Tính hợp lý của nghiệm hệ phuơng trình và khái niệm 'condition number' của một ma trận (kỳ 1) _ Math2IT.pdf

1/15/2015 Tnhhplcanghimhphungtrnhvkhinim'conditionnumber'camtmatrn(k1)|Math2IT

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Dinh Anh Thi | 31.10.14 | Hiu ton hc | Ton cao cp | 7Like

Tnh hp l ca nghim h phung trnh v khinim 'condition number' ca mt ma trn (k 1)

Bn s tm hiu th no l mt h phung trnh iu kin yu v iu kin mnh, chun ca mtma trn, condition number ca ma trn l g v nh hng ca n ti nghim h phung trnh rasao?

Bi vit tham kho t trang Mathematics for College v bi ging ca Dheeraj Bhardwaj. Bncng c th ti bn pdf nguyn gc (bng ting Anh) ca trang ny ti a ch

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k 1 ny, bn s bit c

S khc nhau gia h phung trnh iu kin yu (ill-conditioned systems of equations) vh phung trnh iu kin tt (well-conditioned ).nh ngha v tm chun ca mt matrix (norm of a matrix).

k tip theo, bn s bit c:

nh ngha v cch nh gi condition number ca mt ma trn kh nghch.

Lu : nguyn tc ca blog l khng dch thut ng ting nc ngoi nhm m bo chnh xc v gip bn c th t tm kim thm ti liu tham kho nu mun.

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Mi quan h gia condition number ca mt ma trn vi cc iu kin mnh yu ca mth phung trnh tuyn tnh v bn c th tin bao nhiu phn vo nghim ca cc h ny?

Chng ta cng bt u bi vit bng vic xt dng tng qut ca h phung trnh nh sau:

H phung trnh iu kin yu v mnh l g?Mt h phung trnh c gi l ill-conditioned nu s thay i nh cc h s ca ma trn hoc s thay i nh trong v phi dn n s thay i ln trong vector nghim .

Mt h phung trnh c gi l well-conditioned nu s thay i nh cc h s ca ma trn hoc s thay i nh trong v phi dn n s thay i nh trong vector nghim .

V d 1 - ill-conditioned system of equationsXt h phung trnh sau y

H phung trnh ny c nghim l

Thay i mt cht v phi ca h

s cho ta nghim mi l

Cn nu ta thay i mt cht cc s trong ma trn

ta s nhn c nghim

4

5

4

5

4

5

4

5

4

5



C hai nghim mi nhn c ny iu khc xa nghim c d ta ch thay mt cht t gi tr cch s. Do , ta ni h phung trnh trong trng hp ny ill-conditioned.

V d 2 - well-conditioned system of equationsXt h phung trnh

H ny c nghim

Cng tung t trn, ta tin hnh thay i mt cht t v phi ca h

khi nghim mi l

Cn nu ta thay i trong ma trn v tri

v nghim mi l

C hai nghim mi nhn c ny iu thay i rt t so vi nghim c khi c s thay i t v phi hay ma trn h s. Ta ni h ny well-conditioned.

Nu h phung trnh l ill-conditioned, ta khng nn tin tng qu nhiu vo nghim. S d nh

4

5

4

5

4

5

4

5

4

5

4

5

4

5

iu g s xy ra nu h phung trnh l ill-conditioned hay well-conditioned?



vy v bn th ngh xem, mi khi bn gii mt phung trnh no , hoc bn chuyn t bi tonthc t sang vic phi gii mt h phung trnh tuyn tnh, bn mun th vi nhiu h s khcnhau thy s thay i ca nghim. Trong khi bn ch thay i rt rt t h s, vy m nghimli sai khc mt s rt ln. Bn c hi lng c iu ny khng?

k tip theo, chng ta s i tm hiu chng ta c th tin bao nhiu phn vo nghim ca mtphung trnh, cch tin l nh th no. Mi ma trn kh nghch u c mt condition number vmt machine epsilon, da vo hai lng ny, chng ta c th nh lng xem bao nhiu con strong kt qu nghim l ng tin cy.

c th tnh c condition number ca mt ma trn nghch o, ta cn phi bit trc chunca ma mn l g v lm cch no c th xc nh c n.

Chun ca ma trnCng ging nh nh thc (determinant) ca ma trn, chun ca ma trn cng l mt i lngv hng (scalar number). Tuy nhin, chun th lun dung phi c nh ngha cho tt ccc ma trn (bt k ma trn vung hay ma trn ch nht, ma trn kh nghch hay khng).

Chun ca ma trn c nh ngha tung ng vi chun ca vector c cho bi

Chun ca ma trn o c sc nh hng ca s thay i trong ma trn n s thay i trongvector nghim.

Mt trong nhng nh ngha ph bin ca chun ma trn l tng tr tuyt ln nht theo ct(maximum absolute column sum). Chun ny c tnh bng cch ly tng tt c cc h s trnmi ct ri so snh gia cc gi tr vi nhau, gi tr no ln nht chnh l chun ca ma trn. ivi ma trn dng ct,

nh ngha th hai l tng tr tuyt ln nht theo dng (maximum absolute row sum),

V d 3Vi ma trn

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Khi y,

Mi bn n xem k tip theo.

7peoplelikethis.Bethefirstofyourfriends.Like

Dinh Anh Thi : Tc gi blog, xut thn l sinh vin s phm ton, hin ang l nghincu sinh v Ton ng dng ti Php. Chnh v nim am m Ton hc, tin hc ng dngv qua trng lp s phm nn Thi rt quan tm n vic ng dng Ton hc, Tin hcvo trong cuc sng thc t cng nh vic truyn t kin thc Ton cho hc sinh theomt phung php hin i v tch cc nht, gy hng th v hiu su cho ngi hc,ngi c ni chung. Mi chi tit xin lin h [email protected]. Blog c nhnAnh Thi blog.

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31.10.14 , Hiu ton hc Ton cao cp

Vit bi cng blog bng cch lin h [email protected].



Tnh hp l ca nghim hphung trnh v khi nim

'condition number' ca mt ma trn (k1)

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Tính hợp lý của nghiệm hệ phuơng trình và khái niệm 'condition number' của một ma trận (kỳ 1) _ Math2IT.pdf