J. M. McDonough

University of Kentucky Lexington, KY 40506

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[JMe efFefnM\qLYQwme®sLnM\qL^eJMLi¡Ã[KJMe®skE,hqLic|Y]^_HLiYQJXE,_,_,IconoJpYQnMEHeaefYQ[K[_,E,]^YQnMEHea-efnM\KL:]rea noJpY]rnMEHeay¡OYQ[K[EdaqG[JpE,aK]^EH[_HL coeu6L8uE,_,_[KJMLicoLia n|EHn2\qL^JpLP¶¦·¹¸-¸¡º-»>L î JpconewcoL^JpsLnM\KYQn-feJ"Ya b&cM± IYQJML¡OYQnoJpE g m:Xu6LA\KY®sLIm ö½¼'ë ç ëe¾:Xu|\qL^JML¼ E,c|YhKE,YQGeaY_¡OYQnoJZE g u|EHnM\.Li_HLi¡CLia nMcLi± IKY_noenM\qecoLefnM\qL/¡OYE,ahKE,YQGeaKY_efwm:`YaKh ç YahD¾ YQJMLJpLico[mLi]rnMEHsLi_Hb9_Heu6L^JYahIK[K[mL^JnoJpEdYaqGIK_,YQJF¡OYQnoJpE,]rLic²u|E,nM\:¶^L^JpehE,YQGeaKY_,c^YaKh/u\qecoL«aqeaq¶^L^JMe ¬Li_HLi¡OLiatnMc

ññ ! #"%$'&()+*,-.).-*/$01232.456YQJpLnM\qL8aqL^GYQnMEHsLicef²nM\qL8]reJMJpLico[meaKhKE,aqG:Li_HLi¡CLia nMcefm9P|«,¿enoL8nM\KYQnnM\qLicML8¡OYQnoJpE,]rLic2YQJpL8aqen nM\qL ç YaKh¾%efFnM\qLÀÁ|j°hqLi]re¡C[mecMEHnME,ea-nM\qL^eJMLi¡ cMnMYQnoLih.LiYQJp_dEHL^JiP ¬³2\qLiaA«,¼Bë ç ë!¾¬ª"öo;KYaKh

øýþ ö¨¼Ã þ « ç Xľ8¬ª ø Xq¼Â þ ;4 ø ð «NQPSRkN®¬

´ n2E,cLiYcpE,_Hb]Z\qLi]M¦LihnM\KYQn|nM\KE,c îqg Lihkjl[meE,a nJpL^[KJMLicoLia nMYQnMEHea"E,c2L g Y]rnM_HbnM\qL9feJp¡ ef9«NQPSñ#¬eJ:«NQPSñ߬^P³§eGIKYQJpYa noL^LO]rea sL^JMGLiaK]rLOefnM\KL/E,noL^JpYQnMEHeaycM]p\KLi¡CL"«NQPSRkN®¬Xu6L/¡IKcon î aKh]reaKhKEHnME,eaKc|IKahqL^Ju\KE,]p\

nM\KL]reJMJMLicM[5eahKE,aqGŧE,[cM]p\EHno¶O]reaKconMYa n\Yc9¡OYQGaKE,nMIKhqLO_HLicMc9nM\KYaxIaKEHn°bPDqJMe¡ «NQPSRkN®¬:EHn8fe_,_Heu|cnM\KYQnnM\KLƧEH[cM]p\KE,no¶]reahKEHnMEHea-E,c

é «m¬²ëv«êq¬ é öÈÇÇ ¼  þ « ç X¯¾¬ªX!¼  þ ;ëD¼  þ « ç Xľ8¬ªêëI¼  þ ;1ÇÇö ÇÇ ¼  þ « ç X¯¾¬r«ëêq¬ ÇÇì ÇÇ ¼  þ « ç X¯¾¬¡ÇÇ é ëê é ð´LiaK]rLKnM\KLƧEH[cM]p\KE,no¶]reaconMYatnXE,c É

ö ÇÇ ¼Ã þ « ç Xľ8¬ ÇÇ îcMeO]reatsL^JMGLia]rL:efY]rewE§EHnoL^JpYQnMEHea-E,cWGIYQJpYatnoL^Lih.u|\qLiaqL^sL^J é ¼  þ « ç X¯¾8¬ é ¤=NQP´ f²u6L8aqeu=nMYQ¦L é6Êkénoew5LnM\KL8E,a î aKE,n°b j°aqeJp¡-UnM\qLia É

ö&¡OY g NG = G Ð ûþû G = G ð

³|\ IKciqu6L î ahnM\KYQn|E,aeJphKL^J2feJÉ

¤ Nnoe\qe_,hUuL9¡IKcon2\KY®sL

G = G ¨ Ð ûþû G = G í ö NQîpñUî ð^ð^ð îM¸ ð

³|\KYQn2E,c^KhKEdYQGeaKY_hqe¡OEdaKYaK]rLE,cJMLi±UIKEHJpLih`P²³|\KE,c]reaK]^_,IhqLicnM\qL[KJpe ef P­a=Ed¡C[meJMnMYatn-]reaKcMLi± IqLia]rLAef:nM\E,c-E,cnM\KYQn^nªb [Ed]^Y_,_HbXeaK_Hb1cM[YQJpcoLAcobkconoLi¡Oc.]^Ya=wmLcoe_HsLih4wUb

E,noL^JpYQnMEHea`P­XcEdcLiYcMEd_HbOcoL^Lia`qE,anM\KE,c]^YcoL¡OYatbefnM\qLXnoL^Jp¡OcWE,aOnM\KLXcMI¡O¡OYQnMEHea E,a˱Pm«NQPSñ߬YQJMLX¶^L^JMeqYah-\qLiaK]rL8aqL^Lih-aqenwmLXL^s¯Y_dIKYQnoLih`P³2\E,c^Uef²]reIqJZcoLqJMLihKIK]rLicWwmenM\-conoeJZYQGL9YaKh-YQJpEHnM\¡CL^nME,]E,aYhKhKEHnME,eanoe[KJMe®skE,hKEdaqGYOw5L^nonoL^Je[K[meJMnMIKaKE,n°bOfeJY]Z\KEHL^skE,aqGOhKE,YQGeaY_`hqe¡OE,aKYa]rLPÌ g[ÍÎQÎlÏUÐ »1@Ѳ» kl®»U½ g ¼°|Li]^Y_,_mfJMe¡BnM\qLY_HGeJpEHnM\K¡BfeJ2Y]rewE`E,noL^JpYQnMEHea nM\KYQnnM\qLJpEHG\ nj°\KYaKhcMEdhqLefnM\qL| Li±UIKYQnMEHeaA«NQPSñ߬WE,cL^sQY_,IKYQnoLih IKcMEdaqG8eaK_Hb/JMLicpIK_HnMcfJMe¡#nM\qL|[KJpL^sUEHeIcEHnoL^JpYQnME,ea`tL^sLia nM\qeIqG\¡CeJMLJMLi]rLiatn«YaKh[KJMLicMI¡OYQw_Hb¡OeJMLY]^]^IqJZYQnoL¬6LiconME,¡OYQnoLicefY_,_coe_,IqnMEHea]re¡C[meaqLia nMc[KJpLi]rLihKE,aqG:nM\qL2 eaqL\KYisLY_HJMLiYhKbCw5L^Lia ]^Y_,]^Ikj_dYQnoLih`P¦ÒIqJE,a nMIKEHnMEHeazcMIqGGLiconMcnM\KYQnuLC¡OEHG\ newKnMYEdaY¡CeJpL:JpYQ[E,hK_Hb-]rea sL^JMGLiatn:EHnoL^JpYQnMEHeaycM]Z\qLi¡CLCEHfuL:IcoL:aqL^u JMLicMIK_,nMc|YcXcoe eaYcnM\qL^b"YQJML9¦kaqeu|a`JZYQnM\qL^JnM\Ya.uWYEHnME,aqGnoe ]re¡C[_HL^noL9nM\qL]^IqJMJMLia ncMu6L^L^[nM\KJMeIqG\"Y_,_mLi±UIKYQnMEHeacefFnM\qL8cobkconoLi¡-P´ n2nMIqJpacWeIqn2nM\KYQn2nM\E,cE,c^KE,afY]rn^IcMIKY_,_HbnoJpIqLKYaKh"E,n[KJMe®skE,hqLicnM\KL8wYcME,c2feJnM\qL9¡CL^nM\qekh-¦kaqe®ua.YcnÓ Ë ÌZÌxÔÕ¥D B^ÈC roÚ PW´ a.nM\KE,c2cM]Z\qLi¡CL8nM\KL9GLiaqL^JpY_CLi±UIKYQnMEHeaEdcWGEHsLia"wUb

H øWýþ ö N= ; ë  þ ûþ = øWýþ ëÐ û ýþ = ø0 î «NQPSRñ¬

ÑÒHÑÒ&ÓÔWÕÖWײØiÔWÙ>ÔWÚÕØÙÛ6Ü y ÓÝÓ§×FÛ6ÞyÓ ñRö NQîpñUî ð^ð^ð îM¸xP-LaqenoLnM\YQn9nM\KE,c9E,c9aqenconoJpE,]rnM_,bY îqg Lihqjl[5eEdatn9Y_,GeJpEHnM\K¡<wmLi]^YIKcMLwmenM\A ø YaKh øWýþ YQ[K[mLiYQJE,anM\qLCEHnoL^JpYQnMEHeayfIKaK]rnME,ea`P8´ nEdcfYEHJp_,b-LiYcob\qeu6L^sL^J®noe-LlÖ5Li]rn:YJpLiYQJMJpYaqGLi¡CLia nnM\KYQnfeJp¡OY_d_Hb9Y]p\EHL^sLicnM\qL îKg Lihkjl[meE,a nFfeJZ¡-P²³|\KE,cFE,cE,¡C[meJMnMYa nFfeJnM\qL^eJML^nME,]^Y_KcMnMIKhKEHLicFwIKnFEdcFef5_,E,nonM_HL6sQY_,IqLfeJ2[KJZY]rnME,]^Y_]re¡C[IKnMYQnMEHea`coeCuL8u|Ed_,_maqen[IqJpcpIqLEHn2\qL^JMLP

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øWýþ öã«N|ëAÞ6¬ª ø XqÞ öô ø0 XqÞ ëH ø öô ø0 XqÞwàC ð

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Ð û ýþ = ø0äãå ë ø î «NQPSRR¬ö NQîpñUî ð^ð^ð îM¸ ð

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L:aqeu=[KJMLicoLia n|Y/[cMLiIKhqeQj°_,YaqGIYQGL9Y_HGeJpEHnM\¡>feJ|Ed¡C[_HLi¡CLia nME,aqGiÒ|°9P)@ ¼½¡º²¾ ¿À ç  ¥KË µpµ rÌpÌm æÜWægrÎhrÈHitÚ Qɤ B£ZQËq Þî|W Q5Ƨ = [sèé1ê ÈHiê d Ú QÈDË6ÎÌZÌ& l £ë © Bì& ¸ ^Ú QK̪Ƨ öokî § = [èÕéÌ]iÍOi g ö¨

ñ¯ ! #"%$'&()+*,-.).-*/$01232.456­ BíWæQQÈ ËK¦¥Ü ¬ ^QÎÍ/ËUÈHª Fö NQî¸à/ ö N= náâ ; ë  þ ûþ = H øýþ ëÐ û ýþ = H ø ãå ëH ø

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Preferred Method( Storage

Requirement TotalArithmeticSystem Matrix

Nonsparse)

Sparse

Sparse, compactlybanded

Direct elimination

Iteration, e.g., SOR

Sparse, band LU-decom-position

O(N)

O(N)

O(N )2

O(N)

O(N )3*

O(N )1.5

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Least Squares FitExperimental Dataf(x)

x

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