The Eigenvalue Problem of the Symmetric Toeplitz Matrix

Transcription

1 he Egevalue Poblem of he Smmec oelz Max

2 bsac I hs assgme, he mehods ad algohms fo solvg he egevalue oblem of smmec oelz max ae suded. Fs he oelz ssem s oduced. he he mehods ha ca localze he egevalues of oelz max ae suded. Fall, algohms ha ca solve he egevalue oblem of smmec max ae eseed.

3 Smbols ad Noaos m deoes he veco sace of all m -b- eal maces: m a a am a a m, Whe a caal lee s used o deoe a max e.g.,, B,, he coesodg lowecase lee wh he subsc efes o he, comoe e.g., a, b,. he -b- maces ae sad o be squae. he -b- de max s deoed b I ad s -h colum b e : e. I,,,,, Whe he dmeso s clea fom coex, we sml we I ad e, esecvel. If ad B sasf B I, he B s he vese of ad s deoed b. If exss, he s sad o be osgula; ohewse s sgula. deoes. If a, he s deema s gve b de = a. Fo we have 3

4 whee de = a de s a -b- max obaed b deleg he fs ow ad -h colum of. m deoes m, ad fo hese colum vecos we cusomal use lowecase lees ad deoe dvdual comoes wh sgle subscs. m hus, f, x, ad x, he a x.,, m he se of all lea combaos of as he sa of a,,a : a m,, a s a subsace efeed o sa a,, a =,, a m If S s a subsace, he hee exs deede basc vecos a,,a S such ha a S sa a,,. ll bases fo a subsace S have he same umbe of elemes. hs umbe s he dmeso of S ad s deoed b dms. hee ae wo moa subsaces assocaed wh a max age of s defed b R ad he ull sace of b m x fo some N x x. x m. he If a,, a he R sa a,,. a 4

5 he a of a max s defed b a dm R. m se of vecos x,, x s ohogoal f x x wheeve ad ohoomal f x x. Moe geeall, a colleco of subsaces S,, S m of s muuall ohogoal f x wheeve x S ad S fo. he ohogoal comleme of a subsace S m S m s defed b x fo all x S. If m ad we we dag,,, m m, he a s dagoal ad a fo,,. We sa ha s smmec f sew-smmec f osve defe f x x, x o-egave defe f x x, x defe f x x fo some x ohogoal f I loe f fo some. demoe f. osve f a fo all ad o-egave f a fo all ad dagoall doma f a a fo all emuao f,, e s e s.,. whee s,, s s a emuao of,,, 5

6 Coes Ioduco 7 Basc Facs o Egevalues 8 3 oelz Ssems 9 4 Localzao of he Egevalues of oelz Maces 4. he Embeddg 4. Egesucue Bouds fo he Egevalues Omum Values fo he m 8 5 he Smmec Egevalue Poblem 5. Mahemacal Poees udelg smmec egeoblem 5. dagoalzao ad he Smmec QR lgohm 5.3 he Sgula Value Decomoso Jacob Mehods Some Secal Mehods 8 6 Laczos Mehods 3 6. Devao ad Covegece Poees 3 6. Paccal Laczos Pocedues lcaos o Lea Equaos 35 7 Cocluso 36 Bblogah 37 6

7 Ioduco Ma Paccal oblems egeeg ad hscs lead o egevalue oblems. call, all hese oblems, a ovedeemed ssem of equaos s gve, sa equaos fo uows,, of he fom f,, ; x; :. f,, ; F. whch he fucos f also deed o a addoal aamee. Usuall,. has a soluo x,, ol fo secfc values,,,, of hs aamee. hese values ae called egevalues of he egevalue oblem., ad a coesodg soluo x x of. egesoluo belogg o he egevalue. he followg chaes ovde he ma heoecal esuls ad algohms o he egevalue oblem fo smmec max. Chae oduces he basc facs o egevalues. Chae 3 oduces oelz Ssems. Chae 4 dscusses how o solve he egevalue oblem fo smmec max. Chae 5 s eel devoed o solvg of sase max oblems. he dscusso evolves aoud he Laczos mehod. We show how vaous sase egevalue oblems ca be solved usg hs moa algohms. Chae 6 s localzao of he egevalues of oelz maces usg addve decomoso, embeddg cculas, ad he Floue asfom. 7

