Jacobi Similarity Transformation for SVD and Tikhonov Regularization for Least Squares Problem: The Theoretical foundation
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1 IOSR Joul of Mhemcs IOSR-JM e-issn: ISSN: X Volume Issue 5 Ve VII Se - Oc6 PP wwwosoulsog Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: he heoecl foudo Sehe Ehdmhe Uwmus Deme of Mhemcl Sceces Kog Se Uvesy Aygb Kog Se Absc: he e eses soluo o Les sues euo s dve by Sgul Vlue Decomoso SVD obed fom usg Jcob Smly sfomo whou ecouse o hoov egulzo mee We use he LU decomoso d QR lgohm s bses of comso fo he obed esuls A olyoml f of ode fve fo he Les sues oblem ws doed Smle umecl d oblem ws obed s my souce od Kog Se Uvesy Aygb showg he fucol elosh bewee emeue d Relve Humdy I s suggesed h he SVD s umeclly bcwd sble s esed o by he LU decomoso mehod elve o sml esuls comued by QR d ohe ow mehods Keywods: les sues euo sgul vlue decomoso Jcob smly sfomo lgohm vese oblems AMS subec Clssfco 65G4 65F5 I Ioduco I he e we cosde les sues euo s lcble o modellg my Scefc oblems usg SVD obed fom Jcob Smly sfomo []Les sues euo belogs o he clss of mos oweful vese oblems whch dels wh he ocess of clculg fom gve se of obsevos he cusl fcos whch oduce hem Such suos wee ledy ofe ecoueed ocs d couscs commuco heoy d lguge ocessg clculo of he desy of he eh mesuemes of s gvy feld medcl mgg comue vso oceoghy soomy emoe sesg mche leg d o-desucve esg [] Bsc ools fo dscussos e vese fuco heoems fo se vlued ms eguly os fo Lschz fuco chcezo of gel d mec eguly d Lschz behvou of mulfuco coeco wh closed cove gh heoem[3 4] he dsce bewee eguly d sog eguly of se vlued m F : X Y whch he Jcob m my be vesged fo oeess oologcl sce foms he bss of ou dscusso hs c be vefed usg heoem due o [5] whch ses h evl m [A] s sogly egul f md A] ] d [ ] [ A whch my be we u o fco whee s he sgul vlue of he m A decesg ode he elve dsce fo whch m A s sgul o he se of sgul A s s gul A mces some om s defed s ds A m A A he ecocl of codo umbe c be eeed s mesue of eess o sguly of he m A d heefoe ds A K A As esul [6] oved h fo such sgul m A A hee holds he esme: A h s fo b A A A K A b A A b A K A we hve Ioduced hus o he dscusso s he well ow Holde s om defed he fom DOI: wwwosoulsog 76 Pge
2 Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: Whee s udesood [7] h C 3 4 C m e d e e e e s he cocl bss By fuhe eoso o Holde euly would yeld h: e R hs mles lm ; fo e hs bgs us o he coce of Lschz couy fo f me logous o Rem egble fuco [8] whch eles h fo evey om R hee s b g d g d b such h : f [ ] f d f d Md M I follows h fo cove fucol defed o D R we would eec h g f s couously dffeeble o ] [ fo whch f f g g g d f f [ ][ ] Wheefom esos due o [9] mlced h f f f [ ] M em vld Moe moly whe he m A s sue m he loghm of A s defed by he euo log A A I A I I d d s codo umbe K A K A log A log he emg he e s cegozed s follows: I seco he secl decomoso v Jcob smly sfomo s gve heoecl foudo he Gves ohogol m le oo fo he cosuco of oul QR decomoso s useful ool fo umecl comuo s hghlghed Seco 3 he e dscusses he sgul vlue decomoso SVD fo he les sues oblem s well s hoov egulzo mees he cse of sguly I seco4 we demose wh umecl emle wh descbed mehods hee mgh be eed o comue zeos of olyomls; heefoe seco 5 he e eses evl Newo-Le mehods fo smuleously efg ll zeos of olyoml euo wh cul efeece o he wo of [] Numecl lluso s demosed wh he mehods seco 6 Flly cocluso s dw bsed o he segh of ou fdgs II he Secl Decomoso v Jcob Smly sfomo he moce of comug egevlues d egevecos of el o comle m co be oveemhszed ou esech whch s ofe me hyscs d egeeg cces good emle s vbo oblems d lyss of vces sscs he Jcob smly sfomo ws deved by Jcob 946 d s lcble o el symmec m A [ ] I he cse he m s o symmec oe mus fs sfom he m A o symmec cse he Jcob smly sfomo m s bsed d DOI: wwwosoulsog 77 Pge
3 Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: o he heoem of cl es whch eles h ech symmec m A c be sfomed by mes of ohogol m Q o sml dgol m D such h Q A Q dg he m Q s ohogol d s defed he fom: he lgohm fo Jcob smly sfomo s evously used [] s fuhe descbed s follows u o he dscusso: Fo m of ode : Defe ode of ccucy If he we egevlues e DOI: wwwosoulsog 78 Pge so Else efom he followg oeos begg wh se Choose v If Se m wh < 4 Else ob he vlue fo by seg Whee d sg of sg of 4 v Ob he ew m A fom he Old m A by usg he followg udg fomul: Fo ech whee d Se cos s By symmey cos s By symmey cos s cos s By symmey
4 Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: s s cos cos By symmey e cos s cos s by v d symmey Elsewhee v Ree oeos sg fom se Fsh Fsh Ed I mus be oed ou h he sue oo of dgol m mus be flly comued ssumg o symmec m ws oglly sfomed o symmec m s he lgohm bove I well develoed sese he Gves Roo Ohogol m [] whch s efed Jcob Smly sfomo obed by Gves 954 fo develog he QR decomoso of symmec m whee cos d s esecvely s comued he fom: cos s Moe defve fuhe use of Gves oo s educo of symmec m A o -dgol fom whee Sum seuece becomes hd ool fo he culzo of olyoml euo o whch y of Newo s mehods s lcble he gves oo s fe bu Jcob s s fe fo ehusve dscusso o hs ese edes e efeed o []We lso oed ou h Householde Refleco c be used s leve ug comuol cos o cosdeo All he sme Gves le oo s wohwhle he sudes of oboc ms mulos III he Les Sues Poblem he dscezo of ll-osed oblem ofe leds o le les sues oblem he fom Fd m m A b such h m A b A R s he les sues soluo of he oblem I mos eemel wo whch s ofe he cse cce he m A my hve lge umbe of sgul vlues vey close o zeo hs ushes he ose whch s ese b he gh hd sde o be mlfed he seudo-vese soluo A b 3 wh esul effec of huge codo umbe d heeby edeg he ome soluo useless [3]We hus oduce he Sgul vlue decomoso SVD o he comuo s dydc decomoso of A : A U V dg whee U U U ] U s m [ m m wh ohogol colum vecos whee V s uy m As esul he soluo ocess by SVD s he fom: c v whee c u b 3 I he bsece of e sguly o sguly SVD s umeclly bcwd sble O he ohe hd f hee s sguly he soluo sysem hee s hus eed fo develome of good umbe of egulzo echues [4] fo whch comes o focus he ychoov egulzo h elces soluo of ogl sysem m A m b L A R LR by he eby sysem A A I A 34 b 33 DOI: wwwosoulsog 79 Pge
5 Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: he mee s egulo whch cools he wegh gve o he mmzo of he mmzo of he esdul A L L elve o d b A I s suosed h L m he m L s ce m whch my be e s dey m o dscee omo o some devve oeo [6]Noe h he mmzo of oblem 33 he ssumo Null A Null L O s mo fo useful uose beg ohogol d R ue dgol he ue gul mes fo hee dffee cses e ouled below: I he me beg we defe he QR decomoso wh m Q m R m< R mm m 3 m> R o O m I oe ssg he eed hoov mehod wh esdul [6d 7] s he euo A A I A b A = 35 I he mlemeo of mehod 35 s dvsble s sg o by g A A I A b d Ideed mhemclly euo 35 hs fed cocve guge [5] ecessy codo fo Lschz couy fo whch he Bch Fed o heoem s deely mlced We shll o delve o dels hee hs dscusso DOI: wwwosoulsog 8 Pge
