CS537. Numerical Analysis

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1 CS537 Nuercl Alss Lecture 6 Lest Sures d Curve Ftt Professor Ju Zh Deprtet of Coputer Scece Uverst of Ketuc Leto KY Aprl 6 5

2 Method of Lest Sures Coputer ded dt collectos hve produced treedous out of dt tht re possle to uderstd wthout soe sort of postprocess Gve set of dt If we ssue tht the dt for ler fuctol relto we c wrte the fucto s Wth the coeffcets d to e detered For ech prs we c reuest For =. Note tht f > we hve ore th two ler eutos to detere ust two uows

3 Lest Sures Ft M sple pots whch e ccurte. Not ood for terpolto. 3

4 Over Detered Sste I eerl we hve ore eutos th uows. here s o ect soluto to the prole. However we could detere soluto tht zes the totl error Suppose the ler euto s ve s If the pot s o the strht le defed the fucto we hve I ost cses pot s ot o the le we hve For =. r s the curve ftt error r 4

5 Mze the Errors Most sple pots re ot o the curve. Hopefull the totl dstce etwee the sple pots d the ftt curve s zed. 5

6 Mz otl Error Oe c reso tht f the su of the errors s zed the dt should ft the le s est s t c he totl error c e represeted s r We c ze the ove fuctol to select the coeffcets d. hs c e solved the techues of ler pror hs s l pproto or pproto he shortco of ths forulto s tht the fucto of the totl error s ot dfferetle. M tools clculus ot e used 6

7 Method of Lest Sures A ltertve s to ze dfferet error fucto whch s cotuousl dfferetle hs s lso clled l pproto. It s specl cse of the l p pproto wth the l p or s defed s p where = s desol vector p Fro sttstcl cosdertos f the errors follow orl prolt dstruto the zto of φ produces est estte of d / p p 7

8 How to Copute Mu 8 We use techue clculus to detere the etree pot of fucto hs ves us two eutos he re clled the orl eutos d c e wrtte eplctl s whch c e solved for d

9 Lest Sures Soluto 9 he soluto of the prevous two two ler sste c e solved s Where Lots of sple coputtos Ler Eple d d d

10 Lest Sures Ft he dt the prevous slde led to whch c e solved for =.6 d = 3.

11 Nopolol Eple We c ft tle opolol fucto

12 Bss Fuctos A eerl lest sures ftt c e wrtte s c I whch the fuctos re clled ss fuctos. he re ow d ept fed Gve set of dt we wt to fd the vlues of c to ze the totl error s c c c c We set the prtl dervtves to e zero c

13 Bss Fuctos II 3 he prtl dervtves re for Sett the sulteousl zero we hve hs s the orl euto whch s sste of ler eutos wth uows c c c he coeffcets of the ler sste re he coeffcet tr s osulr f the ss fucto re lerl depedet. he ss fuctos should e pproprte for the prole uesto d e the result coeffcet tr well codtoed c c c

14 4 Orthoorl Bss Fuctos Gve set of ss fuctos { } the set of ll fuctos tht re ler cotos of the ss fuctos re We re loo for prtculr Є G such tht the ftt totl error s zed A + fuctos tht re lerl depedet c e used s ss fuctos. Dfferet choces of ss fuctos e the orl euto For eser or ore dffcult to solve } { c G tht such : c

15 Choose Bss Fuctos We s ss { } hs the propert of orthoorlt f I ths cse the orl euto s splfed s whch c e evluted strhtforwrdl c he Gr Schdt procedure c e used to orthoorlze ve ss order to hve the ove propert. hs procedure e epesve We c lso choose soe ss fuctos so tht the coeffcet tr s es to solve ot ecessrl s dett tr 5

16 6 Polol Bss Fuctos Cosder G s the spce of ll polols of deree. We turll choose A polol G c e represeted s he sple ss s however ot ver ood sce the re too uch le Assue we hve the dt restrcted the tervl [ ] wth We c defe set of Cheshev polols tht for ood ss c c

17 Orthool Polols Orthoorl ss polols ssocted wth the Leeder polols 7

18 8 Cheshev Polols he frst few Cheshev polols re he Cheshev polols c e eerted recursvel s he c lso e wrtte s A fucto c e represeted s ler coto of the Cheshev polols A fucto f wrtte s ler coto of the Cheshev polols c e evluted effcetl rccos cos c f

19 Cheshev Polols he frst few Cheshev polols 9

20 Evlut Cheshev Polols o evlute f for ve We use cwrd recurso procedure he Cheshev polols re defed o the tervl [ ] we would lso le the scsss { } to le the tervl [ ].e. { } = d { } =. If the le dfferet tervl [] we c use trsforto to p the tervl [ ] oto [] c f w w f w w c w w w z

21 Evlut Cheshev Polols w w w w w w w w w w w w w w w w w w w w w w w w c f

22 Alorth of Polol Ftt. Fd the sllest tervl cot ll wth = { } d = { }. Me trsforto to the tervl [ ] us the p 3. Decde o the order of the polols roud 8 or 4. Us the Cheshev polols s ss eerte the + + orl eutos for z z c z z

23 3 Alorth II 5. Use euto solv route to solve the orl eutos for coeffcets c c c to ot the fucto 6. rsfor the fucto c to the orl vrle s he coputtol etesve prt s to for the coeffcet tr of the orl euto Specfc procedures re detled oo c f f : : : z z z A Ac d let

24 Polol Reresso Assue the dt collected cot errors the procedure for sooth dt s to reove the eperetl errors s uch s possle Sooth dt s dfferet fro terpolto sce the ltter ssues tht the dt re ccurte Gve tle of eperetl dt We wt to fd polol tht represets the orl dt fetures We hve where ε s the oservtol error P N P N N 4

25 Polol Reresso II We c use the ethod of lest sures solv sste of orl eutos to detere P. A utt clled vrce p c e coputed to see how ood the pproto s If the orl dt rell represet polol of deree N wth ose the N N We c copute σ σ utl we see for soe N tht σ N σ N+ σ N+ the we chose the polol P N s the oe represet the orl dt tred he drwc s tht we eed to copute p p

26 A Eple of Reresso A reltoshp etwee the hours studed d the test scores

27 Ier Product Let two fuctos f d whose dos cot { } we defe f f s the er product of the fuctos f d A er product of two fuctos hs the follow propertes. f f. f f uless f for ll 3. f f where s sclr 4. f h f f h 7

28 8 Orthool Polols A set of fuctos s orthool f f = for two dfferet fuctos the set We c eerte set of orthool fuctos s For where he polols { - } c sp ler spce whch the re ss

29 9 Orthool Polols We c chec the orthoolt of polols s he re prt c e proved us ducto for Assue

30 Well Defed? We eed to show If ths s ot the cse the hs es tht It follows tht t hs + root. If s sller th the we ow the s the zero polol whch s ot true sce d [ lower order ter s ] for 3

31 Represet Fucto A polol of deree the sped ler spce c e represeted s p If we for the er product wth respect to o oth sdes For d us the fct tht = f wh? we hve Hece for = whch re the eeded coeffcets p p p 3

32 Solv Icosstet Eutos A sste of ler eutos of the for wth > s cosstet f there s o possle vector to e the resdul zero. here s o soluto the covetol sese stsf ths sste It s of soe terest pplctos to fd the vector tht zes the or resdul We c te the prtl dervtves wth respect to d set the eul to zero to ot the orl eutos for 3

33 Drect Fctorztos he orl eutos oted c e solved Guss elto d t s the soluto of the orl sste the lest sures sese If we wrte the orl ler sste s A = It s lso possle to drectl fctor the tr A s A = Q R where Q s + + orthool tr stsf Q Q = I d R s upper trulr + + tr stsf r > d r = for <. We the hve R = Q whch c e solved c susttuto 33

34 34 Sulr Vlue Decoposto he QR fctorzto c e oted lorth clled odfed Gr Schdt procedure to orthoolze the row vectors A ore volved lorth eeds to copute the Sulr Vlue Decoposto of the tr A s I whch U d V re orthool.e. Ad Σ s + + dol tr hv oetve etres V U A I V V I U U r o

35 35 Pseudo Iverse If A s osulr sure tr wth We c copute the true verse of A.e. A to solve the ler sste s If A s + + rectulr tr we frst copute the sulr vlue decoposto of A s the for of We the vert Σ s V U A A A r

36 Pseudo Iverse II We c defe the pseudo verse of A s Ad the soluto of the rectulr ler sste s defed to e It c e show tht ths defto of soluto does ze the resdul or of the orl cosstet sste Note tht QR fctorzto d Sulr Vlue Decoposto re ore epesve to perfor cses th solv the orl eutos. I cse tht the tr A s sprse tr we e le to solve t ore effcetl us cert tertve ethods For the orl euto A A s opto A A V V U U 36

37 Proof I Cosder sste of ler eutos A = d A s tr. he l resdul soluto of the sste s A V U wth Proof. Let e vector. Defe Us the orthoollt propertes we hve A V f U f V Note tht s dol tr V UV d U c U f A f UV U f c 37

38 Fro prevous pe we hve Proof r c r c o ze the epresso we eed to ze the frst ters o the rht hd sde def c for r c Other copoets of c re uspecfed. We c set So we set d c for r V V c V U A 38

39 39 Soe Propertes here re soe terest propertes for the pseudoverse. he re clled Perose Propertes. Proof: A A A A AA AA AA A A A AA A A V U V U V U U V V U A AA

Sequences and Series

Sequences and Series Secto 9. Sequeces d Seres You c thk of sequece s fucto whose dom s the set of postve tegers. f ( ), f (), f (),... f ( ),... Defto of Sequece A fte sequece s fucto whose dom s the set of postve tegers.