8 Basc Facs o Egevalues umbe C s called a egevalue of he max f hee s a veco x such ha x x. Eve such veco s called a gh egeveco of assocaed wh he egevalue. he se of all egevalues s called he secum of. he se foms a lea subsace of L : { x I x } C of dmeso a I, ad a umbe C s a egevalue of ecsel whe L,.e., whe ad hus I s sgula: de I. I s easl see ha : de I s a h-degee olomal of he fom. I s called he chaacesc olomal of he max. Is zeos ae he egevalues of. If s a egevalue of, he s also a egevalue of. 8

9 3 oelz Ssems he class of oelz maces s exemel moa, fo a umbe of heoecal ad accal easos. hese maces ase auall a vae of oblems, cludg goomec mome oblems, omum fleg, lea edco ad secal esmao. he egeaalss of oelz maces s a as ofe equed sgal ocessg ad cool. Fo examle, he sochasc ocesses he sgal ocessg alcao lead o he coelao ax whch s smmec, osve defe ad he oelz sucue. I s accall moa o solve oelz ssems effce wa. Maces whose ees ae cosa alog each dagoal se ma alcaos ad ae called oelz maces. Fomall, s oelz f hee exs scalas,,,, such ha a fo all ad. hus s oelz oelz maces ae esmmec maces. We sa ha B s esmmec f s smmec abou s oheas-souhwes dagoal. I hs chae we show how o solve oelz ssems O me. he dscusso wll be esced o he moa case whe s also smmec ad osve defe. ssume ha we have scalas such ha he maces,, 9

10 ,, ae all osve defe. hee s o loss of geeal omalzg he dagoal. wo algohms wll be descbed:. Dub s algohm fo he Yule-Wale oblem,,.. Levso s algohm fo he geeal.h.s. x b oblem. Dub s algohm lgohm 3. Dub 96. [Golub, Max Comuaos,.7] Gve eal umbes,, ad ha s osve defe, he, followg algohm comues such ha,,. Fo,,! Fo,, z z,, hs algohm eques flos.

11 Levso s algohm lgohm 3. Levso 947. [Golub, Max Comuaos,.8] Gve b, scalas,,, ad ha s osve defe, he followg algohm comues x such ha b x.,, Fo b x he If,,,,! x v x x v x b!,,,, - z z hs algohm eques flos.

12 4 Localzao of he Egevalues of oelz Maces hs chae exloes he close elaosh bewee oelz ad ccula maces. ve smle embeddg esul, ad a somewha less dec addve decomoso of ehas heoecal ees, eable he devao of bouds fo all egevalues of smmec oelz maces. he bouds have a umbe of desable chaacescs, cludg low comuaoal comlex, whch s ol O log oeaos e boud. hs s a cosequece of he use of he fas Foue asfom algohm. Dese hs, he bouds ae shae ha a few ohe bouds Slea ad Ladau, 978; Hez, 99, ad ofe gve gh esmaes fo he egevalues. 4. he Embeddg Suose M deoes he se of " maces ove he comlex feld. he oaos ad meas ha s oegave defe ad osve defe, esecvel. aba oelz max M wh elemes #, ca be embedded a ccula C M m f m. I geeal, hs leads o a max C whch wll coa l m aba aamees. he embeddg s defed b he elemes c of he fs ow of C, c $ $ #, # # l, l # m. 4.. m I ca be vefed ha he max C so defed s deed ccula.

13 3 he esuls gve hs chae could accou fo hema oelz maces ove he comlex feld, bu, fo smlc, ol eal " smmec oelz maces wll be cosdeed. hese maces ae defed b he fs ows #. he modfcaos fo he hema oelz case ae val ad wll be omed. o llusae he embeddg 4.. wo examles wll be gve, he fs beg based o he oelz max I s covee o defe a max S b S whee s a aba aamee. Embeddg a ccula of ode sx, usg 4.., leads o he max C C S S 4.. Fo he secod examle, le he desed ode of he ccula be egh. lg 4.. elds he ccula

