The Exact Methods to Compute The Matrix Exponential

Transcription

1 OSR Jual Mahas OSRJM SSN: 858 SSN: 65X Vlu ssu V V Ja b 6 86 wwwsualsg Th Ea Mhds Cu Th Ma Eal Mhad bdullah Sala V C Ba Da Mahas& sass Yashwa Mahavdyalaya Swa Raaad Th Mahwada Uvsy Nadd da Da Mahas & Sass Yshwa Mahavdyalaya Swa Raaad Th Mahwada Uvsy Nadd da bsa: hs a w s sval ds hds ha allw us u h al a aly Ths hds lud alulag gvalus ad Lala ass a wll w ad a d h lss Oh hd wll w s d h lau ha d ludg h alula gvs ad whh vd gal ulas alabl ay a Kywds: Eal a us a LagagSylvs la uz Sal ula Lala as Cug Ma Nug Ma du Th al a s a vy usul l slvg la syss s d vds a ula lsd slus wh h hl hs a b aalyzd llably ad b svably a la sys [] Th a sval hds alulag h a al h uaally [8] Hwv a hal vw s a w s hs a u ulas vlvg h alula galzd Lala as ad gvs hav b usd a lag au bs ad hs as hs w s vd alav hds wll w dly dda Th a h hds [] a s sg bu d h ls ass baus s aaly la Egh ass dvl hds alula ass baus s aaly la Egh ass dvl hds alula h a al vd als hw aly h lssw hds s ass ad h s w ass h sv bblgahy d Th al a! Ds d Rsuls l a dd by dd by! DO: /5886 wwwsualsg ag! T s h vg hs ss w d sly h bus a a sz a : ds h h lu ad : h h w s asy s ha s sasy as llw : : W wll us hs sadad v baus a ds v sa all s a quval a y s w hw us aws h bus a a du Gv h as ad B h h du hc B wh s : B : had l s w ba uga C h w hav : B : B : C C ald w : Rall h CauhyShwaz qualy : B : : B: B

2 Th Ea Mhds T Cu Th Ma Eal lyg s qualy a squa a s asy ddu : all ally w us a h vg h llwg l: s su bsv ha sass:!!! l = ad hus dsad ha alady s wll dd ay squa a wh sa y s usul b hw h a al bhavs ud dva: d d d d!!!!!!!!! 5 5! 5! Usg du hd ad va d llws h ula: d K d N ha h ula h s dvav ls ha h u bl h llwg s d sys: Tw suls w la algba ha w usd blw a h llwg hs Th Shu Tagulaza Th s slu al valu ay squa a h s a uay a U suh ha UTU s u agula add h s a dagal a T a h gvalus Th Cayly Hal h L a squa a ad s haas lyal h Nw w wll dsuss sval hds by dals alula u ad d h a al D Dagalzabl Ma Gv a dagal a D dag s say ha dag s u all h w hav D D dag!! dag! DO: /5886 wwwsualsg ag

3 Th Ea Mhds T Cu Th Ma Eal s als a dagal a Nw h as ha s a a dagalzabl s w ha h ss a vbl a d by h gvs ad a dagal a D d by h ds gvalus suh ha Th w hav:! D D! D Nw s asy vy h dy D! D all dgly s val d h al a a dagalzabl a vdd ha vusly w d all h gvalus wh sdg gvs Ths as s wll w V N Dagalzabl Ma Sus s dagalzabl a whh s ssbl d laly dd gvs h a hs as a us h Jda Sus s h Jda wh h as a Th Wh dag dag Th J Thus h bl s d h a al a Jda bl wh h Jda bl has h J N M ad gal a J N as s h h u dagal ad s h ull h ds h a By usg h abv ss w hav Ths a b w J!! N N N!! ally w g h al a as llw: Th dag J J N! J Ths as s als wll w O way dsa hs ula s sd h sys s d J wh al d O h had w w ha h slu hs sys s gv J by uh hs sys s asy slv sag las qua whh s duld h vy la sd qua s slvd by va h hd gag a V Tagula Ma: L S s a u agula a lw agula a sla dvl s d ad w as h su a dagal a wh a l a: S D N Rall ha a a N s alld a l h s a sv g suh ha N Th salls sv g whh hs qualy hlds s alld h d l wsh a ssug h w y al as s [] B B B B s D N D N N Nw w a us h abv ula alula: DO: /5886 wwwsualsg ag

4 Th Ea Mhds T Cu Th Ma Eal DO: /5886 wwwsualsg 5 ag W w hw alula h a al a dagal a s w dsuss hw g N Su ha wh N l h ss hs a bs s h ub a quay b addd ah s budd by d l N Ths a h sa s ld by h dg s al lyal b ha all l a has all z gvalus W a galz hs hd ay a T d hs sly aly h h Shu agula USU wh U s a uay a ad S s u agula Ths ylds: U U S ad h w w hw d V uz s Sal ula: [5] uz dsbs w hds alula Ths a basd h a ha s a lyal whs s a sala us ha a b ud usvly by slvg a sl sys la dal quas h s d W shw ly h sd hd baus s as udsad ad l Th 6 Gv a a sz sus w w all s gvalus ssaly ds bu lsd a s abay d Th hlds wh s dy wh ad a slus h dal sys : u ad d Gv u s h s w a a ad h sd a b dsbd as h gh sd qualy sls