6 Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: IV Numecl Eemes Emle he eemel d wee g s my souce fom RODAN Aygb Kog Se Uvesy s showed ble SN DAEIME ble EMPERAURE X O RELAIVE HUMIDIY % Y I s suggesed h emeue s deede vble X d h Relve Humdy s he deede vble Y We use wdows 7 veso of MALAB fo he clculos d se of esuls flog o hmec s dslyed ble ble showg Resuls comued fo he cosdeed mehods decomoso QR decomoso Sgul vlue hoov decomoso mehod LU mehod fo 3 ^ e+6* mehod fo 3 ^ e+9* ^ e+6* Regulzo mehod 34 wh [] ^ e * DOI: wwwosoulsog 8 Pge
7 Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: I ble comued esuls fo LU decomoso d Sgul Vlue decomoso SVD e umeclly bcwd sble wh hgh degee of closeess whe comed wh esuls fom QR Fcozo O he ohe hd he hoov egulzo mehod seemed o hve efomed vey ooly hs cse whch ws ue ueeced We oo he mee he evl of [ ] d foud ou h hee ws o sgfc dffeece he esuls s ll comued esuls by hoov eguled mee mehod he evl sed wee he sme he gh llusg he bove esuls c be demosed usg MALAB lo fuco s dslyed Fgues -3 Ech of he mehods he fgues ws efeed o Sees 3 d 4 esecvely h s o sy he LU mehod Sees QR mehod sees SVD mehod sees 3 d hoov egulzo mehod sees 4Comuolly he ll-osed oblem mlced h A A he secum of A A h fo some hee s m hece he m A A cl vecos deve he Jod fom of dco h egevlues of A A [] A A e ubouded hs mes s degeeed hus he egevecos d Fgue Fgue DOI: wwwosoulsog 8 Pge
8 Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: Fgue 3 V Comuo of olyoml zeos by he Ievl Newo Oeo Suosg he coeffces eg he olyoml f fo he les sues oblems e eessed he fom of ucey such h: P whee e el o comle 5 If we omlze he bove olyoml euo moc olyoml wh ledg coeffce uy s obed he Newo oeo fo sgle vble euo s lcble o he wo s gve he fom 5 Le be o emy cove subse of Bch sce d le be coco of o self If hee s N such h N : N comc se emy closed cove se K such h: K K d N d N hs les wo os he hee ess o K K s he comleme of I s suosed h of os s seed h s hey e K sce d Ldelof codo fo s : R eeds o couous fuco o If we ele closely comubly om holds By eze eeso heoem would follow h s oml d closed hus evey couous fuco wh Hh-Bch eeso heoem he we would l eguly sces wh omly v Uysoh s mezo heoem h s ffoded by ychooff so h he closed gh heoem d ufom boudedess cle he lls of Be s cegoy e foce fo whch he bsc cle fo fdg zeos of olyoml holds I s eeced he u bll B * D s lso comc he we s oology he heoecl foudo s ele meoed he begg dedced o he wo [] s s follows Fo moc olyoml of degee whose zeos of esecve mullces e ow we defe h P 5 Ioduce he followg oos he sese of [67] h: v v v ; DOI: wwwosoulsog 83 Pge
9 Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: DOI: wwwosoulsog 84 Pge ; ; By fuhe usg des due o [96d 7] would be obed h 53 he fed o elo of Lgueel s ye mehod c be develoed [6] wheefom: 54 I he cse of smle zeos fomul eessed euo 54 s he fml Lgueel s mehod of hd ode of covegece he choce of he wo oos o be e fom he wo dss euo 54 s guded by he dscusso deled [6] Imlemeo of he bove Ccul evl hmec by smuleously efg he zeos of he olyoml euo P whee he mdo d dus of he efg ds should esec d g o cosdeo h ^ s ew omo o he zeos of P s we he fom: f 55 md I I Euo 54 S S S S f We oed h veso of he eve mehod of Hlley ye he cse of smle zeos euo 5 s gve by he euo: = 56 he umbe of mullces of he oos gve ds R ID he sese of [7] s he Lgouell lmg fomul s deved [8] he fom: lm lm lm d d D d d 56 Whee deoes he ege closes o he el umbe
10 Jcob Smly sfomo fo SVD d hoov Regulzo fo Les Sues Poblem: VI Cocluso he e suded use of sgul vlue decomoso obed fom Jcob smly sfomo fo he soluo of les sues oblem We comed oes wh sml esuls comued by he LU Fcozo d QR lgohm Smle umecl emle ws e s my souce d fom RODAN Cee fo Lowe Amoshec Sudes Kog Se Uvesy Aygb Kog Se Nge fo he emeue d Relve Humdy fo eod of hy fou mohs whch ws he me ge he Cee fo Lowe moshec Sudes ws esblshed A olyoml f of ode fve ws used Numecl esuls comued by he bove med mehods e gve ble We lso llused he esuls comued fo he uose of foecsg s show Fgues -3 I s suggesed h hee s sog coelo he d s see fom Fgue 3 s esul of coley bewee LU decomoso d SVD A effce evl bsed mehod fo smuleously efg zeos of olyoml whe coeffces e eessed he fom of ucey ws dscussed he sese Peovc [ 6] fo he uose of esy ccessbly o he edes I comug wh Jcob mehod oe fs sfoms o symmec m o symmec m I s ecommeded h whe m A s ely dgol Jcob smly sfomo mehod wll lwys hve ue hd ove QR mehod ovded h oe sog ceo s secfed fo Jcob s mehod [6] Refeeces [] Wlso JH 965: he Algebc Egevlue Poblem Cldo Pess Ofod [] Wled : Ivese Poblem he fee ecycloed ; Auhoy Cool:NDL:57748 Jue 5 [3] Uwmus SE 5 Gh Comleo cluso sooe fo Ievl Les Sues Euo Amec Joul of Mhemcs d Sscs [4] Neume A 99: Ievl mehods fo sysems of euos Cmbdge Uvesy Pess Cmbdge [5] Rum SM 996: he dsce bewee eguly d sog eguly I G Alefeld d B Lg edos Scefc Comug d Vlded Numecs Vol 9 of Mhemcl Resech P 5-7 [6] Boc A 9: Numecl Mehods Scefc Comug Vol SIAM Phldelh [7] L W 8: Lecue Noes of M Comuos Deme of Mhemcs Nol sg Hu Uvesy Hschu w 343ROC [8] Oeg JM Rhebold WC : Ieve Soluo of Nole euos sevel vbles Clsscs Aled Mhemcs SIAM Phldelh USA [9] Neume A: Ioduco o Numecl lyss Cmbdge Uvesy PessCmbdge [] Roh J 5: A Hd boo of esuls o evl le oblems Czech Acdemy of Sceces Pgue Czech Reublc Euoe Uo;5wwwcscsczoh [] Peovc MRcc L d Mlosevc D 6:Lgueel-Le mehods fo he smuleous omo of olyoml mulle zeos; Yugoslv Joul of Oeos Resech [] Soe J d Bulsch R 98: Ioduco o Numecl Alyss Sge Velg New Yo [3] Uwmus SE d Ou FO : Comuo of Egevlues of Hem M v Gves Ple Roos Nge Joul of Aled Scece Vol P -6 [4] Golub G d V-Lo F983:M Comuoshe Johs Hos Uvesy PessBlmoe Myld [5] Sef W 8: ol vo egulzo fo le ll-osed vese oblems: Eesos d lco PhD hess Gdue College Azo Se Uvesy USA [6] Uwmus SE4:Regulzo of No-smooh fuco whose Jcob s ely sgul o sgul Amec Ieol Joul of Coemoy Scefc Resech-8 ISSN [7] Mye G 998: Eslo-flo wh cocve evl fucos Alcos of Mhemcs [8] Peovc MS 989: Ieve mehod fo smuleous cluso of olyoml zeos 387 Sge Velg [9] Peovc MS d covc S 995: Zeo fdg mehods of Fouh Ode Joul of Comuol d Aled Mhemcs No 64 P 9-94 [] Fme MR d Lozu G 977: A Algohm fo he ol o Pl Fcozo of olyoml Mhs Poc Cmb PhlSoc No 8 P DOI: wwwosoulsog 85 Pge
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Addition & Subtraction of Polynomials
Addiion & Sucion of Polynomil Addiion of Polynomil: Adding wo o moe olynomil i imly me of dding like em. The following ocedue hould e ued o dd olynomil 1. Remove enhee if hee e enhee. Add imil em. Wie