14 4 I hs case, he umbe of aba aamees would be 3 l. Howeve, a smme cosa has bee efoced uo he ccula, ad educed ha umbe dow o wo. I geeal, a smme cosa educes he umbe of fee aamees fom l o / l o / l, deedg o whehe l s odd o eve. 4. Egesucue umbe of asecs eag o he egevalues ad egevecos of he bloc-maces aeag 4.. ae dscussed hs seco. he esuls wll subsequel be used o oba bouds fo he egevalues of smmec oelz maces. I he followg, s assumed ha m. hs mles ha C has eve ode ad ol oe fee aamee. I ca be vefed ha he max M F I I I I F, Whee M I, s ua ad educes M C o bloc-dagoal fom, wh bloc-dagoal maces S ad S. Noe ha F s he bloc-aalog of he Floue max of ode wo, whch dagoalzes all ode wo cculas, ad ha C s bloc-ccula, as ca be see fom 4...

15 hus, C f ad ol f S ad S, a fac ha wll be used he sequel. Sce S s oelz, boh S ad S ae oelz maces. I s eas o see ha S s ccula wheeas S s o. I s ossble o exlcl fd he egevecos ad egevalues of hese wo maces. o do hs, ecall ha x w, w, w,, w, / Whee w e, ad, ae ohogoal egevecos of C M. We wll deoe b he coesodg egevalues, whch ae us he dscee Foue asfom DF coeffces of he fs ow of C, / c e. 4.. Paog x wo blocs x u v ad e-mullg b C leads o u Sv Su v u v Usg he fac ha u oe obas v S u S u u u hs shows how he egevecos ad egevalues of S ad S ae elaed o hose of C : he eve ad odd coeffces of he DF of he fs ow of C ae he egevalues of S ad S, esecvel. he coesodg egevecos ae u #, ad u #, esecvel. Noe ha u w, w, w,, w,. 5

16 whee w / e. lhough he vecos he ses u # ad u # sasf he ohogoal codos u u ad u u, oe has u u.,,, 4.3 Bouds fo he Egevalues Le m deoe he odeed egevalues of, ad deoe he odeed egevalues of. Fo smlc, assume ha m. hs seco descbes how o oba bouds fo all egevalues of. wo dffee mehods ae used, he fs beg based o he embeddg meoed seco 4.. I s well-ow ha he egevalues of a of he " cal submaces of a smmec " max M seaae he egevalues of M self. hs s exessed b he classcal elacg equales Ho ad Johso, 99. Ieag hem mes oe s lead o he elaos # #, # 4.3. Iequales vald fo ohe m could also be eadl obaed. he ae he DF coeffces of he fs ow of C see 4... hus, f s a owe of wo, he ca be foud usg he fas Foue asfom FF algohm, ad he umbe of ecessa ahmec oeaos wll be oughl oooal o log e boud. Eve whe s o a comose umbe, m ca be ae as he fs owe of wo exceedg, ad he FF sll aled. he secod mehod deeds o he esuls gve seco 4.3. he max s exessed as 6

17 S S, whch eables he esuls o he egevalues of S o be used. I ca be see, cosdeg he quadac fom assocaed wh, ha he maxmum egevalues of cao exceed he sum of he maxmum egevalues of S. Smlal, he mmum egevalue of ca o be less ha he sum of he mmum egevalues of S. hus max # # max #, 4.3. m # m #, Whee ow he sad fo he egevalues of C wh he aual odeg, exessed b 4... hese ae bee bouds ha he oes whch follow fom 4.3. amel, # max # m # I fac, usg a heoem due o Wel ad ogall deved he coex of egal equaos, oe ca gve ue ad lowe bouds fo all egevalues of. he heoem asses ha, f ad B ae hema, ad f he egevalues ad B ae soed b ceasg ode, he odeed egevalues of B, B, sasf B # B, B # B. Fo deals abou he heoem see Ho ad Johso 99. Seg, fo examle, S ad B S, ad usg he fac ha he egevalues of ad B ae he eve ad odd subsequeces of he egevalues of he 7

