1 Convergence of the Arnoldi method for eigenvalue problems

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1 Lecture otes umercal lear algebra Arold method covergece Covergece of the Arold method for egevalue problems Recall that, uless t breaks dow, k steps of the Arold method geerates a orthogoal bass of a Krylov subspace, represeted by a matrx Q = (q,..., q k ) C k such that Q Q = I ad spa(q,..., q k ) = K k (A, b) := spa(b, Ab,..., A k b). The egevalue approxmatos (called Rtz values) are subsequetly foud from the egevalues of H = Q AQ. The matrx H C k k s a Hesseberg matrx ad ca be geerated as a by-product of the Arold method. We call a par (µ, Qv) a Rtz par ad Qv a Rtz vector, f v ad µ safsfy Hv = µv.. Boud for subspace-egevector agle As a frst dcator of the covergece we wll characterze the followg quatty where error egevector x (I QQ )x (.) Ax = λ x. Recall: Q C k s a orthogoal matrx whch meas that Q Q = I C k k. However, I = QQ C. It s very atural to assocate the accuracy of the egevector wth ths quatty from a geometrc perspectve. The dcator the rght-had sde of (.) s called (the orm of) the orthogoal complemet of the projecto of x oto the space spaed by Q ad t ca be terpreted as the se of the caocal agle betwee the Krylov subspace ad a egevector. For the momet, we wll oly justfy ths dcator wth ths geometrc reasog ad the followg observato: Lecture otes - Elas Jarlebrg - Autum 205 verso:

2 Lecture otes umercal lear algebra Arold method covergece Lemma.. Suppoe (λ, x ) s a egepar A. If the Krylov subspace cotas the egevector (x K k (A, b)), the the dcator vashes (I QQ )x = 0 ad there s at least oe Rtz value µ such that µ = λ. I words: Suppose the Krylov subspace cotas the egevector (x K k (A, b)). The, there exsts a vector z C k such that x = Qz. Moreover, ths s a egevector of H such that the Arold method wll geerate a exact egevalue of A. Moreover, the dcator s (I QQ )x = (I QQ )Qz = 0. The Arold method produces a exact approxmato f the Krylov subspace cotas a egevector, or equvaletly the dcator s zero. If, smlar to above, x x K k (A, b), we expect the dcator to be small ad a egevalue of H also to be close λ. The dcator ca be bouded as follows, where we assume dagoalzablty of the matrx. Theorem..2 Suppose A C s dagoalzable ad let the matrx X = (x,..., x ) C ad dagoal matrx Λ C be the Jorda decomposto such that A = XΛX. Suppose α,..., α C\{0} are such that b = α x + + α x (.2) Recall: The egevectors of a dagoalzable matrx form a bass of C. ad ε (m) := m max( p(λ ),..., p(λ ), p(λ + ),..., p(λ ) ) p(λ )= where P deotes polyomals of degree. Suppose the Arold method does ot break dow whe appled to A ad started wth b. Let Q C m be the orthogoal bass geerated after m teratos. The, where (I QQ )x ξ ε (m), (.3) ξ = j = α j α. The dcator ca be bouded by a product cosstg of two scalar values: ε (m) whch oly depeds o the egevalues ad terato umber; ad ξ oly depedg o the startg vector ad egevectors. Proof The proof cossts of three steps.. Cosder ay vector u C. The m z C m u Qz 2 Lecture otes - Elas Jarlebrg - Autum verso:

3 Lecture otes umercal lear algebra Arold method covergece s a lear least squares problem wth a soluto gve by the ormal equatos Q u = Q Qz. Hece, z = Q u. Ths mples that (for ay vector u) we have m u Qz z C m 2 = u QQ u = (I QQ )u 2. Although we ultmately wat to boud the left-had sde of (.3), the proof s smplfed by cosderatos of a scalg the left-had sde of (.3) wth α as follows: (I QQ )α x = m z C m α x Qz = m y K m (A,b) α x y Apply step reversely wth u = α x Now ote that the space K m (A, b) ca be characterzed wth polyomals. It s easy to verfy that y K m (A, b) s equvalet to the exstace of a polyomal p P m such that y = p(a)b. Cosequetly, (I QQ )α x = m α x p(a)b. 3. The fal step cossts of sertg the expaso of b terms of egevectors (.2) ad applyg approprate bouds: (I QQ )α x = m α x p(a) = m m p(λ )= = m p(λ )= = m ( = ( p(λ )= j = j = α x α x α x α x j = α j ) m α j p(λ j )x j p(λ )= α j ) ε (m) α j x j α j p(λ j )x j α j p(λ j )x j j = α j p(λ j )x j max j = ( p(λ j) ) Sce x egevector, p(a)x = p(λ )x For ay two sets S Z: m z Z g(z) m z S g(z) Lecture otes - Elas Jarlebrg - Autum verso:

4 Lecture otes umercal lear algebra Arold method covergece The cocluso of the theorem s establshed by dvdg the equato by α. Note that b = ad x = = x =. Hece the coeffcets α,..., α are balaced. I partcular they satsfy ad = α x + + α x α + + α. ξ = α α j α From ths we ca easly detfy a very good stuato ad a very bad stuato. Suppose for all j =, α j = δ ad suppose δ s small. We have that ξ = ( )δ α. Due to balacg α caot be small. Hece, ξ s small, showg fast covergece for ths egevalue. O the other had, f α (the compoet of the startg vector the drecto of the th egevector) s very small, we have ξ whch mples that the rght-had sde of (.3) s large ad we have slow covergece. Ths serves as a justfcato for a more geeral property. Rule-of-thumb. Startg vector depedecy. The Arold method for egevalue problems wll favor egevectors whch have large compoets the startg vector. The word favors s purposely vague. It should be terpreted as the stuato that oe observes ofte practce, but certaly ot always. If we have a partcular structure the matrx or startg vector, we mght observe covergece to other egevalues... Boudg ε (m) I the characterzato of the dcator Theorem..2 above we troduced the quatty ε (m). Ths quatty bouds (up to a costat) the error egevector x at terato m. Although ε (m) s defed through Thk: ε (m) a polyomal optmzato problem, whch s complcated to solve, t s surprsgly easy to use ths to obta bouds provdg qualtatve uderstadg of the covergece of the Arold method for egevalue problems. We llustrate the power wth a specfc boud. measures how dffcult t s to push dow a polyomal pots λ j, for all j = ad mata p(λ ) =. Lecture otes - Elas Jarlebrg - Autum verso:

5 Lecture otes umercal lear algebra Arold method covergece Corollary..3 Suppose C(ρ, c) C s a dsk cetered at c C wth radus ρ such that t cotas all egevalues but λ. That s, λ 2,..., λ C(ρ, c) ad λ C(ρ, c). The, ( ) ε (m) ρ m. λ c Proof The proof cossts of selectg a partcular polyomal the polyomal optmzato problem, ε (m) := m max( p(λ ),..., p(λ ), p(λ + ),..., p(λ ) ) p(λ )= = max q(λ j ), j = for ay q P m satsfyg q(λ ) =, partcular q(z) = λ c m (z c)m. Hece, from the defto of ρ ad c we have that ε (m) λ max c m > λ c m ρ m λ c m. The result ca be tutvely terpreted as follows. If we ca costruct a small dsc that ecloses all egevalues but oe egevalue we expect fast (at least lear geometrc) covergece for that egevalue. Ths ca be acheved for a egevalue whch s well separated from the rest of the egevalues ad also a outer part of the spectrum. We call ths extreme solated egevalues. Rule-of-thumb. Egevalue depedecy. Arold s method for egevalue problems favors covergece to extreme solated egevalues. Note the dfferece betwee a extreme egevalue ad the egevalues whch are largest modulus (absolute value). The Arold method wll favor extreme whereas the power method wll essetally always coverge to the egevalue largest modulus..2 Lterature ad further readg The proof ad reasog above s spred by [5]. Other covergece bouds volvg Schur factorzatos, that lead to smlar qualtatve Lecture otes - Elas Jarlebrg - Autum verso:

6 Lecture otes umercal lear algebra Arold method covergece uderstadg ca be foud [6], where also complcatos of the o-geerc cases are dscussed. There are also further characterzatos of covergece ad the coecto wth potetal theory [4]. I the above reasog we characterzed the agle betwee the subspace ad the egevector. Although ths serves as a very accurate predcto of the error practce, t does ot drectly gve a rgorous boud o the accuracy of Rtz par. Several approaches to descrbe the covergece of Rtz values ad Rtz vectors have bee doe for stace [2, 3]. There s also cosderable research o the effect of roudg errors Krylov methods. Ulke may other umercal methods, the effect of fte arthmetc ca mprove the performace of the algorthm. See also the recet summary of the covergece of the Arold method for egevalue problems []..3 Bblography [] M. Bellalj, Y. Saad, ad H. Sadok. Further aalyss of the Arold process for egevalue problems. SIAM J. Numer. Aal., 48(2): , 200. [2] Z. Ja. The covergece of geeralzed Laczos methods for large usymmetrc egeproblems. SIAM J. Matrx Aal. Appl., 6(3): , 995. [3] Z. Ja ad G. W. Stewart. O the covergece of rtz values, rtz vectors, ad refed rtz vectors. Techcal report, 999. [4] A. B. Kujlaars. Covergece aalyss of Krylov subspace teratos wth methods from potetal theory. SIAM Rev., 48():3 40, [5] Y. Saad. Numercal methods for large egevalue problems. SIAM publcatos, 20. [6] G.W. Stewart. Matrx Algorthms volume 2: egesystems. SIAM publcatos, 200. Lecture otes - Elas Jarlebrg - Autum verso: