[SUHVVLRQV � � � � � � � ����JHQHUDWLRQ����!� � � � � � � � � � � � � ����JHQHUDWLRQ���!� � � � � � � � � � � � � ����LGHQWLW\�IXQFWLRQ �LG� � � � � � � � � ����LWHUDWHG�FRPSRVLWLRQ � � � � � � � � VHH ##LWHUDWHG�FRPSRVLWLRQ��##� � � � � � ����NHUQHO ∼ � � � � � � � � � � � � � � � � � � ���QXPEHUV�DV ∼ � � � � � � � � � � � � � � ����URRWV � � � � � � � � � � � � � � � � � � � � ����

,QGH[��

Page 512: Mupad Tutorial

Bg]^q

*JFG � � � � � � � � � � � � � � � � � � � � ����� $���JHQHUDO�SORW�SULQFLSOHV � � � � � � �����±�����JHQHUDWH � � � � � � � � � � � � � � � � � � � � � � ����JHQLGHQW � � � � � � � � � � � � � � � � ����� �����JHRPHWULF�VHULHV � � � � � � � � � � � � � � � � ����JHWSURS � � � � � � � � � � � � ����� ����� ����JOREDO�YDULDEOHV � � � � � � � � � � � � � � � � � ����*ROGEDFK�FRQMHFWXUH � � � � � � � � � � � � ����JUDSKLFDO�WUHHV � � � � � � � � � � � � �����±�����JUDSKLFV � � � � � � � � � � � � � � � � ����±������

DQLPDWLRQV � � � � � � � � � VHH DQLPDWLRQVDWWULEXWHV � � � � � � � � � � � � �����±�����

$GDSWLYH0HVK � � � ������ ������ �����$[HV � � � � � � � � ������ ������ �����$[HV7LWOH)RQW � � � � � � � � � � ������$[HV7LWOHV � � � � � � � � ������ �����$[HV9LVLEOH � � � � � � � ������ �����%RUGHU:LGWK � � � � � � � � � � � � �����&DPHUD'LUHFWLRQ � � � � � � � � ������FKDQJH ∼ LQ�H[LVWLQJ�REMHFWV � �����&RORU � � � � � � � � � � � � � ������ �����&RORUV � � � ����� ������ ������ �����&RRUGLQDWH7\SH � � � � � � � � � � �����GHIDXOW�YDOXHV � � � � � � � �����±�����)LOO&RORU � � � � � � � � � � � � � � �����)LOO&RORU)XQFWLRQ � � � � � � � �����)LOO&RORU7\SH � � � � � � � � � � � �����)LOOHG � � � � � � � � � � � � � � � � � �����)RRWHU � ������ ������ ������ ������)RRWHU)RQW � � � � � � ������� ������)UDPHV � � � � � � � � � � � � � � � � � � �� � � � ������ ������ ������ ������ �����³IXOO\�TXDOLILHG´ � � � � � � � � � � �����*ULG9LVLEOH � � � � � � � � � � � � � �� � � � � � � � � ������ ������ ������ �����+HDGHU � � ������ ������ ������ ������

������ ������+HDGHU)RQW � � � � � � ������� ������+HLJKW � � � � � � � ������ ������ �����KHOS�SDJHV�IRU ∼ � � � � � �����±�����KLQWV � � � � � � � � � � � � � �����±�����LQKHULWDQFH�RI ∼ � � � � � �����±�����

/D\RXW � � � � � � � � � � � � � � � � ������/HJHQG � � � � � � � � � � ������� ������/HJHQG$OLJQPHQW � � � � � � � � ������/HJHQG(QWU\ � � � � � ������� ������/HJHQG)RQW � � � � � � ������� ������/HJHQG3ODFHPHQW � � � � � � � � ������/HJHQG7H[W � � � � � � ������� ������/HJHQG9LVLEOH � � � � � � � � � � � � �� � � � � � � ������ ������ ������� ������/LQH&RORU � � � � � � � � � � � � � � �����/LQH&RORU)XQFWLRQ � � � � � � � �����/LQH&RORU7\SH � � � � � � � � � � � �����/LQHV9LVLEOH � � � � � � � ������ �����0HVK � � � � � � � � � ������ ������ �����1DPH � � � � � � � � � � � � � � � � � ������3RLQW&RORU � � � � � � � � � � � � � �����6FDOLQJ � � � � � � ������ ������ ������VSHOOLQJ � � � � � � � � � � � � � � � � �����6XEJULG9LVLEOH � � � � � � � � � � � �� � � � � � � � � ������ ������ ������ �����6XEPHVK � � � � � � � � � � � ������ �����7H[W)RQW � � � � � � � � � � � � � � ������7LFNV/DEHO)RQW � � � � � � � � � ������7LFNV1XPEHU � � � � � � � ������ �����7LPH%HJLQ � � � � � � � � � ������ �����7LPH(QG � � � � � � � � � � � ������ �����7LPH5DQJH � � � � � � � � � ������ �����7LWOH � � � � � � � � ������ ������ �����7LWOH)RQW � � � � � � � � � � � � � ������7LWOH3RVLWLRQ � � � � � � ������ �����W\SH�VSHFLILF ∼ � � � � � � � � � � � �����9LHZLQJ%R[<5DQJH � � � � ������ �����9LHZLQJ%R[=5DQJH � � � ������ �����9LVLEOH$IWHU � � � � � � � � � � � � �����9LVLEOH$IWHU(QG � � � � � � � � � �����9LVLEOH%HIRUH � � � � � � � � � � � �����9LVLEOH%HIRUH%HJLQ � � � � � � �����9LVLEOH)URP7R � � � � � � � � � � � �����:LGWK � � � � � � � � ������ ������ �����;7LFNV%HWZHHQ � � � � � � � � � � � �����;7LFNV'LVWDQFH � � � � � � � � � � �����<7LFNV%HWZHHQ � � � � � � � � � � � �����<7LFNV'LVWDQFH � � � � � � � � � � �����

,QGH[��

Page 513: Mupad Tutorial

Bg]^q

EDWFK�PRGH � � � � � � � � ������±������FDPHUDV�LQ��' � � � � � � � ������±������DQLPDWHG � � � � � � � � � � � � � � ������

FDQYDV � � � � � � � � � � � � � � � � � � � �����FDSWLRQ � � � � � � � � � � � � � � � � � � � �� � � � VHH JUDSKLFV� DWWULEXWHV� )RRWHU

FRORUV � � � � � � � � � � � � � � � � VHH FRORUVHYDOXDWLRQ�PHVK � � � � � � VHH JUDSKLFV�

DWWULEXWHV� 0HVK XQG 6XEPHVKHYDOXDWLRQ�PHVK� DGDSWLYH � � � � � � �� � � � � � � � � � VHH JUDSKLFV� DWWULEXWHV�

$GDSWLYH0HVKIRQWV � � � � � � � � � � � � � ������±������IRRWHU � � � � � � � � � � � � � � � � � � � � �� � � � VHH JUDSKLFV� DWWULEXWHV� )RRWHU

JHQHUDO�SORW�SULQFLSOHV � � �����±�����JUDSKLFDO�WUHHV � � � � � � � � �����±�����LPSOLFLW ∼ � � � � � � � � � � � � � � � �����

JUDSKV�RI�IXQFWLRQV��SORW� � ����� ����JDSV�LQ�GHILQLWLRQ � � � � � ����� �����SLHFHZLVH�GHILQHG�IXQFWLRQV � � � �� � � � � � � � � � � � � � � � � � � ����� �����VLQJXODULWLHV � � � � � � � � ������ �����

JULG�OLQHV � � � VHH JUDSKLFV� DWWULEXWHV�*ULG9LVLEOH DQG 6XEJULG9LVLEOH

JURXSV�RI�SULPLWLYHV � � � � � � � � � � �� � � � � � � � � � � � � ������ �����±�����

KHDGHU � � � � � � � � � � � � � � � � � � � � �� � � � VHH JUDSKLFV� DWWULEXWHV� +HDGHU

+69 FRORUV � � � � � � � � � � � �����±�����LPDJH�VL]H � � VHH JUDSKLFV� DWWULEXWHV�

+HLJKW DQG :LGWKLPSRUWLQJ�SLFWXUHV � � � � ������±������OHJHQGV � � � � � � � � � � � ������±������OLJKW�VRXUFHV � � � � � � � � � � � � � � �����ORJDULWKPLF�SORWV � � � � � � � � � � � �����REMHFW�EURZVHU � � � � � � � � �����±�����2SHQ*/GULYHUV � � � � � � � � � � � � � � � � ������

3ODWR¶V�SRO\KHGUD � � � � � � � � � � � �����SOD\LQJ�DQLPDWLRQV � � � � � � � � � � �����SULPLWLYHV �����±������ VHH�DOVR SORW

�JUDSKLFV OLEUDU\�

SURSHUW\�LQVSHFWRU � � � � � �����±�����5*% FRORUV � � � � � � � � � � �����±�����VDYLQJ�DQG�H[SRUWLQJ � ������±������EDWFK�PRGH � � � � � � ������±������LQWHUDFWLYH � � � � � � � � � � � � � ������

6LHUSLQVNL�WULDQJOH � � � � � ������ $���VL]H�RI�LPDJH VHH JUDSKLFV� DWWULEXWHV�

+HLJKW DQG :LGWKWLWOH��SHU�REMHFW� � � � � � � � � � � � � � �� � � � � VHH JUDSKLFV� DWWULEXWHV� 7LWOH

WLWOH�SRVLWLRQ VHH JUDSKLFV� DWWULEXWHV�7LWOH3RVLWLRQ

WUDQVIRUPDWLRQ � � � � � � � � �����±�����WUDQVIRUPDWLRQV � � � � � � � � � � � � �����

JUDSKV�RI�IXQFWLRQV��SORW� � � � � � � � � � ����JUHDWHVW�FRPPRQ�GLYLVRU VHH JFG DQG LJFG*U|EQHU�EDVHV � � � � � � � � � � � � � VHH OLEUDU\JURXSV�RI�SULPLWLYHV � � � � � � � � �����±�����

+KDUGZDUH�IORDW�DUUD\V � � � � � � � � ����±����KDV � � � � � � � � � � � ����� ����� ������ $���KHOS � � � � � � � � � � � � � � � � � � � � � � ���� %��KHOS � � � � � � � � � � � � � � � � � � � � � ���� ���+H[DKHGURQ � � � � � � � � � � � � � � � � � � �����+LOEHUW�PDWUL[��OLQDOJ��KLOEHUW� � � � �

� � � � � � � � � � ����� ����� ����� ����LQYHUVH ∼ �OLQDOJ��LQYKLOEHUW� �� � � � � � � � � � � � � � � � � � � � � � � � ����

KLQW � � � � � � VHH JUDSKLFV� DWWULEXWHV� KLQWV+,6725< � � � � � � � � � � � � � � � � � � � � � � ����KLVWRU\��� DQG ODVW� � � � � � � � � � ����±����

HYDOXDWLRQ�RI ODVW � � � � � � � � � � � ����KROG � � � ���� ���� ���� ���� ���� ���� �����

����� ����� ����� ������ ������ $����$���

GH�O¶+RVSLWDO � � � � � � � � � � � � � � � � � � � ���KXOO � � � � � � � � � � � � � � � ����� ����� ����K\SHUWH[W � � � � � � � � � � � � � � � � � � � � � ���

,QGH[��

Page 514: Mupad Tutorial

Bg]^q

,, � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���,FRVDKHGURQ � � � � � � � � � � � � � � � � � � �����LG � � � � � � � � � � � � � � � � � � � � � � � � � � ����LGHQWLILHUV � � � � � � � � � � � � � ����� ���±����

DVVLJQPHQW � � � � � � � � � � � � ���±����DVVXPSWLRQV � � � � VHH DVVXPH DQG LVFRQFDWHQDWLRQ � � � � � � � � � � � � � ����GHOHWLQJ�WKH�YDOXH � � � � � � � � � � � � ���G\QDPLFDO�JHQHUDWLRQ � � � � � � � � ����HYDOXDWLRQ � � � � � � � � � � � � � � � � � � ���JHQHUDWLRQ�YLD�VWULQJV � � � � � � � � � ����JUHHN�OHWWHUV � � � � � � � � � � � � � � � � ���OLVW��DQDPHV� � � � � � � � � � � � � � � � ����VSHFLDO�FKDUDFWHUV � � � � � � � � � � � � ���VXE��DQG�VXSHUVFULSWHG ∼ � � � � � � ���YDOXHV � � � � � � � � � � � � � � � � � � � � � ���ZLWK�DUELWUDU\�FKDUDFWHUV � � � � � � � ���ZULWH�SURWHFWLRQUHPRYLQJ ∼ �XQSURWHFW� � � � � ����VHWWLQJ ∼ �SURWHFW� � � � � � � � ����

LI � ����� ����� ���� ����� ����� ����� ���������±����� ����� ����� ������

�����±������ ������ ������ ������$���� $���� $���� $���� $���� $���

LIDFWRU � � � � � � � � � ���� ����� ����� ���LJFG � � � � � � � � � � � � � � ����� ������ $���,JQRUH6SHFLDO&DVHV � � � � � � � � � � � � � ����,P � � � � � � � � � � � � � � � � � � � � � � � ����� ���LPDJLQDU\�SDUW � � � � � � � � � � � � � � � � VHH ,PLPDJLQDU\�XQLW � � � � � � � � � � � � � � � � � � ���LPSRUW��UHDGELWPDS � � � � � � � � � � � ������LPSRUW��UHDGGDWD � � � � � � � � � ����� �����LPSRUWLQJ�SLFWXUHV � � � � � � � � ������±������LQ � � � � � � � � � � � � � � � � � � � � � � � ����� ���LQGHWHUPLQDWHV � � � � � � � � � � VHH LGHQWLILHUVLQGHWHUPLQDWHV��LQGHWV� � � � � � � � � � ����LQGHWV � � � � � � � � � ����� ���� $���� $���LQGH[HG�DFFHVV � � � � � � � � � � � � � � � � � ����LQHTXDOLWLHV � � � � � � � � � � � � � � � � � � � � ����

VROYLQJ ∼ � � � � � � � � � � � � � � � � � � ����LQILQLW\ � � � � ���� ����� ����� ����� �����

����� $���

URXQGHG ∼ �5'B,1)� 5'B1,1)� � � ����LQIR � � � � � � � � � � � � ���� ����� ������ ����LQSXW�DQG�RXWSXW � � � � � � � � � � � � ����±����LQW ���� ����� ����� ����� ����� ����� �����

����� ���� ���� ����� ����� ����� ���������� ������ ������ $���� $���� $���

LQWHJUDWLRQ � � � � � � � � � � � � � � � � � VHH LQWFKDQJH�RI�YDULDEOH � � � � � � � � � � � �� � � � � � � � � � VHH LQWOLE��FKDQJHYDU

QXPHULFDO ∼ � � � � � � � � � � � � � � � � �� � � � � � � � VHH QXPHULFDO LQWHJUDWLRQ

LQWHUPHGLDWH�UHVXOW � � � � � � � � � � � � � � ���LQWHUSRODWLRQ � � � � � � � � � � � � � � � � � � �����LQWHUVHFW � � � � � � � � � � � � � � � � ����� $���LQWHUVHFW � � � � � � � � � � � ����� ����� ����LQWHUYDO�DULWKPHWLF � � � � � � � � � � ����±����

XQLRQ� LQWHUVHFW � � � � � � � � � � � ����FRQYHUW�WR�LQWHUYDOV��LQWHUYDO� � ����IDoDGH�GRPDLQ��'RP��)ORDW,9� � ����JHQHUDWH�LQWHUYDOV������ KXOO� � � � �� � � � � � � � � � � � � � � � � � � ����� ����

LQWOLE��FKDQJHYDU � � � � � � � � � ���� $���,QW0RG � � � � � � � � � � � � � � � � � � � � � � ����LUUHGXFLEOH � � � � � � � � � � � � � ����� $���LV � � � � � � � � � � � � � � � � � � � � � � � ���� ����LVSULPH ���� ����� ����� ����� ���� �����

����� ����� $��� $���LV]HUR � � � � ����� ����� ����� ����� ����LWHUDWLRQ�RSHUDWRU � � � � � � � � � � � � � VHH ##LWHUDWLRQ�RSHUDWRU��##� � � � � � � ������ ����LWKSULPH � � � � � � � � � � � � � ����� ����� ����

--DYD9LHZ � � � � � � � � � � � � � � � � � � � ������MSJ � � � � � � � � � � � � � � � � � � � � � � � � ������

.NHUQHO � � � � � � � � � � � � � � � � � � � � � � � � ���

H[WHQGLQJ�WKH ∼ � � � � � � � � � � � � � ���NHUQHO�IXQFWLRQ � � � � � � � � � � � � � � � � � ���

,QGH[��

Page 515: Mupad Tutorial

Bg]^q

//DQGDX�V\PERO��2� � � � � � � � � � ����� $���ODVW � � � � � � � � � � � � � � � � � � � � � � � � VHH �ODVW

HYDOXDWLRQ�RI ∼ � � � � � � � � � � � � � � ����ODVW � � � � � � � � � � � � � � � � ���� ����� $���/DXUHQW�VHULHV � � � � � � � � � � � � � � � � � ����OHJHQGV � � � � � � � � � � � � � � � ������±������OHQJWK � � � � � � � � � � � � � ����� $���� $���/(9(/ � � � � � � � � � � � � � � � � � � � � � � � � ���OHYHO � � � � � � � � � � � � � � ���� ������ $���OLEUDU\ � � � � � � � � � � � � � � � � � � � � � ���±���

H[SRUWLQJ�D ∼ �XVH� � � � � � � � � � � ���IRU�FRORU�QDPHV �5*%� � � � � � � � � �����IRU�GDWD�VWUXFWXUHV �'RP� � � � � � � ����IRU�H[WHUQDO�IRUPDWV �JHQHUDWH� � � ����IRU�*U|EQHU�EDVHV �JURHEQHU� � � ����IRU�LQSXW��LPSRUW� � � � � � � � � � � � ����IRU�OLQHDU�DOJHEUD �OLQDOJ� � � � � ����IRU�QXPEHU�WKHRU\ �QXPOLE� � � � � �� � � � � � � � � � � � � � � � � � � � ����� ���

IRU�QXPHULFDO�DOJRULWKPV �QXPHULF�� � � � � � � � � � � � � � � � � � � � ���� ����

IRU�RUWKRJRQDO�SRO\QRPLDOV�RUWKSRO\� � � � � � � � � � � � � � � � $���IRU�VWDWLVWLFV �VWDWV� � � � � � � � � � ����IRU�VWULQJV �VWULQJOLE� � � � � � � � ����IRU�W\SH�VSHFLILHUV �7\SH� � � � � � � � �� � � � � � � � � � � � � � ����� ������ $���

LQIRUPDWLRQ�RQ�D ∼ �" DQG LQIR� � ���VWDQGDUG ∼ � � � � � � � � � � � � � � � � � ���

OLPLW � ���� ����� ����� ����� ������ $���RQH�VLGHG � � � � � � � � � � � � � � � � � ����

OLPLW�FRPSXWDWLRQ��OLPLW� � � � � � � � � ����OLQDOJ �OLEUDU\�IRU�OLQHDU�DOJHEUD�∼��FKDUSRO\ � � � � � � � � � ����� $���∼��GHW � � � � � � � � � � � � � � ����� $���∼��HLJHQYDOXHV � � ����� ����� $���∼��HLJHQYHFWRUV � � � � � � � � � � $���∼��LQYKLOEHUW � � � � � � � � � � � � ����∼��LV3RV'HI � � � � � � � � � � � � � � ����

OLQHDU�DOJHEUD��OLQDOJ� � � � � � � � � � � � ����OLQHIHHG � � � � � � � � � � � � � � � � � � � � � � � ���

OLVWV � � � � � � � � � � � � � � � � � � � � � ����±����DSSO\LQJ�D�IXQFWLRQ �PDS� � � � � � � ����FRPELQLQJ ∼ �]LS� � � � � � � � � � � ����HPSW\�OLVW � � � � � � � � � � � � � � � � � ����VHOHFWLQJ�DFFRUGLQJ�WR�SURSHUWLHV�VHOHFW� � � � � � � � � � � � � � � � � � ����VSOLWWLQJ�DFFRUGLQJ�WR�SURSHUWLHV�VSOLW� � � � � � � � � � � � � � � � � � � ����

OQ � � � � � � � � � � � � � � � � � � � � � � � � � � � ���ORFDO � � � � � � � � � � � � � � � � � � � � � � � � ����ORFDO�YDULDEOHV��ORFDO� � � � � � � � ����±�����

IRUPDO�SDUDPHWHUV � � � � � � � � � � �����XQLQLWLDOL]HG ∼ � � � � � � � � � � � � � �����

ORJ � � � � � � � � � � � � � � � � � � � � � � � � � ����ORJDULWKP��OQ� ORJ� � � � � � � � � � � ���� ����ORJLFDO�IRUPXOD � � � � � � � � � � � � � � � � �����ORQJ�DUURZ�RSHUDWRU����!� � � � � � � � � � ����ORRSV � � � � � � � � � � � � � � � � � � � � � ����±����

DERUWLQJ ∼ �EUHDN� � � � � � � � � � � � ����IRU � � � � � � � � � � � � � � � � � � � � � � ����UHSHDW � � � � � � � � � � � � � � � � � � � � ����UHWXUQ�YDOXH � � � � � � � � � � � � � � � � ����VNLSSLQJ�FRPPDQGV �QH[W� � � � � � ����ZKLOH � � � � � � � � � � � � � � � � � � � � ����

/RUHQ]�DWWUDFWRU � � � � � � � � � � � � � � � ������

0PDQLSXODWLQJ�H[SUHVVLRQV � � � � � � ���±����PDS � � � ����� ����� ����� ����� ����� �����

����� ����� ���� ���� ���� ����������� $��� $���� $���� $���� $����

$���� $���� $���� $���PDSFRHIIV � � � � � � � � � � � � � � � � � � � � ����PDSV � � � � � � � � � � � � � � � � � � VHH IXQFWLRQVPDWKHPDWLFDO�REMHFWV � � � � � � � � � � � � � � ���PDWULFHV � � � � � � � � � � � � � � � � � ����±����

FKDUDFWHULVWLF�SRO\QRPLDO�OLQDOJ��FKDUSRO\� � � � ����� $���FRPSXWLQJ�ZLWK ∼ � � � � � � ����±����GHIDXOW�FRHIILFLHQW�ULQJ�'RP��([SUHVVLRQ)LHOG� � � � � � ����GHWHUPLQDQW �OLQDOJ��GHW� � � � � ����GLDJRQDO ∼ � � � � � � � � � � � � � � � � ����

,QGH[��

Page 516: Mupad Tutorial

Bg]^q

HLJHQYDOXHVQXPHULFDO � � � � � � � � � � � � � � � �� � � � � � � VHH QXPHULF��HLJHQYDOXHVV\PEROLF � � � � � � � � � � � � � � � � �� � � � � � � � VHH OLQDOJ��HLJHQYDOXHV

HLJHQYHFWRUVQXPHULFDO � � � � � � � � � � � � � � � �� � � � � � VHH QXPHULF��HLJHQYHFWRUVV\PEROLF � � � � � � � � � � � � � � � � �� � � � � � � VHH OLQDOJ��HLJHQYHFWRUV

H[SRQHQWLDO�IXQFWLRQ �H[S� ����� ����H[SRQHQWLDO�IXQFWLRQ�QXPHULF��H[S0DWUL[� � � � � � � � ����IXQFWLRQDO�FDOFXOXV�QXPHULF��I0DWUL[� � � � � � � � � ����+LOEHUW�PDWUL[��OLQDOJ��KLOEHUW�� � � � � � � � � � � � � � � ����� ����� ����

LGHQWLW\ ∼ � � � � � � � � � � � � � � � � ����LQGH[HG�DFFHVV � � � � � � � � � � � � � ����LQLWLDOL]DWLRQ � � � � � � � � � � ����±����LQYHUVH � � � � � � � � � � � � � � ����� ����LQYHUVH�+LOEHUW�PDWUL[�OLQDOJ��LQYKLOEHUW� � � � � � � � ����OLEUDU\�IRU�OLQHDU�DOJHEUD �OLQDOJ� �� � � � � � � � � � � � � � � � � � � � � � � � ����

OLEUDU\�IRU�QXPHULFDO�DOJRULWKPV�QXPHULF� � � � � � � � � � � � � � � � � ����PHWKRGV � � � � � � � � � � � � � ����±����ULQJ�RI�VTXDUH ∼ � � � � � � � � � � � � ����VSDUVH ∼ � � � � � � � � � � � � � ����±����VXEPDWULFHV � � � � � � � � � � � � � � � ����7RHSOLW] ∼ � � � � � � � � � � � � � � � � ����

PDWUL[ � � � � ����� ����� ����� ����� ���������� $���� $���

PD[ � � � � � � � � � � � � � � � � ����� ����� �����0D['HJUHH � � � � � � � � � � � � � � � � � � � � � ���0$;'(37+ � � � � � � � � � � � � � � � � � ����� $���0$;/(9(/ � � � � � � � � � � � � � � � � � � � � � � ���PHDQ�YDOXH��VWDWV��PHDQ� � � � � � � � � ����PHPRU\�PDQDJHPHQW � � � � � � � � � � � � ���0HUVHQQH�SULPHV � � � � � � � � � � � ����� $��PLQ � � � � � � � � � � � � � � � � � � � � � � � � � ����PLQXV � � � � � � � � � � � � � � � � � � � � � � � ����

PRG � � � ����� ����� ����� ����� ����� ����������� $���� $���

UHGHILQH ∼ � � � � � � � � � � � � � � � � � ����PRGS � � � � � � � � � � � � � � � � � � � � � � � � � ����PRGV � � � � � � � � � � � � � � � � � � � � � � � � � ����PRGXODU�H[SRQHQWLDWLRQ �SRZHUPRG� � � ����PRGXOH � � � � � � � � � � � � � � � � � � � � � � � ���0RHELXV�VWULS � � � � � � � � � � � � � � � � � �����0RQWH�&DUOR�VLPXODWLRQ � � � � � � � � � � ����PXOWFRHIIV � � � � � � � � � � � � � � � � � � � ����PXOWLVHW � � � � � � � � � � � � � � � � � � � � � ����

11HZWRQ�LWHUDWLRQ � � � � � � � � � � ������ $���

QXPHULF��IVROYH � � � � � � � � � � � � ���QH[WSULPH � � � � � � � � � � � � � � � � ����±����1,/ � � � � � � � � ����� ����� ����� ����� $���QRSV � � ����±����� ����� ���� ����� �����

����� $���� $���� $���� $���� $���QRUP � � � � � � � � � � � � � � � � � � � � � � � � ����QRUPDO � ����� ����� ����� ����� ���� �����

����� $���� $���� $���QRUPDO�GLVWULEXWLRQ �VWDWV��QRUPDO&')�

� � � � � � � � � � � � � � � � � � � � � � � � ����QRW � � ����� ����� ����� ����� ����� ������

$��� $���QRWHERRN � � � � � � � � � � � � � � � � � � ���� ����QWHUPV � � � � � � � � � � � � � � � � � � � � � � ����QWKFRHII � � � � � � � � � � � � � � � � � � � � � ����QWKWHUP � � � � � � � � � � � � � � � � � � � � � ����QXOO � � � � � � � � � � � � � � � � � � � � � � � � � ����QXOO�REMHFWV � � � � � � � � � � � � � � ����±�����QXOO�� � � � � � � � � � � � � � � � � � ����� ����QXPEHU�WKHRU\��QXPOLE� � � � � � � � ����� ���QXPEHUV � � � � � � � � � � � � � � � � � � � ���±���

DEVROXWH�YDOXH � � � � � � � � � � � VHH DEVEDVLF�DULWKPHWLF � � � � � � � � � � � � � ���FRPSOH[ ∼ � � � ���� VHH�DOVR GRPDLQV�

'20B&203/(;FRPSOH[�FRQMXJDWLRQ � VHH FRQMXJDWHFRPSXWLQJ�ZLWK ∼ � � � � � � � � ���±���GHFLPDO�H[SDQVLRQ�QXPOLE��GHFLPDO� � � � � � � � � � � ���

,QGH[���

Page 517: Mupad Tutorial

Bg]^q

GHQRPLQDWRU � � � � � � � � � � � VHH GHQRPGRPDLQ�W\SHV � � � � � � � � � � � � � � � ���HYHQ�LQWHJHUV �7\SH��(YHQ� � � � � � ����IDFWRULDO � � � � � � � � � � � � � � � � VHH IDFWIORDWLQJ�SRLQW�DSSUR[LPDWLRQ � � � � �� � � � � � � � � � � � � � � � � � � � VHH IORDW

IUDFWLRQDO�SDUW � � � � � � � � � � � VHH IUDFJUHDWHVW�FRPPRQ�GLYLVRU � � � � VHH LJFGL�WK�SULPH��LWKSULPH� � � � � � � � � ����LPDJLQDU\�SDUW � � � � � � � � � � � � VHH ,PLQWHJHUV � � � � � � VHH GRPDLQV� '20B,17LQWHJUDO�SDUW � � � � � � � � � � � � VHH WUXQFi�WK�SULPH � � � � � � � � � � VHH LWKSULPHPRGXODU�H[SRQHQWLDWLRQ �SRZHUPRG�� � � � � � � � � � � � � � � � � � � � � � � � ����

QH[W�SULPH � � � � � � � � � VHH QH[WSULPHQXPHUDWRU � � � � � � � � � � � � � VHH QXPHURGG�LQWHJHUV �7\SH��2GG� � � � � � � ����RSHUDQGV � � � � � � � � � � � � � � � � � � ���RXWSXW�IRUPDW�RI�IORDWLQJ�SRLQWQXPEHUV �3UHI��IORDW)RUPDW� � ����SULPDOLW\�WHVW � � � � � � � � � VHH LVSULPHSULPH�IDFWRUL]DWLRQ � � � � � VHH LIDFWRUTXRWLHQW�PRGXOR��GLY� � � � � � � � � ���UDQGRP ∼ � � VHH UDQGRP DQG IUDQGRPUDWLRQDO ∼ � � � � VHH GRPDLQV� '20B5$7UHDO�DQG�LPDJLQDU\�SDUW VHH UHFWIRUPUHDO�SDUW � � � � � � � � � � � � � � � � � VHH 5HUHPDLQGHU�PRGXOR � � � � � � � � VHH PRGURXQGLQJ � � � � � � � � � � � � � � � � � � �� � � � � VHH FHLO� IORRU� URXQG� WUXQF

VLJQ � � � � � � � � � � � � � � � � � � � VHH VLJQVLPSOLILFDWLRQ�RI�UDGLFDOV �UDGVLPS�� � � � � � � � � � � � � � � � � � � ����� ����

VTXDUH�URRW � � � � � � � � � � � � � VHH VTUWW\SH�VSHFLILHU � � � � � � � � � � � � � � � ����

QXPHU � � � � � � � � � � � � � � � � � � � � � � � � ���QXPHUDWRU��QXPHU� � � � � � � � � � � � � � � � ���QXPHULF �QXPHULFV�OLEUDU\�∼��GHW � � � � � � � � � � � � � � � � � � ����∼��HLJHQYDOXHV � � � � � � � � � � � � �� � � � � � � � � � � ���� ����� ����� �����∼��HLJHQYHFWRUV � � � � � � � � � � ����

∼��H[S0DWUL[ � � � � � � � � � � � � � ����∼��I0DWUL[ � � � � � � � � � � � � � � � ����∼��IVROYH � � � � � � � � � � � � � � ���� ���∼��LQW � � � � � � � � � � � ���� ���� $���∼��LQYHUVH � � � � � � � � � � � � � � � ����∼��RGHVROYH � � � � � � � � � � � � � � � ����∼��TXDGUDWXUH � � � � � � � � � � � � � ���∼��UHDOURRWV � � � � � � ���� ���� ����∼��VLQJXODUYDOXHV � � � � � � � � � ����∼��VLQJXODUYHFWRUV � � � � � � � � ����∼��VROYH � � � � � � � � � � � � � � � � � ���

QXPHULFDO�FRPSXWDWLRQV � � � � � � � � � � � �� � � � � � � � � ���� ���� ���� ���� ����

SUHFLVLRQ � � � � � � � � � � � � � VHH ',*,76QXPHULFDO�LQWHJUDWLRQ � � � � ���� ���� $���QXPHULFDO�LQWHJUDWLRQ �QXPHULF��LQW DQG

QXPHULF��TXDGUDWXUH� � � � � � �����QXPOLE �OLEUDU\�IRU�QXPEHU�WKHRU\�∼��GHFLPDO � � � � � � � � � � � � � � � � ���∼��ILERQDFFL � � � � � � � � � � � � � �����∼��SULPHGLYLVRUV � � � � � � � � � ����∼��SURYHSULPH � � � � � � � � � � � � ����

22 � � � � � � � � � � � � � � � � � � � � � � ����� $���REMHFW�EURZVHU � � � � � � � � � � � � �����±�����REMHFW�RULHQWHG � � � � � � � � � � � � � � � � � ���2FWDKHGURQ � � � � � � � � � � � � � � � � � � � �����RGH � � � � � � � � � � � � � � � � � � � � ����� $���RS � � ���� ���±���� ����±����� ����� �����

����� ����� ����� ����� ����� ���������� ����� ���±���� ���� ���� ����������� ������ ������ $��� $��±$����$���±$���� $���� $���� $���� $����

$���� $���2SHQ*/ � � � � � � � � � � � � � � � � � � � � ������

GULYHUV � � � � � � � � � � � � � � � � � � ������RSHUDQGV � � � � � � � � � � � � � � � ���±���� ���

0�WK�RSHUDQG � � � � � � � � � � � ����� ���DFFHVVLQJ ∼ � � � � � � � � � � � � � � VHH RSQXPEHU�RI ∼ � � � � � � � � � � � � VHH QRSVRI�VHULHV�H[SDQVLRQV � � � � � � � � � ����

RSHUDWRUV � � � � � � � � � � � � � � � � ����±����

,QGH[���

Page 518: Mupad Tutorial

Bg]^q

SULRULWLHV � � � � � � � � � � � � � � � � � ����RSWLRQ�HVFDSH � � � � � � � � � � � ����� �����RSWLRQ�UHPHPEHU � � � � � ���� �����±�����RU � � � � ����� ����� ����� ������ $��� $���25'(5 � � � � � � � � � � � � � � � � � � � � � � � ����RUWKSRO\��FKHE\VKHY� � � � � � � � � � � $���26 FRPPDQG��V\VWHP� � � � � � � � � � � � �����RXWSXW

PDQLSXODWLRQ �3UHI��RXWSXW� � � � ����RI�IORDWLQJ�SRLQW�QXPEHUV�3UHI��IORDW)RUPDW� � � � � � � � ����RUGHU�RI�WHUPV � � � � � � � � � � � � � � ����VXSSUHVVLQJ ∼ � � � � � � � � � � � � � � ���

RXWSXW�DQG�LQSXW � � � � � � � � � � � � ����±����SUHWW\�SULQW � � � � � � � � � � � � � � � � ����

RYHUORDG � � � � � � � � � � � � � � � � � � � � � � ���

3SDUVHU � � � � � � � � � � � � � � � � � � � � � � � � ���SDUWIUDF � � � � � � � � � � � � � � � � � � ����� ���SDUWLDO�IUDFWLRQ�GHFRPSRVLWLRQ

�SDUWIUDF� � � � � � � � � � � � � � � � � ���3DVFDO � � � � � � � � � � � � � � � � � � � � � � � � � ���3, � � � � � � � � � � � � � � � � � � � � � � � � � � � ���π � � � � � � � � � � � � � � � � � � � � � � � � � � � ���SLHFHZLVH � � � � � � � ����� ����� ����� �����3ODWR¶V�SRO\KHGUD � � � � � � � � � � � � � � � �����SORW �JUDSKLFV�OLEUDU\�∼��$PELHQW/LJKW � � � � � � � � � � �����∼��$UF�G � � � � � � � � � � � � � � � � �����∼��$UF�G � � � � � � � � � � � � � � � � �����∼��$UURZ�G � � � � � � � � � � � � � � � �����∼��$UURZ�G � � � � � � � � � � � � � � � �����∼��%DUV�G � � � � � � � � � � � � � � � � �����∼��%DUV�G � � � � � � � � � � � � � � � � �����∼��%R[ � � � � � � � � � � � � � � � � � � �����∼��%R[SORW � � � � � � � � � � � � � � � �����∼��&DPHUD � � � � � � � � � � ������ ������∼��&DQYDV � � � � � � � � � � � ������ �����∼��&LUFOH�G � � � � � � � � � � � � � � �����∼��&LUFOH�G � � � � � � � � � � � � � � �����∼��&OLSSLQJ%R[ � � � � � � � � � � � �����∼��&RQH � � � � � � � � � � � � � � � � � �����

∼��&RQIRUPDO � � � � � � � � � � � � � �����∼��&RRUGLQDWH6\VWHP�G � � � � � � �� � � � � � � � � � � � � � � � � � ������ �����∼��&RRUGLQDWH6\VWHP�G � � � � � � �� � � � � � � � � � � � � � � � � � ������ �����∼��&XUYH�G � � � � � � � � � � � � � � � �����∼��&XUYH�G � � � � � � � � � � � � � � � �����∼��&\OLQGHU � � � � � � � � � � � � � � �����∼��&\OLQGULFDO � � � � � � � � � � � �����∼��'HQVLW\ � � � � � � � � � � � � � � � �����∼��'LVWDQW/LJKW � � � � � � � � � � �����∼��'RGHFDKHGURQ � � � � � � � � � � �����∼��(OOLSVH�G � � � � � � � � � � � � � �����∼��(OOLSVH�G � � � � � � � � � � � � � �����∼��(OOLSVRLG � � � � � � � � � � � � � �����∼��)XQFWLRQ�G � � � � � � � � � � � � �����∼��)XQFWLRQ�G � � � � � � � � � � � � �����∼��JHW'HIDXOW � � � � � � � � � � � � �����∼��*URXS�G � � � � � � � � � � � � � � � �����∼��*URXS�G � � � � � � � � � � � � � � � �����∼��+DWFK � � � � � � � � � � � � � � � � �����∼��+H[DKHGURQ � � � � � � � � � � � � �����∼��+LVWRJUDP�G � � � � � � � � � � � �����∼��,FRVDKHGURQ � � � � � � � � � � � �����∼��,PSOLFLW�G � � � � � � � � � � � � �����∼��,PSOLFLW�G � � � � � � � � � � � � �����∼��,QHTXDOLW\ � � � � � � � � � � � � �����∼��,WHUDWLRQ � � � � � � � � � � � � � �����∼��/LQH�G � � � � � � � � � � � � � � � � �����∼��/LQH�G � � � � � � � � � � � � � � � � �����∼��/LVWSORW � � � � � � � � � � � � � � �����∼��/V\V � � � � � � � � � � � � � � � � � �����∼��0DWUL[SORW � � � � � � � � � � � � �����∼��2FWDKHGURQ � � � � � � � � � � � � �����∼��2GH�G � � � � � � � � � � � � � � � � �����∼��2GH�G � � � � � � � � � � ������ ������∼��3DUDOOHORJUDP�G � � � � � � � � �����∼��3DUDOOHORJUDP�G � � � � � � � � �����∼��3LHFKDUW�G � � � � � � � � � � � � �����∼��3LHFKDUW�G � � � � � � � ������ �����∼��3ODQH � � � � � � � � � � � � � � � � �����∼��3RLQW�G � � � � � � � � � � � � � � � �����∼��3RLQW�G � � � � � � � � � � � � � � � �����

,QGH[���

Page 519: Mupad Tutorial

Bg]^q

∼��3RLQW/LJKW � � � � � � � � � � � � �����∼��3RLQW/LVW�G � � � � � � � � � � � �����∼��3RLQW/LVW�G � � � � � � � � � � � �����∼��3RODU � � � � � � � � � � � � � � � � �����∼��3RO\JRQ�G � � � � � � � � � � � � � �����∼��3RO\JRQ�G � � � � � � � � � � � � � �����∼��3ULVP � � � � � � � � � � � � � � � � �����∼��3\UDPLG � � � � � � � � � � � � � � � �����∼��443ORW � � � � � � � � � � � � � � � � �����∼��5DVWHU � � � � � � � � � � ������ ������∼��5HFWDQJOH � � � � � � � � � � � � � �����∼��5HIOHFW�G � � � � � � � � � � � � � �����∼��5HIOHFW�G � � � � � � � � � � � � � �����∼��5RRWORFXV � � � � � � � � � � � � � �����∼��5RWDWH�G � � � � � � � � � ������ �����∼��5RWDWH�G � � � � � � � � � ������ �����∼��6FDOH�G � � � � � � � � � � ������ �����∼��6FDOH�G � � � � � � � � � � ������ �����∼��6FDWWHUSORW � � � � � � � � � � � �����∼��6FHQH�G � � � � � � � � � � ������ �����∼��6FHQH�G � � � � � � � � � � ������ �����∼��6HTXHQFH � � � � � � � � � � � � � � �����∼��VHW'HIDXOW � � � � � � � � � � � � �����∼��6SDUVH0DWUL[SORW � � � � � � � �����∼��6SKHUH � � � � � � � � � � � � � � � � �����∼��6SKHULFDO � � � � � � � � � � � � � �����∼��6SRW/LJKW � � � � � � � � � � � � � �����∼��6WUHDPOLQHV�G � � � � � � � � � �����∼��6XP � � � � � � � � � � � � � � � � � � �����∼��6XUIDFH � � � � � � � � � � � � � � � �����∼��6XUIDFH6HW � � � � � � � � � � � � �����∼��6XUIDFH67/ � � � � � � ������ ������∼��6ZHHS � � � � � � � � � � � � � � � � �����∼��7HWUDKHGURQ � � � � � � � � � � � �����∼��7H[W�G � � � � � � � � � � � ������ �����∼��7H[W�G � � � � � � � � � � � ������ �����∼��7UDQVIRUP�G � � � � � � ������ �����∼��7UDQVIRUP�G � � � � � � ������ �����∼��7UDQVODWH�G � � � � � � ������ �����∼��7UDQVODWH�G � � � � � � ������ �����∼��7XEH � � � � � � � � � � � � � � � � � �����∼��7XUWOH � � � � � � � � � � � � � � � � �����∼��9HFWRU)LHOG�G � � � � � � � � � �����

∼��9HFWRU)LHOG�G � � � � � � � � � �����∼��:DWHUPDQ � � � � � � � � � � � � � � �����∼��;5RWDWH � � � � � � � � � � � � � � � �����∼��=5RWDWH � � � � � � � � � � � � � � � �����

SORW � � � � � � � � � � ����� ����� ����� �����SORWIXQF�G � � � � � � � � � ����� ������� $���SQJ � � � � � � � � � � � � � � � � � � � � � � � � ������SRO\ � � � � � � � � � � ����� ����±����� $���SRO\�OLVW � � � � � � � � � � � � � � � � � � � � ����SRO\QRPLDOV � � � � � � � � � � � � � � ����±����

DFFHVVLQJ�WHUPV��QWKWHUP� � � � � ����DSSO\LQJ�D�IXQFWLRQ�WR�FRHIILFLHQWV�PDSFRHIIV� � � � � � � � � � � � � � � ����DULWKPHWLF � � � � � � � � � � � � � � � � ����&KHE\VKHY ∼ � � � � � � � � � � ����� $���FRHIILFLHQWV

FRHII � � � � � � � � � � � � � � � � � � ����QWKFRHII � � � � � � � � � � � � � � � ����

FRPSXWLQJ�ZLWK ∼ � � � � � � ����±����FRQYHUVLRQ�RI�D�SRO\QRPLDOWR�D�OLVW �SRO\�OLVW� � � � � � � � ����WR�DQ�H[SUHVVLRQ �H[SU� � � � � � ����

GHIDXOW�ULQJ��([SU� � � � � � � � � � � ����GHILQLWLRQ��SRO\� � � � � � � � ����±����GHJUHH��GHJUHH� � � � � � � � � � � � � ����GLYLVLRQ�ZLWK�UHPDLQGHU �GLYLGH� �� � � � � � � � � � � � � � � � � � � � � � � � ����

HYDOXDWLRQ��HYDOS� � � � � � � � � � � ����IDFWRUL]DWLRQ �IDFWRU� � � � � � � � � ����*U|EQHU�EDVHV �JURHEQHU� � � � � � ����JUHDWHVW�FRPPRQ�GLYLVRU �JFG� � ����LUUHGXFLELOLW\�WHVW �LUUHGXFLEOH� � �� � � � � � � � � � � � � � � � � � � � � � � � ����

OLEUDU\�IRU�RUWKRJRQDO ∼ �RUWKSRO\�� � � � � � � � � � � � � � � � � � � � � � � � $���

PXOWLSOLFDWLRQ�E\�D�VFDODU�IDFWRU�PXOWFRHIIV� � � � � � � � � � � � � � � ����QXPEHU�RI�WHUPV��QWHUPV� � � � � � ����RSHUDQGV � � � � � � � � � � � � � � � � � ����

3RVWVFULSW � � � � � � � � � � � � � � � � � � � ������SRZHUPRG � � � � � � � � � � � � � � � � � � � � � � ����3UHI �OLEUDU\�IRU�XVHU�SUHIHUHQFHV� � � � �

� � � � � � � � � � � � � � � � � � � � ����±����

,QGH[���

Page 520: Mupad Tutorial

Bg]^q

∼��IORDW)RUPDW � � � � � � � � � � � � ����∼��RXWSXW � � � � � � � � � � � � � � � � � ����∼��SRVW,QSXW � � � � � � � � � � � � � � ����∼��SRVW2XWSXW � � � � � � � � � � � � � ����∼��UHSRUW � � � � � � � � � � � � � � � � � ����UHVHWWLQJ�SUHIHUHQFHV � � � � � � � � � ����

35(77<35,17 � � � � � � � � � � � � � � � � � � � ����SULPLWLYHV � � � � � � � � � � � � � � � �����±�����SULQW ����� ����� ����� ����±����� �����

����� ����±����� ����� ������ $���SUREDELOLW\ � � � � � � � � � � � � � � � ����±����SUREDELOLW\�GHQVLW\�IXQFWLRQ � � � � � � � ����SURFHGXUHV � � � � � � � � � � � � � � � ����±�����

FDOO � � � � � � � � � � � � � � � � � � � � � � ����E\�QDPH � � � � � � � � � � � � � � � � �����E\�YDOXH � � � � � � � � � � � � � � � � �����

FRPPHQWV��� � �� � � � � � � � � � � � ����GHILQLWLRQ � � � � � � � � � � � � � ����±����HYDOXDWLRQ�OHYHO � � � � � � � �����±�����JOREDO�YDULDEOHV � � � � � � � � � � � � � ����LQSXW�SDUDPHWHUV � � � � � � �����±�����ORFDO�YDULDEOHV��ORFDO� � � � ����±�����IRUPDO�SDUDPHWHUV � � � � � � � � �����ZLWKRXW�D�YDOXH � � � � � � � � � � � �����

0DWUL[3URGXFW � � � � � � � � � � � � �����RSWLRQ�HVFDSH � � � � � � � ����� �����RSWLRQ�UHPHPEHU � � ���� �����±�����SURFQDPH � � � � � � � � � � � � � � � � � � ����UHFXUVLRQ�GHSWK �0$;'(37+� � � � � � �� � � � � � � � � � � � � � � � � � � ����� $���

UHWXUQ�YDOXH��UHWXUQ� � � � � � ����±����VFRSLQJ � � � � � � � � � � � � � �����±�����VXESURFHGXUHV � � � � � � � � �����±�����V\PEROLF�GLIIHUHQWLDWLRQ �����±�����V\PEROLF�UHWXUQ�YDOXHV � � � � ����±����WULYLDO�IXQFWLRQ � � � � � � � � � � � � � �����W\SH�GHFODUDWLRQ � � � � � � � � � � � � �����YDULDEOH�QXPEHU�RI�DUJXPHQWV�DUJV� � � � � � � � � � � � � � �����±�����

SURFQDPH � � � � � � ����� ������ ������ $���SURGXFW � � � � � � � � � � � � � � � � � � � � � � ����SURJ��H[SUWUHH � � � � � � � � � � � � � � � ����SURSHUWLHV � � � � � � � � � � � � � � � � � � � � ����

SURSHUW\�LQVSHFWRU � � � � � � � � � �����±�����SURWHFW � � � � � � � � � � � � � � � � � � � � � ����3XLVHX[�VHULHV � � � � � � � � � � � � � � � � � ����

4Q � � � VHH 4B DQG GRPDLQ� 'RP��5DWLRQDO4B � � � � � � � � � � � � � � � � � � � � � � � � � � � ���TXDGUDWXUH � � � � VHH QXPHULFDO LQWHJUDWLRQTXDQWLOH�IXQFWLRQ � � � � � � � � � � � � � � � ����TXRWLHQW�PRGXOR��GLY� � � � � � � � � ���� ����

5R � � � � � � � VHH 5B DQG GRPDLQ� 'RP��5HDO5B � � � � � � � � � � � � � � � � � � � � � � � � � � � ���5B � � � � � � � � � � � � � � � � � � � � � � ����� ����UDGLFDO�VLPSOLILFDWLRQ �UDGVLPS� � � � � ����UDGVLPS � � � � � � � � � � � � � � � � � ����� ����UDQGRP � � � � � � � � � � � � � ����� ����� $���UDQGRP�QXPEHU�JHQHUDWRUV��UDQGRP�

IUDQGRP� VWDWV��QRUPDO5DQGRP�� � � � � � � � � � � � � � � � � � � � ����±����

UDQGRP�QXPEHUVQRUPDOO\�GLVWULEXWHG�VWDWV��QRUPDO5DQGRP� � � � � � � ����RWKHU�GLVWULEXWLRQV � � � � � � � � � � ����XQLIRUPO\�GLVWULEXWHG �IUDQGRP� ����XQLIRUPO\�GLVWULEXWHG �UDQGRP� � � ����

UDQJH � � � � � � � VHH H[SUHVVLRQV� UDQJH ����5'B,1) � � � � � � � � � � � � � � � � � � � � � � ����5'B1,1) � � � � � � � � � � � � � � � � � � � � � ����5H � � � � � � � � � � � � � � � � � � ����� ���� ����UHDG � � � � � � � � � � � � � � � � � � � ����� �����UHDGLQJ�GDWD �LPSRUW��UHDGGDWD� � � � � ����5HDO � � � � � � � � � � � � � � � � � � � � � � � � � ���UHDO�SDUW � � � � � � � � � � � � � � � � � � � � VHH 5HUHF � � � � � � � � � � � � � � � � � � � � ������ $���UHFWIRUP � � � � � � � � � � � � � � � � � ����� ����UHFXUUHQFH�HTXDWLRQV��UHF� � � � � � � � � � ����UHPDLQGHU�PRGXOR��PRG� � � � � � � ���� ����

UHGHILQLWLRQ � � � � � � � � � � � � � � � � ����RSWLRQ�UHPHPEHU � � � � � � � � � �����±�����UHSHDW � � � � � � � � � � � � � � � � � � � � � � $���

,QGH[���

Page 521: Mupad Tutorial

Bg]^q

UHVHW � � � � � � � � � � � � � � � � � � ����� �����UHVLGXH�FODVV�ULQJ � � � � � � � � � � � � � � � � �

� � � � � VHH GRPDLQ� 'RP��,QWHJHU0RGUHVWDUWLQJ�D�VHVVLRQ��UHVHW��� � � ���� ������5HWXUQ! � � � � � � � � � � � � � � � � � � � � � � ���UHWXUQ � � ����� ����� ������ ������ ������

$���UHYHUW � � � � � � � � � ����� $���� $���� $���UHZULWH

WDUJHWV � � � � � � � � � � � � � � � � � � � ����UHZULWH � � � � � � � � � � � � � � � � � � � � � � ���5*%��&RORU1DPHV � � � � � � � � � � � � � � �����5*%��SORW&RORU3DOHWWH � � � � � � � � � �����ULQJV � � � � � � � � � � � � � � � � � � � � ����±����5RRW2I � � � � � � � � � � � � � � � � � � � � � � � ���URRWV � � � � � � � � � � � � � � � � � � � � � � � � ����URXQG � � � � � � � � � � � � � � � � � � � � � � � � ���URXQGLQJ�HUURUV � � � � � � � � � � � � � � � � � ���URZ�YHFWRUV � � � � � � � � � � � � � � � � � � � ����UXQWLPH�LQIRUPDWLRQ�DERXW�DOJRULWKPV

�XVHULQIR DQG VHWXVHULQIR� � � ����

6VDYLQJ�JUDSKLFV � � � � � � � � � ������±������

EDWFK�PRGH � � � � � � � � ������±������LQWHUDFWLYH � � � � � � � � � � � � � � � ������

VFRSLQJ � � � � � � � � � � � � � � � � � �����±�����VFUHHQ�RXWSXW��SULQW� � � � � � � � � � � � � ����

PDQLSXODWLRQ �3UHI��RXWSXW� � � � ����VHOHFW � � � � ����±����� ����� ����� �����

����� ����� $��� $���� $���� $����$���

VHPLFRORQ � � � � � � � � � � � � � � � � � � � � � ���VHTXHQFH � � � � � � � � � � � � � � � � � � � � � � ����VHTXHQFH�JHQHUDWRU���� � � � � � � � ����� ����

LQ � � � � � � � � � � � � � � � � � � � � � � ����VHTXHQFH�JHQHUDWRU���� � � � � � � � � � � � � ����VHTXHQFHV � � � � � � � � � � � � � � � � ����±����

GHOHWLQJ�HOHPHQWV �GHOHWH� � � � � ����LQGH[HG�DFFHVV � � � � � � � � � � � � � ����RI�LQWHJHUV � � � � � � � � � � � � � � � � ����

VHULHV � � � � � � � � � � � � � ����� ����� $���

VHULHV�H[SDQVLRQV � � � � � � � � � � � � � � � � �� � � � � � � ����±����� VHH�DOVR VHULHV

DV\PSWRWLF�H[SDQVLRQ �VHULHV� � � �� � � � � � � � � � � � � � � � � � � ����� ����

FRQYHUVLRQ�WR�D�VXP �H[SU� � � � � ����/DXUHQW�VHULHV��VHULHV� � � � � � � ����RI�LQYHUVH�IXQFWLRQV �UHYHUW� � � ����RSHUDQGV � � � � � � � � � � � � � � � � � ����SRLQW�RI�H[SDQVLRQ � � � � � � � � � � ����3XLVHX[�VHULHV��VHULHV� � � � � � � � ����7D\ORU�VHULHV��WD\ORU� � � � � � � � � ����WUXQFDWLRQ��2� � � � � � � � � � � � � � � ����RUGHU��25'(5� � � � � � � � � � � � � ����

VHVVLRQUHVWDUW��UHVHW��� � � � � � � � ���� �����VDYLQJ�D ∼ �ZULWH� � � � � � � � � � � � ����

VHWV � � � � � � � � � � � � � � � � � � � � � ����±����DSSO\LQJ�D�IXQFWLRQ �PDS� � � � � � ����EDVLF�VHWV��VROYHOLE��%DVLF6HW� � ���GLIIHUHQFH��PLQXV� � � � � � � � � � � ����HOHPHQWV��RS� � � � � � � � � � � � � � � ����HPSW\�VHW��^`� ∅� � � � � � � � � � � � � ����H[FKDQJLQJ�HOHPHQWV � � � � � � � � ����LPDJH�VHW �'RP��,PDJH6HW� � � � � � ���LQWHUVHFWLRQ��LQWHUVHFW� � � � � � � ����LQWHUYDOV��'RP��,QWHUYDO� � � � � � ����QXPEHU�RI�HOHPHQWV �QRSV� � � � � ����RUGHU�RI�WKH�HOHPHQWV � � � � � � � � ����UHPRYLQJ�HOHPHQWV � � � � � � � � � � ����VHOHFWLQJ�DFFRUGLQJ�WR�SURSHUWLHV�VHOHFW� � � � � � � � � � � � � � � � � � ����VHW�YDOXHG�IXQFWLRQV � � � � � � � � � ����VSOLWWLQJ�DFFRUGLQJ�WR�SURSHUWLHV�VSOLW� � � � � � � � � � � � � � � � � � � ����XQLRQ��XQLRQ� � � � � � � � � � � � � � ����

VHWXVHULQIR � � � � � � � � � � � � � � � � � � � ����6LHUSLQVNL�WULDQJOH � � � � � � � � � ������ $���VLJQ � � � � � � � � � � � � � � � � � � � � � ���� ����VLPSOLILFDWLRQ � � � � � � � � � � � � � � � � � � ���

DXWRPDWLF ∼ � � � � � � � � � � � ����±����RI�H[SUHVVLRQV��VLPSOLI\� � ����±����RI�UDGLFDOV �UDGVLPS� � � � � ����� ����

,QGH[���

Page 522: Mupad Tutorial

Bg]^q

6LPSOLI\ � � � ����� ����� ����� ����� �����$���� $���

VLPSOLI\ � � � ����� ����� ����� $���� $���VLQH�LQWHJUDO��6L� � � � � � � � � � � � � � � � � ���VROXWLRQ

DSSUR[LPDWH � � � � � � � � � � � VHH IORDWVROYH � � � ���� ����� ����� ����� ���� �����

����� ���� ���� ���±���� ���� ���������� ����� ����� $��� $���� $����

$���VROYLQJ�HTXDWLRQV � � � � � � � � � VHH HTXDWLRQVVRUW � � � � � � � � � � � ����� ����� $���� $���

XVHU�GHILQHG�RUGHU � � � � � � � � � � ����VRXUFH�FRGH��H[SRVH� � � � � � � � � � � ���� ���VRXUFH�FRGH�GHEXJJHU � � � � � � � � � � � � � ���VSDUVH�PDWULFHV � � � � � � � � � � � � ����±����VSOLW � � � � � � � � � ����� ����� ����� $���VTUW � � � � � � � � � � � � � � � � � � � � � � ���� ���VTXDUH�URRW��VTUW� � � � � � � � � � � � � ���� ���VWDQGDUG�GHYLDWLRQ �VWDWV��VWGHY� � � ����VWDWLVWLFV � � � � � � � � � � � � � � � � � ����±����χ2�WHVW �VWDWV��FV*2)7� � � � � � � � ����HTXLSUREDEOH�FHOOV�VWDWV��HTXLSUREDEOH&HOOV� � ����IUHTXHQFLHV �'RP��0XOWLVHW� � � � �� � � � � � � � � � � � � � � � � � � ����� $���

OLEUDU\�IRU ∼ �VWDWV� � � � � � � � � � ����PHDQ�YDOXH �VWDWV��PHDQ� � � � � ����QRUPDO�GLVWULEXWLRQ�VWDWV��QRUPDO&')� � � � � � � � � ����TXDQWLOH �VWDWV��QRUPDO4XDQWLOH�� � � � � � � � � � � � � � � � � � � � � � � � ����

UDQGRP�QXPEHUV��IUDQGRP� UDQGRP�VWDWV��QRUPDO5DQGRP� � � � � � � �� � � � � � � � � � � � � � � ����� ����� ����

VWDQGDUG�GHYLDWLRQ �VWDWV��VWGHY�� � � � � � � � � � � � � � � � � � � � � � � � ����

YDULDQFH �VWDWV��YDULDQFH� � � � ����VWDWV �OLEUDU\�IRU�VWDWLVWLFV�∼��FV*2)7 � � � � � � � � � � � � � � � � � ����∼��HTXLSUREDEOH&HOOV � � � � � � � ����∼��PHDQ � � � � � � � � � � � � � � � � � ����∼��QRUPDO&') � � � � � � � � � � � � � � ����

∼��QRUPDO4XDQWLOH � � � � � � � � � � ����∼��QRUPDO5DQGRP � � � � � � � � � � ����∼��VWGHY � � � � � � � � � � � � � � � � ����∼��YDULDQFH � � � � � � � � � � � � � � ����

VWULQJV � � � � � � � � � � � � � � � � � � ����±����FRQYHUWLQJ�H[SUHVVLRQV�WR ∼�H[SU�WH[W� � � � � � � � � � ����� ����H[WUDFWLQJ�V\PEROV � � � � � � � � � � ����OHQJWK��OHQJWK� � � � � � � � � � � � � � ����OLEUDU\��VWULQJOLE� � � � � � � � � � ����VFUHHQ�RXWSXW��SULQW� � � � � � � � � ����VXEVWULQJV � � � � � � � � � � � � � � � � ����

VXESURFHGXUHV � � � � � � � � � � � � �����±�����VXEV � ����� ����� ���� ���±���� ���� �����

����� $���� $���� $���� $���VXEVH[ � � � � � � � � � � � � � � � � � � � � � � � ���VXEVRS � � � � � ����� ���� ���� ������ $���VXEVWLWXWLRQ � � � � � � � � � � � � � � � � � ���±���

HQIRUFHG�HYDOXDWLRQ �HYDO� � � � � � ���LQ�VXPV�DQG�SURGXFWV �VXEVH[� � � ���PXOWLSOH ∼V � � � � � � � � � � � � � � � � ���RI�RSHUDQGV��VXEVRS� � � � � � � � � � ���RI�VXEH[SUHVVLRQV��VXEV� � � � � � � � ���RI�V\VWHP�IXQFWLRQV � � � � � � � � � � � ���VLPXOWDQHRXV ∼ � � � � � � � � � � � � � ���

VXP � � � � � � � � � � � � � � � � � � � � � � � � � � ����6\PERO � � � � � � � � � � � � � � � � � � � � � � � ���V\PEROLF�FRPSXWDWLRQV � � � � � � � ���� ����V\PEROLF�PDQLSXODWLRQ � � � � � � � � � � � � � ���V\VWHP � � � � � � � � � � � � � � � � � � � � � � �����

7WDEOH ����� ����� ����� ����� ����� $����

$���WDEOHV � � � � � � � � � � � � � � � � � � � ����±����

DFFHVVLQJ�HOHPHQWV � � � � � � � � � � � ����DSSO\LQJ�D�IXQFWLRQ��PDS� � � � � � ����GHOHWLQJ�HQWULHV �GHOHWH� � � � � � � � ����H[SOLFLW�JHQHUDWLRQ � � � � � � � VHH WDEOHLPSOLFLW�JHQHUDWLRQ � � � � � � � � � � ����RSHUDQGV � � � � � � � � � � � � � � � � � � ����TXHU\LQJ�LQGLFHV �FRQWDLQV� � � � ����

,QGH[���

Page 523: Mupad Tutorial

Bg]^q

VHOHFWLQJ�DFFRUGLQJ�WR�SURSHUWLHV�VHOHFW� � � � � � � � � � � � � � � � � � ����VSOLWWLQJ�DFFRUGLQJ�WR�SURSHUWLHV�VSOLW� � � � � � � � � � � � � � � � � � � ����

WDQ � � � � � � � � � � � � � � � � � � � � � ����� $���WDQJHQW � � � � � � � � � � � � � � � � � � � � � � �����WD\ORU � � � ����� ����� ����� ����� ������

$���±$���WHVWW\SH � � � � ����� ����� ����� ����� �����

����� $���� $���7HWUDKHGURQ � � � � � � � � � � � � � � � � � � �����7(;7:,'7+ � � � � � � � � � � � � � � � � � � � � � ����WLPH � ����� ����� ����� �����±������ $���WUDQVIRUPDWLRQV � � � � � � � � � � � �����±�����WUDQVIRUPLQJ�H[SUHVVLRQV � � � � � � ���±����758( � � � � � � � � � � � � � � � � � � � � � � � � ����WUXQF � � � � � � � � � � � � � � � � � � � � ���� $��7\SH �OLEUDU\�IRU�W\SH�FKHFNLQJ�∼��$Q\7\SH � � � � � � � � � � � � � � � $���∼��(YHQ � � � � � � � � � � � � � � � � � ����∼��,QWHUYDO � � � � � � � � � � � � � � ����∼��/LVW2I � � � � � � � � � � � � � � � � $���∼��1HJ,QW � � � � � � � � � � � � � � � � � ����∼��1RQ1HJ,QW � � � � � � � � � � � � � �����∼��1XPHULF � � � � � � � ����� ����� ����∼��2GG � � � � � � � � � � � � � � � � � � � ����∼��3RV,QW � � � � � � � � � � � � � � � � � ����∼��3RVLWLYH � � � � � � � � � � � � � � ����∼��5HDO � � � � � � � � � � � � � � � � � ����∼��5HVLGXH � � � � � � � � � � � � � � � ����

W\SH � � � � � � � � � � � � � � � � � � � � � � � � � ����W\SHV�RI�0X3$' REMHFWV � � � � � � ����±����

GRPDLQ�W\SH��GRPW\SH� � � � � � � � � ���OLEUDU\��7\SH� � � � � ����� ������ $���W\SH�FKHFNLQJ��WHVWW\SH� � � � � � � ����W\SH�TXHU\��W\SH� � � � � � � � � � � � � ����

W\SHVHW�H[SUHVVLRQV � � � � � � � � � � � � � � ����

8XQDVVXPH � � � � � � � � � � � � � � � � � � � � � ����XQGHILQHG � � � � � � � � � � � � � � � ������ ����XQLRQ � � � � � ����� ����� ����� $���� $���81.12:1 � � � � � � � � � � � � ����� ����� ����

XQNQRZQV � � � � � � � � � � � � � � VHH LGHQWLILHUVXQSURWHFW � � � � � � � � � � � � � � � � � � � � ����XVH � � � � � � � � � � � � � � � � � � � � � ���� �����

9YDULDQFH��VWDWV��YDULDQFH� � � � � � � � ����9HFWRU)RUPDW � � � � � � � � � � � � � � � � � � ���YHFWRUV � � � � � � � � � � � � � � � � � � ����±����

FROXPQ ∼ � � � � � � � � � � � � � ����� ����LQGH[HG�DFFHVV � � � � � � � � � � � � � ����LQLWLDOL]DWLRQ � � � � � � � � � � � � � � ����URZ ∼ � � � � � � � � � � � � � � � � � � � ����

::DWHUPDQ�SRO\KHGUD � � � � � � � � � � � � �����ZRUNVKHHW � � � � � � � � � � � � � � � � � � � � � � ���ZULWH � � � � � � � � � � � � � � � � � � � ����� ����ZULWH�SURWHFWLRQ

UHPRYLQJ ∼ �XQSURWHFW� � � � � � � ����VHWWLQJ ∼ �SURWHFW� � � � � � � � � � ����

=Z � � � � VHH =B DQG GRPDLQ� 'RP��,QWHJHU=B � � � � � � � � � � � � � � � � � � � � � � � � � � � ���=B � � � � � � � � � � � � � � � � � � ����� ����� ����]LS � � ����� ����� ����� ����� ����� �����

$���� $���� $���Zn � � � � � � VHH GRPDLQ� 'RP��,QWHJHU0RG

,QGH[���