5 Th Ea Mhds T Cu Th Ma Eal Nw as s u ha uh has h ss bas s dud Bu h CaylyHal h S w bad ha ls ha ally as s llw DO: /5886 wwwsualsg 6 ag by uquss slus s a ala hs hd w d ulas h al a a a h wh gvalus dg h au h gvalus w hav h ass sudy a Ral ad ds gvalus: W us slv h sys quas Slvg h s qua whh s always duld w g h sd usg h hd gag a w ba Csquly w ahvd h llwg ula b Ral ad qual gvalus: hs as slv h sys quas w g ad S w g h ula CCl gvalus: h as C wh gvalus bl usg h sa ula as h as al ad ds gvalus Bu has al s ad s gvalus wuld b l ugas l say a b a b wh b hs as slv h sys quas w g a s b s b a s b b wh s a slu sys wh al d ad havg al s bvusly w s a al slu S h al a h Sal ula s b sdd al slu ha s R R R ad h luds ha a s b S ass sl: a s b b a [] sl shws hw ba l ulas h a al ass: a all gvalus a qual h llwg

6 Th Ea Mhds T Cu Th Ma Eal b all gvalus a ds has ly w ds gvalus wh ully algba Whl hs ass d v all ssbl alavs all gvalus a a hs ulas wll shw by sly ad baus hy hl us d all ssbl ulas al as lss ha qual sz shuld b d ha h uz's sal ula als hl us ddu hs ulas bu h way bad by sl s ul Th : s a a wh all s gvalus qual h w hav! : s h as ad u w w! h h CaylyHal h ls ad s h h s v Th : s a a L wh ds gvalus h w hav wh L a h Lagag la s gv by L : lhugh hs h s a sal as h la ula LagagSylvs whh gvs us ds W d h llwg a u sala vaabl as llw T v L wll shw ha sass h dal qua wh al d = a w bsvd ha sasy L By h Cay lyhal h w hav L ah ad s sass h dal qua add ally llw L uquss slus Th : L b a a wh w gvalus d ad wh has ully ad has ully Th hlds :! sala vs d h as! Tayl ss s d s DO: /5886 wwwsualsg ag

7 Th Ea Mhds T Cu Th Ma Eal DO: /5886 wwwsualsg 8 ag! Nw w valuad ad vly w a hs ss!!!!!! Nw w w h sd h las ss s bsv usg h CaylyHal h w hav qualy ha s By du llws ad s By lag hs lash h sd su w ba!!! whh ls h s w lud h ala h a al ulas ay a sz adg h ully s gvalus: a gvalu wh algba ully h hav b d gvalus hav d gvalus wh has algba ully w hav la LagagSylvs ad algh Gaah:

8 Th Ea Mhds T Cu Th Ma Eal DO: /5886 wwwsualsg ag Th hd llusad Gaah [] ly hls us alula h a al bu als valua ay aaly a u s w h as wh h gvalus a a a d h h gal as wh h a ulls Th 8: s a lyal a ad h gvalus a d h a b dsd as z h a h gvalus ad z s a a s gv by z : By h Cay lyhal a b dud a lyal dg l say Ths ss lyal a b ad va la Lagag whh T alula w ully h gh bh sd qualy by h h gv v sdg Ths du gvs us v v v v v wh h sd qualy s bad by sdg v wh h gvalus a d w hav v v ad qualy s asly ddud by as Th all gh gvs us Wh h gvalus a ds algh Gaah s [] a b usd ad h aalyal u as! wh s h ub ds gvalus s h ully h h gvalu s h dvav wh s valuad h h gvalu ad a h su as ha a ud d ay aaly u T llusa h us hs ula wh a al Eal 8: L us d h a

9 Th Ea Mhds T Cu Th Ma Eal DO: /5886 wwwsualsg 8 ag 6 wh ully ad wh ully Th w hav!!! Rall ha hs as s vald ay aaly u u sua w a sd h a vs H a h s b sdd T d h su as usg w lyal us s h a s sz h Cayly Hal h sas ha us b hav a lyal u ad h wll us h llwg a h sala vs s Th s a W ba h qua h sala vs s Th s a W ba h qua h sala vs s Th s a 8 6 W ba h qua 8 6 Csquly w us slv h al a sys 8 6 Th h slu s Nw u all gh w s ha