18 ccula C, leads mmedael o ue ad lowe bouds fo he egevalues of, S S S S # he bouds 4.3. ad ae secal cases of hese equaos. 4.4 Omum Values fo he he bouds gve befoe deed o he l m aamees, ad so s aual o as wha values of he lead o bes-ossble esuls. hs oblem s addessed hs seco. ga, he oao # m m deoes he egevalues of he ccula C, wh he odeg gve b 4... he aveage values of he egevalues s deoed b m m, If bes-ossble s cosdeed o mea ha he qua m,, l, s o be mmzed, he aswe s que smle. Oe ca he show ha he all mus be zeo. Whe hs haes, he egevalues of he ccula ae cluseed as closel as ossble he Eucldea om aoud. Fo bev, a dealed dscusso wll be omed. Howeve, oe ha % / m % e, ad ha. I s eas o see ha / %, fo,,, l, leads o % 8

19 m, e / m whch mles c m, ha s,. hus, as saed befoe, he values whch ede he bouds bes-ossble ae,,, l, deedel of he max. 9

20 5 he Smmec Egevalue Poblem 5. Mahemacal Poees udelg smmec egeoblem Smme smlfes he eal egevalue oblem x x wo was: esues ha all of s egevalues ae eal, ad esues ha hee s a ohoomal bass of egevecos. Hee ae some mahemacal oees of smmec egevalue oblem. heoem 5..: Real Schu Decomoso fo Smmec Maces. [Golub, Max Comuaos,.68] If s smmec, he hee exss a ohogoal Q such ha Q Q dag,,. heoem 5..: Coua-Fsche Mmax Chaacezao. [Golub, Max s smmec, he fo,, Comuaos,.69] If max m, dm S S whee deoes he -h lages egevalue of., heoem 5..3: Welad-Hoffma. [Golub, Max Comuaos,.7] If ad E ae -b- smmec maces, he E # E. F heoem 5..4: [Golub, Max Comuaos,.7] Suose ad -b- smmec maces ad ha Q Q, Q E ae s a ohogoal max such ha R Q s a vaa subsace fo. Pao he maces Q Q ad Q EQ as followg:

21 If E Q E Q, Q EQ. E E m E E ad E # /, he hee exss a max P sasfg P # E / such ha he colums of Q Q Q bass fo a subsace ha s vaa fo / P I P P E. fom a ohoomal heoem 5..5: [Golub, Max Comuaos,.7] Suose S ae smmec ad ha, ad Q QS E whee Q sasfes Q Q I. he hee exs egevalues,, such ha S E.,, # heoem 5..6: [Golub, Max Comuaos,.73] Suose S ae smmec ad ha, ad X X S F whee ha X sasfes X. he hee exs,, such F S #. X heoem 5..7: [Golub, Max Comuaos,.73] If ad Q sasfes Q Q I, he s smmec

22 S m Q Q S F Q Q Q I Q Q Q Q F F. heoem 5..8: [Golub, Max Comuaos,.74] Suose smmec ad Q sasfes Q Q I. If s Z Q Q Z dag,, D s he Schu decomoso of Q Q ad Q Z,,, he I Q Q Q Ze # I Q Q Q fo,,. he ae called Rz values, he ae called Rz vecos, ad he, ae called Rz as. heoem 5..9: Slvese Law of Iea. [Golub, Max Comuaos,.74] If s smmec ad X s osgula, he ad X X have he same ea. 5. dagoalzao ad he Smmec QR lgohm he mehods fo acuall comug he egevalues ad egevecos of a max usuall ae eceded b a educo se, whch he max s asfomed o a smla max B havg he same egevalues as. he max B b has a smle sucue ha B s ehe a dagoal max, b fo, o a Hessebeg max, b fo, so ha he sadad mehods fo comug egevalues ad egevecos ae comuaoall less exesve whe aled o B ha whe aled o. lgohm 5..: Householde dagoalzao. [Golub, Max Comuaos,.77] Gve a -b- smmec max, he followg algohm ovewes wh U o U, whee s dagoal ad U P P s he oduc of Householde asfomaos.

23 Fo,, Deeme a Householde max x a, P a : P P, dag, P I P 3 hs algohm eques 3 flos. P such ha lgohm 5..: Imlc Smmec Q-R Se wh Wlso Shf. [Golub, Max Comuaos,.8] Gve a ueduced smmec dagoal max, he followg algohm ovewes wh Z Z, whee Z J J s a oduc of Gves oaos wh he oe ha I s ue agula ad s ha egevalue of s alg -b- cal submax close o. d,, x z Fo,, / / d sg d d, Deeme c cos ad s s c s s c J J, J J,, If he x, z,. x * z such ha hs algohm eques abou 4 flos ad squae oos. If a gve ohogoal max Q s ovewe wh QJ J, he a addoal 4 flos ae eeded. Z 3

24 lgohm 5..3: Smmec QR lgohm. [Golub, Max Comuaos,.8] Gve a -b- smmec max ad, a small mulle of he u oudoff, he followg algohm ovewes wh Q Q D E whee Q s ohogoal, D s dagoal, ad E sasfes E & u. Use lgohm 5.., comue he dagoalzao P P P P Reea Se a, ad a, o zeo f a, a, # a a, fo a,,. Fd he lages q ad he smalles such ha f he 33 s dagoal ad If q he qu. l lgohm 5.. o Go o Reea. dag I, Z, I 33 q q has o zeo subdagoal elemes., q dag I, Z, I 3 hs algohm eques abou 3 flos f Q s o accumulaed ad abou 3 5 flos f Q s accumulaed. q 5.3 he Sgula Value Decomoso lgohm 5.3.: QR wh Colum Pvog. [Golub, Max Comuaos, m.65] Gve, he followg algohm comues he facozao ' QR defed b R R ' QR R m 4

25 whee a,q s ohogoal, R s ue agula, ad ' s a emuao. he emuao ' e c,, e c s deemed accodg o he saeg v max v,, v. he eleme a s ovewe b #. c m a,, Fo,,,, Deeme # # so max. If he qu else Iechage c ad c, ad, ad a ad a fo,, m. Deeme a Householde Q ~ a * ~ Q. am ~ dag I, Q,, a such ha 3 hs algohm eques m m / 3 flos whee a. lgohm 5.3.: Golub-Kaha SVD Se. [Golub, Max Comuaos,.9] Gve a bdagoal max B havg o zeos o s dagoal o suedagoal, he followg algohm ovewes B wh he bdagoal max B U BV whee U ad V ae ohogoal ad V s esseall he ohogoal max ha would be obaed b alg lgohm 5.. o B B. 5

26 Le be he egevalue of he alg -b- submax of ha s close o. z Fo,, Deeme c cos ad s s B BJ,, b z b, c s s c such ha z * Deeme c cos ad s s B J,, If he b z b,, B c s s c * z such ha B B effce mlemeao of hs algohm would soe B s dagoal ad suedagoal vecos a,, a ad f,, f esecvel ad would eque flos ad squae oos. ccumulag U eques 4 m flos. ccumulag V eques 4 flos. lgohm 5.3.3: he SVD lgohm. [Golub, Max Comuaos,.93] m Gve m ad, a small mulle of he u oudoff, he followg algohm ovewes wh U m V D E, whee U s m ohogoal, V s ohogoal, D s dagoal, ad E sasfes E & u. Use lgohm 5.3., comue he bdagoalzao U U V V Reea Se a, o zeo f a, # a a, fo a,, Fd he lages q ad he smalles such ha f. 6

27 33 q q m he 33 s dagoal ad has ozeo suedagoal. If q he qu. If a dagoal e s zeo, he zeo he suedagoal e he same ow ad go o Reea. l lgohm 5.3. o, Go o Reea. dag I, U, I qm dag I, Z, I q he amou of wo equed b hs algohm ad s umecal oees ae show able able SVD Flo cous m Reque Golub-Resch SVD Cha SVD 3 m 3 m, V 3 m 4, U m 4m, U 3 7m, U, V 4 3 m 4m 3, U, V 3 7m 3 3 m m m 3 3 m 3 3m 5.4 Jacob Mehods Jacob 846 oosed a mehod fo educg a smmec max o dagoal fom usg wha we have bee callg Gves oaos. lhough hs mehod has esseall bee eclsed b he smmec QP algohm, s moa o udesad because of s sgfca ole aallel comuao. Jacob s mehod s also useful fo fdg he egevalues of eal dagoal smmec maces. 7

28 lgohm 5.4.: Seal Jacob. [Golub, Max Comuaos,.99] Gve a smmec max ad u, he u oudoff, he followg algohm ovewes wh U U D E whee U s ohogoal, D s dagoal, ad E has a zeo dagoal ad sasfes E # : F F : F Do Ul off # Fo,,, Fo q,, Fd J J, q, such ha he, q e of J J s zeo. J J 3 3 hs algohm eques flos e swee. addoal flos ae equed f he ohogoal max U s accumulaed. 5.5 Some Secal Mehods B exlog he ch mahemacal sucue of he smmec egeoblem, s ossble o devse useful aleaves o he smmec QR algohm. Ma of hese echques ae aoae whe ol a few egevalues ad/o egevecos ae desed. hee such mehods ae descbed hs seco: bseco ad Ralegh quoe eao. Bseco Le deoe he leadg -b- cal submax of a b b a b b 3 b a 5.5. ad defe he olomals x,, b x 8

29 x x de xi fo,,,. smle deemaal exaso ca be used o show ha x x a x x b.,, Because x ca be evaluaed O flos, s feasble o fd s oos usg he mehod of bseco. Fo examle, f z ad z, he he eao Do Whle z u z x z / f x he z x else x s guaaeed o covege o a zeo of x,.e., o a egevalue of. he eao coveges leal: eo s aoxmael halved a each se. Somemes s ecessa o comue he -h lages egevalue of fo some escbed value of. hs ca be doe effcel b usg he bseco dea ad he followg classcal esul: heoem 5.5.: Sum Sequece Poe. [Golub, Max Comuaos,.36] If he dagoal max 4.5. s ueduced, he he egevalues of scl seaae he egevalues of :. Moeove, f a deoes he umbe of sg chages he sequece,,, he a equals he umbe of s egevalues ha ae less ha. Coveo: has he oose sg fom f. Suose we wsh o comue. Fom he Geshgo ccle heoem z whee follows ha, 9

30 m a b b m z a b b b b Wh hese sag values, s clea fom he Sum sequece oe ha he eao Do Whle z u z x z / f a x 5.5. he z x x else oduces a sequece of subevals ha ae eeaedl halved legh bu whch alwas coa. Dug he execug of 5.5., fomao abou he locao of ohe egevalues s obaed. B ssemacall eeg ac of hs fomao s ossble o devse a effce scheme fo comug coguous subses of,,,., e.g., If seleced egevalues of a geeal smmec max ae desed, he s ecessa fs o comue he dagoalzao U U befoe he abouve bseco schemes ca be aled. hs ca be doe usg lgohm 5.. o b he Laczos algohm dscussed he ex chae. I eghe case, he coesodg egevecos ca be eadl foud va vese eao, sce dagoal ssems ca be solved O flos. Ralegh Quoe Ieao Suose s smmec ad ha x s a gve ozeo -veco. smle dffeeao eveals ha x x x x x 3

31 mmzes I x. he scala x s called he Ralegh quoe of x. Cleal, f x s a aoxmae egeveco, he x s a easoable choce fo he coesodg egevalue. O he ohe had, f s a aoxmae egevalues, he vese eao heo ells us ha he soluo o I x b wll almos alwas be a good aoxmae egeveco. Combg hese wo deas he aual wa gve se o he Ralegh quoe eao: x gve, x Fo,, x Solve x I z x fo z z / Noe ha fo a we have whee z E z z / z E x z. I follows fom # / z fo some. 3

32 6 Laczos Mehods I hs chae we develo he Laczos mehod, a echque ha s alcable o lage, sase, smmec egeoblems. he mehod volves dagoalzg he gve max. Howeve, ule he Householde aoach, o emedae ad full submaces ae geeaed. Equall moa, fomao abou s exeal egevalues eds o emege log befoe he dagoalzao s comlee. hs maes he Laczos algohm aculal useful suaos whee a few of s lages o smalles egevalues ae desed. 6. Devao ad Covegece Poees Suose s lage, sase, ad smmec ad assume ha a few of s lages ad/o smalles egevalues ae desed. hs oblem ca be solved b a mehod abued o Laczos 95. he mehod geeaes a sequece of dagoal maces wh he oe ha he exeal egevalues of ae ogessvel bee esmaes of s exeal egevalues. lgohm 6..: he Laczos lgohm. [Golub, Max Comuaos,.35] Gve a smmec ad w havg u -om, he followg algohm comues a -b- smmec dagoal max wh he oe ha. he dagoal ad subdagoal elemes of ae soed,, ad, esecvel. v,, Do Whle If he Fo v w v,,,, w, w v /, v 3

33 w v v v w v Noe ha s o aleed dug he ee ocess. hus, ol a ocedue fo comug max-veco oducs volvg eed be suled. If sasel s exloed hs ocedue ad ol flos ae volved each call, he each Laczos se eques aoxmael 4 flos o execue. he egevalues of ca be foud usg he smmec QR algohm o a of he secal mehods of seco Paccal Laczos Pocedues Laczos wh Comlee Reohogoalzao Le,, be gve ad suose ha Householde maces P,, P have bee comued such ha P P,, s ue agula. Deoe he fs colums of P P b q,, q. Now suose ha we ae gve a veco ad wsh o comue a u veco q he deco of w q q sa q,, q. If a Householde max P P,, s ue agula, he follows ha he +-s colum of P P s he desed u veco. P s deemed so If we cooae hese Householde comuaos o he Laczos ocess, we ca oduce Laczos vecos ha ae ohogoal o wog accuac: q gve u veco Deeme P I vv / v v so e P. 33

34 q q Do,, I q q q w P P Deeme q w P P e P I v v / v v such ha P w w,,,,,, q q hs s a examle of a comlee eohogoalzao Laczos scheme. Bloc Laczos Le he smle owe mehod, he Laczos algohm has a bloc aalog. Suose ad cosde he decomoso whee Q Q M B B M B B B M 6.. Q X,, X X s ohogoal, each Comag blocs M, ad each Q Q show ha B s ue agula. X X B X M X B X B fo,,,. Fom he ohogoal of Q follows ha Moeove, M X X.,,, X B R eeses he Q R facozao of 34

35 R X X M X B hese obsevaos sugges ha he bloc dagoal max 6.. ca be geeaed as follows: X M X I. gve, wh X X Fo,, X M R X X M X B B R Q-R facozao X X B X X 6.. s-se Laczos he bloc Laczos algohm 6.. ca be used a eave fasho o calculae seleced egevalues of. o fx deas, suose we wsh o calculae he lages egevalues. If X s a gve max havg ohoomal colums, we ma oceed as follows:. Geeae X,, X s va he bloc Laczos algohm.. Fom s X,, X s X,, X s 3. Comue a ohogoal max U u,, U ad s -b- s, -dagoal max. such ha s U dag,, s whee s X X,, X s u,, u. 4. Se 5. If X X s F. s sll oo lage, go o. hs s he bloc aalog of he s-se Laczos algohm. he same dea ca also be used o comue seveal of s smalles egevalues o a mxue of boh lage ad small egevalues. 6.3 lcaos o Lea Equaos he Laczos eao ca be used o solve lage sase lea equao. he algohm s dscussed [Golub, Max Comuao,.344]. d s deded as he wdel ow mehod of cougae gades. u s 35

36 7 Cocluso I hs ae, we oduced he oelz max whch s exemel moa of solvg he oblems sgal ocessg. fs, we suded a moa mehod whch ca localze he egvalues of oelz max. he we oduced he mehod whch ca dagoalze he smmec max ad s moace. Fall, we dscussed seveal moa algohms whch ca calculae he egevalues of smmec max. I chae 5, we descbe he smmec QR algohm, Raleg quoe eao ad bseco mehod. I chae 6, he Laczos mehods ae oduced. 36

37 Bblogah Gee H. Golub, Chales F. va Loa 983. Max Comuaos, he Johs Hos Uves Pess. Paulo Joge S. G. Feea 994. Localzao of he Egevalues of oelz Maces Usg ddve Decomoso, Embeddg Cculas, ad he Foue asfom, Poc. h IFC Smosum o Ssem Idefcao, Ss ID 94, Coehage, Dema, Jul 994, J. Soe, R. Bulsch 98. Ioduco o Numecal alss, Sge- Velag New Yo Ic. G.Sag 976. Lea lgeba ad Is lcaos, cademc Pess, New Yo. L. Ms 955. Ioduco o Lea lgeba, Oxfod Uves Pess, Lodo. J. M. Oega 97. Numecal alss: Secod Couse, cademc Pess, New Yo.. S. Householde 964. he heo of Maces Numecal alss, Blasdell Publcao Co., New Yo. 37