10 Th Ea Mhds T Cu Th Ma Eal DO: /5886 wwwsualsg 8 ag s h dsd al a shuld b haszg ha ud h su as; w a d asly s s a us aaly us Us udaal slus a la dal qua wh sa s: Th llwg hd avds wd h a s dagalzabl ad s quss us w Cayly Hal h ad w hw slv la hgus sala dal quas d wh sa s l N ha h hd s asy l h s h lyal qua haas qua assad wh h abv qua a usg g algbaally Th llwg hs hl us udsad hw hs hd ws Th s h guaas h s ad uquss a al valu bl a a dal qua whl h sd vds a hd sug h slus a al basd al valu bls sala dal quas Th : L b a sa a wh haas lyal d Th s h ly slu h a dal qua d gv by wh al d : s w v h uquss Sus ad a w slus sasyg h al ds gv D Baus lay hs u sass bu wh al d Ths as ha ah y sass h llwg al valu bl b wh s bvus ha h ly slu s all h val Thus w hav all ad ba uquss Nw w v h s g ha sass h al valu bl Eh h sa a wh haas ly al dslsd hyhss Rall h ula h dvav h al s Equa

11 Th Ea Mhds T Cu Th Ma Eal DO: /5886 wwwsualsg 8 ag W lad h gh sd Equa ba sud by h CaylyHal h ally h ula h dvav s ddud ad sass h al ds Ths ls h Th : L b a sa a wh haas lyal d Th w hav wh a h slus sala dal quas gv by ad sasyg h al ds : D wh slus al valu bls d h h s w shw ha sass qua dd w hav O O Nw w shw ha sass al d whh gv : Th uquss s sasd all L us gv hs al llusa hs hd Eal : W wa d h a al

12 Th Ea Mhds T Cu Th Ma Eal DO: /5886 wwwsualsg 8 ag 6 hs w alula ha haas lyal d W assu ha h Th s sasd 5 Th haas lyal dus h llwg sala dal qua wh sa s h h gal slu s ud basd h haas qua hus h gal slu as h T d w us h al ds by usg hs a w ba T d w us h al ds by usg hs a w ba T d w us h al ds by usg hs a w ba ally w la hs us ula 5 ad g 5 6 a b vd ha h a s dagalzabl bu hs s lva alulas W a avd s alulas h abv al? Th llwg h ss h sag Th : L b a sa a wh haas lyal d h w g h slu wh

13 Th Ea Mhds T Cu Th Ma Eal DO: /5886 wwwsualsg 8 ag B 6 wh B s h valua h a = h a B vdd ha S s a udaal s slus h hgus la dal qua : N ha s h haas qua h dal qua Du Th w hav wh s slu wh al ds w h s s als a udaal s slus s h Wsa W s s as h valu a = uh wh usg h w h uquss slus h al valu bl { w hav ah whh w ddu B s B s vbl s qual 6 whh ls h Csd h a Eal aa h sla h alulas h abula Rall ha h haas lyal s d whh w assa h la dal qua

14 Th Ea Mhds T Cu Th Ma Eal Suh as haas qua s w ba as a udaal s slus Wh hs w h a B wh s alulad B h w g B 6 wh vs B 6 8 By lag hs us h ula 6 DO: /5886 wwwsualsg 85 ag 8 by Th w hav b sasd h al a Eal s vd Lala Tas: h Lala as hd s usual us gg slv al valu bls sala la quas d wh sa s Ths hd s galzd slv h bl al valu lyg Lala as w ba sx s X s ls ha s X s suh ha w g X s s ag vs Lala as w ba L s Thus uquss s ahvd L s alas gg h al a s alld h sa as a als ad sg s h al a s [] V Clus llusad sval hds alula h al a a squa a Ms h d us h alula gvalus galzd a whh has b a sadad hd alg h bl sval ds a h lvl a Whl hs hds a b ald ay a all h a v dalg wh lag as aua s dals uad ubs aad wh s Ths s wh s abl us h u bu as h WVa La h s suabl hd ual la Rs [] J sals & N Mhl; La Syss Bhaus Bs [] T M sl; S El ulas h Eal Ma h a Mahaal Mhly 6 6 :8{ [] R Gaah; h Thy Mas Vlu Chlsa Nw Y 5 [] SH Hu & WK ag; h a al u aal Jual Mahaal Edua S ad Thlgy 6 :65{ [5] E J uz; vdg h Jda Caal h Dsuss La Syss wh Csa Cs Th a Mahaal Mhly 66 : [6] Nhlas J Hgha ad wad H lmhy Cug Ma us Mahs su Mahaal Ss Th Uvsy Mahs SSN [] R H ad C R Jhs Ma alyss Cabdg Uvsy ss85 SBN [8] R Blla du Ma alyss d d Nw Y: MGaw Hll6 d by SM hladlha5

15 Th Ea Mhds T Cu Th Ma Eal [] Syd Muhaad Ghua Th Cua Ma us aula Th Ma Eal Mas Thss Uvsy BghaEglad6 [] Nahl Salls Th al as Mas Thss Uvsy Gga Gga DO: /5886 wwwsualsg 86 ag