A GLOBAL ARNOLDI METHOD FOR LARGE NON-HERMITIAN EIGENPROBLEMS WITH SPECIAL APPLICATIONS TO MULTIPLE EIGENPROBLEMS. Congying Duan and Zhongxiao Jia*

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1 TAIWANESE JOURNAL OF MATHEMATICS Vol. 15, No. 4, pp , August 2011 Ths paper s avalable onlne at A GLOBAL ARNOLDI METHOD FOR LARGE NON-HERMITIAN EIGENPROBLEMS WITH SPECIAL APPLICATIONS TO MULTIPLE EIGENPROBLEMS Congyng Duan and Zhongxao Ja* Abstract. Global projecton methods have been used for solvng numerous large matrx equatons, but nothng has been known on f and how ths method can be proposed for solvng large egenproblems. In ths paper, a global Arnold method s proposed for large egenproblems. It computes certan F-Rtz pars that are used to approxmate some egenpars. The global Arnold method nherts convergence propertes of the standard Arnold method appled to a larger matrx whose dstnct egenvalues are the egenvalues of the orgnal gven matrx. As an applcaton, assumng that A s dagonalzable, we show that the global Arnold method s able to solve multple egenvalue problems. To be practcal, we develop an mplctly restarted global Arnold algorthm wth certan F-shfts suggested. In partcular, ths algorthm can be adaptvely used to solve multple egenvalue problems. Numercal experments show that the algorthm s effcent for the egenproblem and s relable for qute ll-condtoned multple egenproblems. 1. INTRODUCTION Jblou, Messaoud and Sadok [16] propose a global projecton method for solvng matrx equatons. One of the man ngredents of the global methods s the use of the Frobenus scalar product. Smply speakng, here \global" descrbes the algorthms wth the F-nner product to be defned by (1). They present a global Arnold process that generates an F-orthonormal bass of a matrx Krylov subspace, and based on t they derve the global FOM and the global GMRES methods. Several Receved November 16, 2009, accepted February 23, Communcated by Wen-We Ln Mathematcs Subject Classfcaton: 65F15, 15A18. Key words and phrases: Global Arnold process, Global Arnold method, F-orthonormal, Egenvalue problem, Multple, F-Rtz value, F-Rtz vector, Convergence, Implct restart. Supported by the Natonal Basc Research Program of Chna 2011CB and the Natonal Scence Foundaton of Chna (Nos and ). *Correspondng author. 1497

2 1498 Congyng Duan and Zhongxao Ja authors [13, 14, 30] propose some other global methods, for nstance, the global versons of CG, SCG, CR and CMRH. Over the past several years, numerous global methods have been wdely used for solvng lnear systems wth multple rghthand sdes and matrx equatons, e.g., Sylvester equatons and Rccat equatons [4, 17, 18, 24, 31]. These methods fall nto category of global projecton methods onto certan matrx Krylov subspaces. A convergence analyss on the global GMRES method s made n [5]. These global Krylov subspace algorthms appear effectve when appled for solvng the problems mentoned above. Other applcatons of the global Krylov subspace methods are n model reducton, especally MIMO systems [7, 8, 9, 15]. However, no global projecton method has been proposed for large matrx egenproblems htherto. Whether or not a global projecton method can be proposed for the egenproblems and, f yes, how to develop a practcal algorthm have not been known. For large non-hermtan egenproblems, a major class of methods are orthogonal projecton methods, whch nclude the famous Arnold method [1, 26, 29, 34]. The Arnold method uses the Arnold process to construct an orthonormal bass of the Krylov subspace generated by a sngle startng vector and compute the Rtz pars to approxmate some of the egenpars of a large n n matrx A. Assumng that A s dagonalzable, however, t s known that the Arnold method tself s unable to determne the multplctes of the requred egenvalues and the assocated egenspaces [19, 20, 21]. In order to fgure out these problems, block versons of the Arnold method are proposed [2, 20, 23] that frst use the block Arnold process to construct an orthonormal bass of the block Krylov subspace generated by a set of vectors and then extract the Rtz pars from the block Krylov subspaces to approxmate the desred egenpars. In ths paper, based on the global Arnold process startng wth an n s ntal matrx, we show how to derve a global Arnold method for large unsymmetrc egenproblems and set up a general framework of global projecton methods for egenproblems, also called the F-orthogonal projecton methods. The method computes so-called F-Rtz pars to approxmate some of the egenpars of A. A fundamental dfference wth a usual projecton method s that now there are s F-Rtz vectors assocated wth each F-Rtz value, each of whch can be used as an approxmate egenvector. So we can pck up any one of the F-Rtz vectors for our use for each F-Rtz value! We show that the F-Rtz values are equal to certan usual Rtz values of a larger matrx wth each egenvalue of A as an s multple one over a closely related Krylov subspace and the F-Rtz pars are at least as accurate as the usual Rtz pars. So the global Arnold method nherts convergence propertes of the standard Arnold method. We wll pay specal attenton to the multple egenvalue problem. Under the assumpton that A s dagonalzable, we show that the global Arnold method can be appled to adaptvely determne multplctes of the desred

3 A Global Arnold Method for Large Non-Hermtan Egenproblems 1499 egenvalues and the correspondng egenspaces. To be clearer and more llustratve, we state more on the global method. The global Arnold process constructs an F-orthonormal bass V 1,V 2,...,V m of the matrx Krylov subspace K m (A, V 1 ) generated by an n s ntal matrx V 1 wth the Frobenus norm one, where V =(v 1,v 2,...,v s ), =1, 2,...,mare n s matrces. If we nterpret the bass (V 1,V 2,...,V m ) as ms lnearly ndependent vectors, ths matrx Krylov subspace can be regarded as a usual block Krylov subspace of dmenson ms startng wth the ntal block vector V 1, so that we can decompose t nto the drect sum of the s sngle vector Krylov subspaces K m (A, v 1j ),j=1, 2,...,s of dmenson m. By the F-orthogonal projecton prncple, we can get m approxmate egenvalues λ (m),=1, 2,...,m, called the F-Rtz values wth respect to the matrx Krylov subspace. For each F-Rtz value λ (m), we can obtan an approxmate egenvector ϕ (m) j, j =1, 2,..., s from each sngle vector Krylov subspace. Assume that the F-Rtz value λ (m) and these s correspondng F-Rtz vectors ϕ (m) 1,ϕ(m) 2,...,ϕ(m) s have converged. Then they are all good approxmatons to the egenpars of A. If a requred egenvalue λ s smple, the s F-Rtz vectors are lnearly dependent numercally. So f the multplcty of the desred egenvalue s not concerned, we smply use any one of the s F-Rtz vectors as an approxmate egenvector rather than compute all of them. If λ s d (d <s) multple, the s F-Rtz vectors must be numercally lnearly dependent, from whch we can numercally determne the multplcty d relably and effcently. If λ s d (d s) multple, these s F-Rtz vectors are lnearly ndependent. So λ s at least s multple. We then run the global Arnold method startng wth a new V 1 that are ndependent of the old V 1, and compute the new converged F-Rtz vectors. Addng them to the prevous ϕ (m) j, j=1, 2,..., s, f they are lnearly dependent numercally, then the rank s d ; otherwse, contnue. Proceed n such a way untl d s determned. We wll quanttatvely establsh a sold foundaton for the above clams. Both theory and numercal experments llustrate that the procedure s relable to determne egenvalue multplctes and egenspaces when condton numbers of the desred egenvectors are consderably smaller than recprocals of the resdual norms. Ths means that the procedure s effectve for qute ll-condtoned multple egenproblems. The global Arnold method becomes very expensve n storage and computatonal cost as m ncreases. Therefore, restartng s necessary when the method does not delver the approxmate egenpars wth prescrbed accuracy for a maxmum m allowed. The need for restart was frstly recognzed by Karush [22], and there are varous restartng schemes developed by many researchers, e.g., Page [25], Cullum and Donath [10], Golub and Underwood [12], Saad [27, 28] and Chateln and Ho [6]. All of these schemes are explct restartng. Over the years, the most popular restartng scheme s mplct restartng proposed by Sorensen [33] that

4 1500 Congyng Duan and Zhongxao Ja combnes the mplctly shfted QR teratons wth the Arnold process and leads to a truncated form of the mplctly shfted QR teraton. In ths paper, we extend mplct restartng to the global Arnold process and develop an mplctly restarted global Arnold algorthm (IRGA) wth those unwanted F-Rtz values as shfts, also called exact shfts as n [33]. The rest of the paper s organzed as follows. In the next secton, we brefly revew the global Arnold process and the global FOM and GMRES algorthms. In Secton 3, we propose a global Arnold method for large unsymmetrc egenproblems. In Secton 4, we show how the global Arnold method s used to solve the multple egenproblem. In Secton 5, we dscuss how to mplctly restart the global Arnold method and develop the mplctly restarted global Arnold algorthm wth the exact shfts suggested. Fnally, we report numercal examples to llustrate effcency and relablty of the algorthm n Secton 6. Some notatons to be used are ntroduced. A s an n n large dagonalzable matrx throughout the paper, and λ,ϕ are ts egenvalues and egenvectors wth λ labeled n a desred order. Denote by the spectral norm of a matrx and the vector 2-norm, by F the Frobenus norm of a matrx, by the superscrpt H the conjugate transpose of a matrx or vector and by I the dentty matrx wth the order clear from the context. The egenvectors and ther approxmatons are normalzed to have unt length. 2. THE GLOBAL ARNOLDI PROCESS AND THE GLOBAL FOM AND GMRES Let M n,s denote the complex lnear space of n s rectangular matrces. For two matrces X and Y n M n,s, we defne ther F-nner product by (1) X, Y F = tr(x H Y ), where tr(z) denotes the trace of the square matrx Z. Note that F = 1/2 F. We wll use the notaton X F Y to denote X, Y F =0, meanng that X and Y are F-orthogonal. For a startng matrx V M n,s, the matrx Krylov subspace K m (A, V ) s defned by (2) K m (A, V )=span{v,av,...,a m 1 V }, whch s a subset of M n,s. Z K m (A, V ) means that there are scalars α,= 1, 2,..., m such that (3) Z = m 1 =0 α A V.

5 A Global Arnold Method for Large Non-Hermtan Egenproblems 1501 Let V =(v 1,v 2,...,v s ) and defne a lnear operator vec: M n,s C ns by (4) vec(v )=(v H 1,v H 2,...,v H s ) H. Then we have (5) X, Y F = vec(x),vec(y ), where, denotes the usual l 2 nner product of the complex vector space C ns. Denote by A B the Kronecker product of the matrces A and B. The followng basc propertes hold, e.g., [11]: 1. (A B)(C D) =(AC) (BD). 2. (A B) H = A H B H. 3. If A C n n, X C n m, then vec(ax)=(i m A)vec(X). 4. Each egenvalue λ,=1, 2,...,nof A s an s multple egenvalue of I s A. The followng global Arnold process s based on the modfed Gram-Schmdt process. It constructs an F-orthonormal bass V 1,V 2,...,V m of the matrx Krylov subspace K m (A, V ), that s, tr(v H V j )=δ j for, j =1,...,m, where δ j s the Kronecker delta. Algorthm 1. The global Arnold process 1. V 1 = V/ V F 2. for j =1, 2,...,mdo W := AV j ; for =1, 2,...,j do h,j = W, V F ; W = W h,j V ; end h j+1,j = W F ; V j+1 = W/h j+1,j. end Ths process needs ms matrx A by vector products plus nm 2 s flops. Defne V m =(V 1,V 2,...,V m ) and H m and H m to be the m m and the (m + 1) m Hessenberg matrces whose nonzero entres h,j are defned by Algorthm 1. Then ( (6) H m = H m h m+1,m e H m where e m =(0,...,0, 1) H s the m-th canoncal bass of C m. We have the followng results [16, 24]. ),

6 1502 Congyng Duan and Zhongxao Ja Theorem 1. Let V m, H m and H m be defned as above. Then V 1,V 2,...,V m are an F-orthonormal bass of the matrx Krylov subspace K m (A, V 1 ) and (7) AV m = V m (H m I s )+h m+1,m (0 n s,...,0 n s,v m+1 ) (8) = V m (H m I s )+r m (e H m I s ), (9) AV m = V m+1 ( H m I s ), where 0 n s denotes the n s zero matrx, r m = h m+1,m V m+1. Based on the global Arnold process, Jblou et al. [16] propose a global FOM (GL-FOM) algorthm and a global GMRES (GL-GMRES) algorthm for solvng lnear systems wth multple rght-hand sdes and numerous matrx equatons. Takng the lnear system wth s multple rght-hand sdes as example, we brefly descrbe the GL-FOM and GL-GMRES algorthms. Let X 0 be an ntal guess to the soluton X of AX = B and R 0 = B AX 0 ts assocated resdual. At the m-th terate of the Gl-FOM algorthm, the correcton Z m = V m (y m I s ) s extracted from the matrx Krylov subspace K m (A, R 0 ) such that the resdual R m = B A(X 0 + Z m )=R 0 AZ m satsfes (10) R m F K m (A, R 0 ), where y m s shown to be the soluton of the followng m m lnear system: (11) H m y m = βe 1, where β = R 0 F. In the GL-GMRES algorthm, the correcton Z m = V m (y m I s ) s determned by mposng the resdual mnmzaton condton (12) R m F = mn R y C m 0 AV m (y I s ) F, where y m s shown to be the soluton of the (m +1) m least squares problem (13) y m = arg mn βe y C m 1 H m y, where β = R 0 F. For detals on these two global methods, we refer to [16]. We remark that f s =1then the global Arnold process reduces to the standard Arnold process and the GL-FOM and GL-GMRES methods become the standard FOM (Arnold) and GMRES methods. 3. A GLOBAL ARNOLDI METHOD FOR EIGENPROBLEMS As a frst step towards dervng a global Arnold method for egenproblems, a smple but fundamental key s that we can nterpret the matrx Krylov subspace K m (A, V 1 ) of M n,s as a standard block Krylov subspace of C n startng wth the

7 A Global Arnold Method for Large Non-Hermtan Egenproblems 1503 ntal block vector V 1. Correspondngly, we can nterpret each bass element V,= 1, 2,...,m of the matrx Krylov subspace K m (A, V 1 ) as usual s vectors and each n s element of t as s vectors nstead of an n s matrx. We wll swtch K m (A, V 1 ) between the matrx Krylov subspace and the standard Krylov subspace, followng our need. In the sequel, when speakng of the block Krylov subspace, suppose that V m =(V 1,V 2,...,V m ) s of full column rank. Then Algorthm 1 also generates a bass of the block Krylov subspace K m (A, V 1 ) n C n. However, we should keep n mnd that ths bass s generally not orthogonal. Mathematcally, when K m (A, V 1 ) s regarded as the block Krylov subspace, we mght use the standard orthogonal projecton prncple to solve egenproblems. In ths case, premultplyng (7) by (V H mv m ) 1 V H m gves (14) (V H m V m) 1 V H m AV m = H m I s +(V H m V m ) 1 V H m h m+1,m (0 n s,...,0 n s,v m+1 ), whch s just the projecton matrx of A onto the subspace spanned by the columns (m) of V m, whose ms egenvalues ˆλ are just the Rtz values of A wth respect to ths subspace and the (unnormalzng) Rtz vectors are ˆϕ (m) = V m ŷ (m), where the ŷ (m) are the egenvectors of the projecton matrx assocated wth the egenvalues ˆλ (m). Mathematcally, they are the same as the Rtz values and Rtz vectors when the standard block Arnold process s exploted to generate an orthonormal bass of K m (A, V 1 ). However, computatonally, we do not recommend ths approach. Frstly, t may be unstable as the columns of V m can be nearly lnearly dependent so that Vm H V m s nearly sngular; secondly, t s more costly than the block Arnold method for the same m. Actually, apart from the cost of m-step global Arnold process, we need n(ms) 2 +2(ms) 3 flops for formng Vm HV m and nvertng t, whle the block Arnold process costs ms matrx by vector products plus n(ms) 2 flops, assumng that no reorthogonalzaton s used. So such a procedure costs more than the block Arnold process for the same m. Therefore, we have to abandon t and seek other vable procedures. Defne A = I s A. Then t has each egenvalue λ of A as an s multple egenvalue. Assume that A s dagonalzable and defne D = dag(λ 1,λ 2,...,λ n ) and the egenvector matrx Then Φ Φ... Φ 1 Φ=(ϕ 1,ϕ 2,...,ϕ n ). Φ A Φ D... = Φ D.... D

8 1504 Congyng Duan and Zhongxao Ja From ths egen-decomposton, we get the s correspondng egenvectors of A assocated wth λ that have the form w = (0,...,ϕ H,...,0)H, whose possble nonzero entres are n postons n( 1) + 1 to n, =1, 2,...,s. Any lnear combnaton of them s stll an egenvector of A assocated wth λ and may have no specal structure. The global Arnold process on A startng wth the matrx V 1 s closely related to the standard Arnold process on A startng wth the ntal vector vec(v 1 ). The followng results can be easly justfed [24], and they are the very frst step to propose and understand a global Arnold method for egenproblems. Theorem 2. Let H m and H m be defned as above. Then vec(v 1 ),vec(v 2 ),..., vec(v m ) form an orthonormal bass of the usual Krylov subspace K m ( A,vec(V1 ) ) generated by A and the startng vector vec(v 1 ). Defne the matrx V m = ( vec(v 1 ),vec(v 2 ),...,vec(v m ) ). Then the followng standard Arnold process holds: (15) (16) AV m = V m H m + h m+1,m vec(v m+1 )e H m, = V m H m. Snce vec(v 1 ),vec(v 2 ),...,vec(v m ) are orthonormal, Theorem ( 2 shows that H m s the orthogonal projecton matrx of A onto the subspace K m A,vec(V1 ) ). Let λ (m),y (m), =1, 2,...,m be the egenpars of H m wth y (m) =1. The egenvalues λ (m) are the Rtz values of A wth respect to the subspace K m (A,vec(V 1 )) and the correspondng Rtz vectors are w (m) =(vec(v 1 ),vec(v 2 ),...,vec(v m ))y (m),=1, 2,...,s. So we can smply use the Rtz pars to approxmate some egenpars of A. The resdual norms are computed cheaply by (17) Aw (m) λ (m) w (m) = h m+1m e H m y(m) wthout explctly formng w (m) untl the convergence occurs. However, the stuaton s subtle and by no means so smple as A s much larger than A n sze and all the egenvalues are at least s multple. We note that on one hand each egenvalue of A s an s multple one of A and on the other hand the egenvalues of H m are always smple f t s dagonalzable. In fact, assumng that A s dagonalzable, the standard Arnold method works as f A had only smple egenvalues [19, 20, 21, 26, 27]. Therefore, when a Rtz par defned above

9 A Global Arnold Method for Large Non-Hermtan Egenproblems 1505 converges, we can get only one smple approxmaton to the s multple egenpars of A. So we have to do more. We now propose a better and practcal global Arnold method that works on the orgnal A drectly rather than on the much larger ns ns matrx A. Recall that the global Arnold process constructs an F-orthonormal bass V 1,V 2,...,V m of the matrx Krylov subspace K m (A, V 1 ). When speakng of a block Krylov subspace K m (A, V 1 ), we can smply decompose t nto the drect sum of the s sngle vector Krylov subspaces generated by the startng vectors v 11,v 12,...,v 1s, respectvely: (18) K m (A, V 1 )=K m (A, v 11 ) K m (A, v 12 ) K m (A, v 1s ). Keepng n mnd the assumpton that the columns of V 1,V 2,...,V m are lnearly ndependent. Then n the MATLAB language the columns of Vm j = V m (:,j : s : ms) form a bass of K m (A, v 1j ),j =1, 2,...,s. Snce v 1j,j =1, 2,...,s are supposed to be lnearly ndependent, we get the s dstnct m-dmensonal Krylov subspaces K m (A, v 1j ). Now we stll use λ (m), =1, 2,..., m as approxmatons to some of the egenvalues of A, called the F-Rtz values of A wth respect to the matrx Krylov subspace K m (A, V 1 ), but we compute the s new vectors (19) ϕ (m) j = V j my (m), j =1, 2,...,s, called the F-Rtz vectors of A wth respect to K m (A, V 1 ). For each λ (m) we now have the s correspondng F-Rtz vectors ϕ (m) j K m (A, v 1j ). We shed more lghts on the F-Rtz pars. Suppose that they already converge. Then f λ s smple, these s F-Rtz vectors must be almost parallel as they approxmate the same ϕ. If λ s multple, each of these s F-Rtz vectors s a good approxmate egenvector of A. If we do not care the multplcty of λ and do not determne the whole egenspace of A assocated wth λ, then we can smply use any of the F-Rtz vectors as an approxmate egenvector of A nstead of computng all of them. Let U (m) Then U (m) = V m (y (m) I s ) = (Vmy 1 (m),...,vmy s (m) ) = (ϕ (m) 1,...,ϕ(m) s ). F =1and t s easly verfed that (λ (m),u (m) ), =1, 2,...,m are the solutons of { U (m) V m (20) (A λ (m) I)U (m) F V m. We call the (λ (m),u (m) ) the F-Rtz pars of A wth respect to the matrx Krylov subspace K m (A, V 1 ). Therefore, the global Arnold method falls nto a general framework, and we call t the F-orthogonal projecton. We nvestgate how a F-Rtz value and a F-Rtz vector approxmates an egenvalue and ts assocated egenvector of A by relatng t to the standard Rtz par (λ (m),w (m) ( ) of A wth respect to K m A,vec(V1 ) ).

10 1506 Congyng Duan and Zhongxao Ja Theorem 3. We have (21) (A λ (m) I)U (m) F = h m+1,m e H my (m), (22) Proof. (A λ (m) I)ϕ (m) j h m+1,m e H my (m), j=1, 2,...,s. From (7) we have (A λ (m) = AV m (y (m) = V m (H m y (m) I)U (m) F I s ) λ (m) V m (y (m) I s ) I s )+h m+1,m (0 n s,...,0 n s,v m+1 )(y (m) I s ) λ (m) V m (y (m) I s ) = h m+1,m (0 n s,...,0 n s,v m+1 )(y (m) I s ) = h m+1,m V m+1 (e H m I s)(y (m) I s ) = h m+1,m e H m y(m) V m+1. Therefore, we get whch s just (21). Notng that AV j my (m) λ (m) we have shown (22). (A λ (m) I)U (m) Vmy j (m) AV m (y (m) F = h m+1,m e H my (m), I s ) λ (m) V m (y (m) I s ) F, Recall (17). (21) ndcates that the resdual Frobenus norm of the F-Rtz par (λ (m),u (m) ) s just equal to the resdual 2-norm of the Rtz par (λ (m),w (m) ) obtaned by the standard Arnold method appled to A over K m (A,vec(V 1 )). (22) can be used as a stoppng crteron whch cheaply checks f the global Arnold method converges wthout computng ϕ (m) j by (19) untl convergence occurs. Recall that for the standard Arnold method on A over K m (A,vec(V 1 )) the Rtz vectors w (m) = V m y (m) =[(V 1 m) H,...,(V s m) H ] H y (m) =[(ϕ (m) 1 )H,...,(ϕ (m) s )H ] H wth w (m) = 1 as t s assumed that y (m) = 1. So the ϕ (m) j are unnormalzed and ther norms are always smaller than one. Snce no one of the nonorthonormal bases Vm j s specal, the ϕ (m) j should generally be comparable n

11 A Global Arnold Method for Large Non-Hermtan Egenproblems 1507 sze,.e., ϕ (m) j 1 s. Therefore, we get AVmy j (m) λ (m) 1 [ (AV 1 s my (m) Vmy j (m) λ (m) Vmy 1 (m) ),...,(AVmy s (m) λ (m) Vmy s (m) ) ] F = 1 s AV m (y (m) = 1 s h m+1,m e H my (m), I s ) λ (m) V m (y (m) I s ) F So combnng the above and the proof of Theorem 3, we have (23) r (m) j := (A λ(m) ϕ (m) j I)ϕ (m) j h m+1,m e H my (m). The left-hand sde of the above relaton s the resdual norm of the normalzed F-Rtz par, whle from (17) the rght-hand sde s just the resdual norm of (λ (m),w (m) ) as an approxmate egenpar of A. So, (22) and (23) demonstrates that all the s F-Rtz vectors ϕ (m) j are good approxmate egenvectors of A assocated wth the egenvalue λ f the standard Arnold method appled to A converges. So the global Arnold method nherts convergence propertes of the standard Arnold method and acheve comparable resduals for the same m. For a convergence analyss of the standard Arnold method, we refer to [19, 20, 21, 26, 27]. We now present a basc global Arnold algorthm for egenproblems. Algorthm 2. A basc global Arnold algorthm 1. Let (λ,ϕ ), = 1, 2,...,k be k desred egenpars of A and tol a user prescrbed tolerance. Choose an n s matrx V and take V 1 = V/ V F as the startng matrx. 2. For m = k + 1,k + 2,...untl convergence (a) Construct the F-orthonormal bass V 1,V 2,...,V m by Algorthm 1. (b) Compute the m egenpars (λ (m),y (m) ), =1, 2,..., m of the resultng Hessenberg matrx H m and select k F-Rtz values, say, λ (m),= 1, 2,...,k as approxmatons to the k desred λ,=1, 2,...,k. (c) Form ϕ (m) = ϕ (m) 1 = Vmy 1,=1, 2,...,k. (d) Test convergence of the k approxmate egenpars (λ (m),ϕ (m) ). (e) If they all drop below tol, then go to Step 3. We should menton that f s =1then the global Arnold method s just the standard basc Arnold method.

12 1508 Congyng Duan and Zhongxao Ja 4. MULTIPLE EIGENPROBLEMS As we have seen prevously, under the assumpton that A s dagonalzable, f H m s dagonalzable, F-Rtz values are always smple even f A has multple egenvalues. So the global Arnold method works as f A has only smple egenvalues. As a result, when a desred λ s multple, the method tself cannot determne the multplcty of λ and compute the egenspace assocated wth t. Explotng the theoretcal analyss of [21], we consder how the global Arnold method can be successfully used to solve multple egenvalue problems. Before dscussons, we need some notatons. We always assume that A s an n n dagonalzable matrx and has M dstnct egenvalues λ, where the multplctes of λ are d, =1, 2,...,M. Let P be the d -dmensonal egenspace assocated wth λ and the columns of Φ d =(ϕ 1,ϕ 2,...,ϕ d ) form a bass of P, where ϕ j =1,j =1, 2,...,d. Wrte the n s startng matrx V 1 as V 1 =(v 11,v 12,...,v 1s ). Then gven Φ d, each v 1j, 1 j s, can be unquely expanded as (24) v 1j = b j1 ϕ 1 + b j2 ϕ b jd ϕ d + u j, u j P 1 P 1 P +1 P M. Defne b 11 b b 1d b 21 b b 2d (25) B s = b s1 b s2... b sd Obvously, B s s row rank defcent when s>d. Assume that the matrx B s s of row full rank for s d. We rewrte the above v 1j as (26) v 1j = β j ϕ j + u j,u j P 1 P 1 P +1 P M, where ϕ j, j =1, 2,...,s are also unt length egenvectors assocated wth λ. Under the assumpton on B s, just as {ϕ j } d j=1, { ϕ j} d j=1 s also a bass of P, and for s>d, ϕ j, j = d +1,...,smust be lnearly dependent to ϕ j,j=1, 2,...,d and belong to the span of { ϕ j } d j=1. In other words, defne Φ s =( ϕ 1, ϕ 2,..., ϕ s ) (m) and Φ s =( ϕ (m) 1, ϕ(m) 2,..., ϕ(m) s ). Then Φ s s of full column rank for s d, whle t must be column rank defcent and the smallest sngular value of t s zero for s>d. From now on we omt the tlde n the Greek letters wthout ambguty; furthermore, we assume to have normalzed ϕ (m) j n the dscussons below. We also assume that the columns of Φ s are strongly lnear ndependent for s d, that s, the smallest sngular value of Φ j s not small and moderate for s d. From

13 A Global Arnold Method for Large Non-Hermtan Egenproblems 1509 the vewpont of the probablty theory, ths assumpton s holds for a bass of P generated randomly. The followng theorem from [21] s the theoretcal background for determnng d numercally by the global Arnold method. Theorem 4. Let P 1,...,P M be the spectral projectors assocated wth λ 1,...,λ M, and defne the matrx X j = ( P1 ϕ (m) j P 1 ϕ (m) j,..., P 1 ϕ (m) j P 1 ϕ (m) j, P +1 ϕ (m) j P +1 ϕ (m) j,..., P M ϕ (m) j P M ϕ (m) j ) and g = mn k λ (m) (27) λ k. Then C j = κ(x j) I P g κ(x j)(1 + P ) g, (28) sn (ϕ j,ϕ (m) j ) C j r (m), j where κ(x j ) s the spectral condton number of X j and r (m) j s the resdual defned by (23). Let σ mn (Φ (m) s ) and σ mn(φ s ) be the smallest sngular values of the matrces Φ (m) s and Φ s, respectvely. Then (29) σ mn (Φ (m) s ) σ mn(φ s )+ s max ϕ j ϕ (m) j. 1 j s In partcular, f s>d, then (30) σ mn (Φ (m) s ) s max ϕ j ϕ (m) 1 j s j (31) s C j max 1 j s r(m) j for small r (m) j. The relaton (28) estmates the accuracy of ϕ (m) j n terms of the resdual norm r (m) j, where C j acts as a condton number and measures the condtonng of ϕ j. The bgger t s, the worse condtoned ϕ j s. If one of κ(x j ) and P s bg or the separaton g of the approxmate λ (m) and the other exact egenvalues λ j s very small, ϕ j s ll condtoned. We can see from (31) that f C j s comparable to or 1 bgger than then σ mn(φ (m) max 1 j s r (m) j s ) may not be small. Based on ths theorem, we can decde f Φ (m) s,=1, 2,...,k, are approxmately column rank defcent n the sense of (31) and thus detect the multplctes d and get an approxmate bass of P. It tells us that Φ (m),=1, 2,...,k are approxmately s

14 1510 Congyng Duan and Zhongxao Ja column rank defcent for d < s and have full column rank for d s. So σ mn (Φ (m) s ),=1, 2,...,k are not small for d s. We decde d n such a way: Assume that r (m) j < tol. Then f s s the smallest nteger such that (32) σ mn (Φ (m) s ) s C j max 1 j s r(m) j hold wth a constant C j sgnfcantly smaller than max 1 j s r(m) j, λ s (s 1) multple and d = s 1. In practce, C j s a-pror unknown. We take C j to be consderably less than tol , say tol or smaller, whch means that ϕ j,j =1, 2,...,s can be qute ll condtoned, such as C j =10 3 or 10 5 f max j s r(m) j or So the procedure may fal to determne d f C j s comparable to or bgger than tol 1 but t s defntely relable. Later numercal experments wll ndeed llustrate that (31) s conservatve and our procedure can determne the multplctes of egenvalues for qute ll-condtoned egenproblems, e.g., C j In practce, s s gven. A random V 1 wll make B s defned by (25) satsfy the assumpton on the rank of t. Wth V 1, f we compute an s numercally multple egenvalue, then ths egenvalue s at least s multple, so the determnaton of d s not actually resolved when s d. The followng approach taken from [20, 21] can fgure out ths problem elegantly. Frstly, choose a random startng matrx V (1) 1 wth s 1 columns. If we have found an s 1 multple egenvalue, then we apply Algorthm 2 wth a new startng ntal V (2) 1 wth s 2 columns, whch s chosen randomly. Now we can compute an s 2 s 1 multple egenvalue, whch s numercally equal to the one computed wth s 1,V (1) 1. We then determne the rank of the matrx (Φ (m) s 1, Φ (m) s 2 ) consstng of these s 1 + s 2 converged F-Rtz vectors wth the numercally multple converged F-Rtz values. Note here that when the egenproblem of A s not too ll condtoned, f some sngular values of ths matrx are of the same order as the maxmum of resdual norms of these s 1 + s 2 converged nteror egenpars, then we consder them to be zero numercally. If the numercal rank of the matrx s less than s 1 + s 2, then the multplcty of ths egenvalue s just the rank of such a matrx. Otherwse, we repeat Algorthm 2 wth s 3 s 1,V (3) 1 and so on untl the numercal rank of the matrx consstng of these s 1 + s s q converged F-Rtz vectors startng wth s 1,V (1) 1,s 2,V (2) 1,...,s q,v (q) 1 respectvely, s smaller than s 1 + s s q. Then the multplcty d of the egenvalue has been determned and equals the numercal rank of ths matrx. In summary, we can present a global Arnold algorthm for multple egenproblems. Algorthm 3. A global Arnold algorthm for multple egenproblems

15 A Global Arnold Method for Large Non-Hermtan Egenproblems Defne the set S ={1, 2,...,k}, Ψ=, q =1 and prescrbe a tolerance tol. 2. Choose a startng n s q matrx V 1 wth V 1 F =1. 3. For m = k + 1,k + 2,...untl convergence (a) Construct the F-orthonormal bass V 1,V 2,...,V m by Algorthm 1. (b) Compute the m egenpars (λ (m),y (m) ), =1, 2,..., mof the resultng Hessenberg matrx H m and use λ (m), =1, 2,..., k to approxmate the desred λ,=1, 2,...,k. (c) Test convergence of the approxmate egenpars (λ (m),ϕ (m) j ), =1, 2,...,k; j =1, 2,...,s q. (d) If they all drop below tol, then go to Step For all S, set Φ (m) s columns. =(Ψ,ϕ (m) 1,ϕ(m) 2,...,ϕ(m) s q ) and s the number of ts (a) Compute the numercal rank r s of Φ (m) s for all S; (b) If r s <s, set d = r s and remove from S; (c) Otherwse, let Ψ=Φ (m) s,q = q +1and go to Step AN IMPLICITLY RESTARTED GLOBAL ARNOLDI ALGORITHM The basc global Arnold algorthm becomes very expensve and mpractcal due to excess storage and hgh computatonal cost as m ncreases. So m must be lmted not to be bg. To make the method practcal, restartng s necessary. The mplct restartng technque due to Sorensen [33] s a very successful and popular one. We show how to extend t to the global Arnold process and develop an mplctly restarted global Arnold algorthm (IRGA). Let k be a fxed specfed nteger, usually the number of the desred egenpars of A and the steps m = k + p. Consder the k + p step global Arnold process (33) (34) AV k+p = V k+p (H k+p I s )+r k+p (e H k+p I s) ) = (V k+p,v k+p+1 )( ( Hk+p β k+p e H k+p I s ). We apply shfted QR teratons to ths truncated factorzaton of A. Let µ be a shft and H k+p µi = QR be the QR factorzaton wth Q orthogonal and R upper trangular. Then H k+p I s µi =(H k+p µi) I s =(QR) I s =(Q I s )(R I s )

16 1512 Congyng Duan and Zhongxao Ja and Therefore, we get (RQ) I s + µi =(RQ + µi) I s =(Q H H k+p Q) I s. (A µi)v k+p V k+p (H k+p I s µi) =r k+p (e H k+p I s ), (A µi)v k+p V k+p (Q I s )(R I s )=r k+p (e H k+p I s), (A µi) ( V k+p (Q I s ) ) ( V k+p (Q I s ) )( (RQ) I s ) =rk+p ( (e H k+p Q) I s ), A ( V k+p (Q I s ) ) ( V k+p (Q I s ) )( (RQ) I s + µi ) = r k+p ( (e H k+p Q) I s ),.e., (35) AV k+p (Q I s )= ( ) ( Q H ) H k+p Q V k+p (Q I s ),V k+p+1 β k+p e H k+p Q A successve applcaton of p mplct shfts results n ( ) H + (36) AV + k+p =(V+ k+p,v k+p k+p+1) β k+p e H ˆQ I s, k+p I s. where V + k+p = V k+p( ˆQ I s ), H + k+p = ˆQ H H k+p ˆQ and ˆQ = Q 1 Q 2...Q p, wth Q j the orthogonal matrx assocated wth the shft µ j. Now, partton ( H V + k+p =(V+ k, ˆV k ), H + + ) k+p = k M ˆβ k e 1 e H k Ĥ p and note So we get β k+p e H k+p ˆQ =(0, 0,..., β k+p,b H ). (37) A(V + k, ˆV p )=(V + k, ˆV p,v k+p+1 ) ˆβ k e 1 e H k β k+p e H k H + k Equatng the frst k columns on both sdes of (37) gves (38) AV + k = V+ k (H+ k I s)+r + k (eh k I s ), M Ĥ I s. where V + k+1 =(1/β+ k )r+ k, r+ k ˆV p (e 1 I s ) ˆβ k + V k+p+1 βk+p and β + k = r+ k. Note that tr ( (V + k )H ˆV p (e 1 I s ) ) =0 b T and tr ( (V + k )H V k+p+1 ) =0.

17 A Global Arnold Method for Large Non-Hermtan Egenproblems 1513 So tr ( (V + ) k )H V k+1 =0. Ths s a new k-step global Arnold process startng wth the updated V 1 +, so we do not need to restart from scratch and extend t to an m-step one from step k +1upwards n a standard way. Smlarly to the mplctly restarted Arnold algorthm (IRA) [33], we can use those p unwanted F-Rtz values as shfts, also called the exact shfts. So we have developed an mplctly restarted global Arnold algorthm (IRGA) wth the exact shfts suggested. Algorthm 4. IRGA wth the exact shfts 1. Gven the number k of desred egenpars, choose s, m and let p = m k, and take V 1 = V/ V F as an n s startng matrx. 2. Run the m-step global Arnold process to get [H, V,k]. 3. Compute the egenpars of H, select k egenvalues of H as approxmatons to the desred egenvalues and take the p unwanted egenvalues as shfts. 4. Apply mplct restartng approach wth p shfts, and the update the global Arnold algorthm V VQ, H Q T HQ s returned. 5. Test convergence. If yes, stop; otherwse, go to Step 2 and extend the global Arnold process from step k +1upwards. In order to determne the multplctes of the desred egenvalues, we combne Algorthm 3 wth Algorthms 4 and present the followng algorthm. Algorthm 5. IRGA for multple egenproblems 1. Gven the number k of desred egenpars, choose s, m and let p = m k. Defne the set S = {1, 2,...,k}, Φ=, q =1 and prescrbe a tolerance tol. 2. Take V 1 = V/ V F as an n s q startng matrx. 3. Run the m-step global Arnold process to get [H, V,k]. 4. Compute the egenpars (λ (m),y (m) ), =1, 2,...,m of H, select λ (m),= 1, 2,...,k as approxmatons to the desred egenvalues λ,=1, 2,...,k and take the p unwanted λ (m),= k +1,...,m as shfts. 5. Apply mplct restartng approach wth p shfts, and the update the global Arnold algorthm V VQ, H Q T HQ s returned. 6. Test convergence. If yes, go to Step 7; otherwse, go to Step 3 and extend the global Arnold process from step k +1upwards. 7. For all S, set Φ (m) s =(Ψ,ϕ (m) 1,ϕ(m) 2,...,ϕ(m) s q ) and s the number of ts columns. (a) Compute the numercal rank r s of Φ (m) s for all S; (b) If r s <s, set d = r s and remove from S; (c) Otherwse, let Ψ=Φ (m) s,q = q +1and go to Step 2.

18 1514 Congyng Duan and Zhongxao Ja We comment that f A s real symmetrc then Algorthm 4 naturally works by smplfyng the global Arnold process as a symmetrc global Lanczos process. For s =1, Algorthm 4 mathematcally reduces to the mplctly restarted Arnold algorthm [33], whose standard Matlab code s the functon egs.m. 6. NUMERICAL EXPERIMENTS We present numercal examples to llustrate the effcency and relablty of IRGA. All experments were run on a PC wth 2.2 GHz Intel Core 2 Duo T7500 processor usng MATLAB 7.1 wth the machne precson ɛ = under the Wndow XP operatng system. If the relatve resdual norm r (m) j = (A λ (m) I)ϕ (m) j tol A 1 A 1 wth tol a prescrbed tolerance, then (λ (m),ϕ (m) j ) s accepted to have converged. Based on the prevous analyss, we assume that 10 3 C j C j = max 1 j s r (m) j n (32), whch allows the multple egenproblems to be qute ll condtoned for small tol. Astol dmnshes, Algorthm 5 s more applcable and works for worse condtoned multple egenproblems. We took random V 1 's n all the examples. In all the tables below, we denote by ter the number of restarts and by Resdual norms the above relatve resdual norms. Example 1. Ths test problem s taken from the set of test matrces n netlb, and the matrx A s of form 0 1 n n A has zero dagonal entres and known egenvalues, and s sngular f n s odd. The egenvalues are plus and mnus n 1, n 3, n 5,..., (1 or 0). The egenvalue problem of A s hghly ll condtoned for n = 2000, the computed condton number of the egenvector matrx s 4.5e Moreover, the smallest spectral condton number of an ndvdual egenvalue s large up to 5.6e So t s expected that t may be very dffcult to compute a few largest egenpars of A usng Arnold type algorthms.

19 A Global Arnold Method for Large Non-Hermtan Egenproblems 1515 We perform Algorthm 4 on the matrx A, and requre that the algorthm stops as soon as all actual resdual norms of the approxmatng egenpars are below tol = We want to compute the four largest egenvalues 1999, 1997, 1995, Table 1 shows the results obtaned. The left part of Fgure 1 depcts the convergence curves for m =30. We see the algorthm used almost the same restarts to acheve the prescrbed accuracy for dfferent s. Ths justfes that the global Arnold algorthm has the same convergence speed as the standard Arnold algorthm,.e., s =1, as far as ter s concerned. It s also seen from the table that the bgger m s, fewer restarts the algorthm uses and the algorthm converges faster for exteror egenvalues. Fg. 1. Left: A(2000) wth m =30, s =2; rght: dw8192 wth m =20, s =2.

20 1516 Congyng Duan and Zhongxao Ja Example 2. We test the unsymmetrc matrx DW8192 from [3]. The stoppng crteron and the notaton used are as before. We compute the largest four egenvalues n magntude λ ,λ , λ ,λ , whch are qute clustered but numercally dstnct. So ths problem may not be easy for Arnold type algorthms. Table 2 reports the results, where we see the proposed algorthm solved the problem effectvely and relably. The rght part of Fgure 1 depcts the convergence curves for m =20. Besdes, we have observatons smlar to those for Example 1. Example 3. We test the matrx OLM1000 [3], a real unsymmetrc matrx. The stoppng crteron as well as the notaton used are as before. We want to compute the largest four egenvalues n magntude λ ,λ , λ ,λ , whch are qute clustered but numercally dstnct. So lke Example 1, ths problem may not be easy for the proposed algorthm. Table 3 lsts the results obtaned. The rght part of Fgure 2 depcts the convergence curves for m =30. From both the table and the fgure, we see the algorthm solved the problem effectvely and relably. Agan, we have smlar observatons to those for Example 1. Example 4. Consder convecton dffuson dfferental operator u(x, y)+ρu x (x, y)=λu(x, y)

21 A Global Arnold Method for Large Non-Hermtan Egenproblems 1517 Fg. 2. Left: olm1000 wth m =30, s =2; rght: A 100 wth m =20, s =2. on a square regon [0, 1] [0, 1] wth the boundary condton u(x, y) =0. Takng ρ =1and dscretzng wth centered dfferences yeld a block trdagonal matrx A n = tr( I,B n, I), where B n = tr(b, 4,a) s a trdagonal matrx wth a = 1+1/2(n +1) and b = 1 1/2(n +1), and n s chosen the number of nteror mesh ponts on each sde of the square. A n s of order N = n 2. The egenvalues λ 2 and λ 3 are very clustered and they get closer as n ncreases. We test Algorthm 4 on the matrx A 100 obtaned by takng n = 100. We are nterested n the four egenvalues wth the largest real parts, and the stoppng crteron as well as the notaton used are as before. The ntal V 1 were

22 1518 Congyng Duan and Zhongxao Ja chosen randomly n a normal dstrbuton, and the computed egenvalues are λ ,λ λ ,λ Table 4 lsts the results obtaned and the rght part of Fgure 2 depcts the convergence curves for m =20. In what follows, we show how to use the global Arnold method and Algorthm 5 to solve multple egenproblems and determne the multplctes d of λ,=1, 2,...,k and the assocated egenspaces. Dfferently from that done n Examples 1{4, for each F-Rtz value we now compute all the s F-Rtz vectors by (19) smultaneously. Examples 5{8 reports the numercal results, where n the tables we lst all the resdual norms of the s F-Rtz pars for each F-Rtz value used to approxmate a desred egenvalue. Example 5. In ths example, we test a multple egenproblem. Let B = I 2 A, where A s the one presented n Example 1. The real unsymmetrc matrx B has egenvalues wth multplcty two. Algorthm 5 s run on B. As have been seen from Example 1, the egenvalue problem of A s very hghly ll condtoned and C j s very huge. We use the same stoppng crteron and the notaton as before. Both the number of restarts and CPU tme (n seconds) are used to measure the cost of the algorthm. Table 5 reports the results for varous s and m and the experments of determnng d for m =30, 40, where svd(x) s the set of all the sngular values of the matrx X and Φ (m) s =(Φ (m) s 1,...,Φ (m) s q ) n all the tables. We see from Table 5 that Algorthm 5 has found λ,d,=1, 2, 3, 4 relably. Ths s very surprsng snce the true C j are much larger than 1 tol. It s observed that for the same m and dfferent s the algorthm used almost the same restarts and the bgger s s, the more costly the algorthm s. However, f one s requred to

23 A Global Arnold Method for Large Non-Hermtan Egenproblems 1519 determne d and compute a bass of P, then IRGA for s > 1 s preferable and advantageous to IRGA for s = 1,.e., IRA, snce t uses less CPU tme. Note that for the same m, runnng IRGA for s > 1 once s less costly than runnng IRA s tmes. For example, we have to run IRGA for s = 1 three tmes to acheve the am but only need to run IRGA for each s = 1, 2 once or IRGA for s = 3 once, whle the latter s cheaper than the former. Example 6. The matrx B = A I2, where A s the one presented n Example 2. Ths real unsymmetrc matrx B has egenvalues wth multplcty

24 1520 Congyng Duan and Zhongxao Ja two, and the four desred egenvalues are clustered. We fnd g 1 g and g 3 g So C 1j 1 g , C 2j 1 g , C 3j 1 g , C 4j g The egenvectors assocated wth λ 1,...,λ 4 may be moderately ll condtoned. Algorthm 5 s run on B. The stoppng crteron and the notaton used are as before. Table 6 lsts the results. We see from Table 6 that Algorthm 5 has found d, =1, 2, 3, 4 relably. Other observatons are smlar to those for Example 5. Example 7. We construct a matrx B = A I 2, where A s the one presented n Example 3. The matrx B has egenvalues wth multplcty two

25 A Global Arnold Method for Large Non-Hermtan Egenproblems 1521 and the four desred egenvalues are clustered. We fnd g1 g and g3 g , so C1j g , C2j g , C3j 1 1 g , C4j g It s clear that the egenvectors assocated wth λ1,..., λ4 are qute ll-condtoned Algorthm 5 s run on B. The stoppng crteron and the notaton used are as before. Table 7 reports the results. We see from Table 7 that for ths ll-condtoned problem Algorthm 5 has found d, = 1, 2, 3, 4 relably. Other observatons are smlar to those for Example 5. Example 8. Ths test matrx B = A I2 s a matrx, where A

26 1522 Congyng Duan and Zhongxao Ja s the one presented n Example 4. The matrx B has egenvalues wth multplcty two and the four desred egenvalues are clustered. We fnd g , g2 g and g , so C1j g , C2j g , C3j g , C4j g It s seen that the egenvectors assocated wth λ1, λ4 may be moderately ll condtoned but the ones wth λ2, λ3 are defntely very ll condtoned. Algorthm 5 s run on B. The stoppng crteron and the notaton used are as before. Table 8 lsts the results. We see from Table 8 that for ths very ll-condtoned problem Algorthm 5 has found d, = 1, 2, 3, 4 relably. Other observatons are smlar to those for Example 5.

27 A Global Arnold Method for Large Non-Hermtan Egenproblems 1523 REFERENCES 1. W. E. Arnold, The prncple of mnmzed teraton n the soluton of the matrx egenvalue problem, Quart. Appl. Math., 9 (1951), Z. Ba, J. Demmel, J. Dongarra, A. Ruhe and H. A. van der Vorst (eds.), Templates for the Soluton of Algebrac Egenvalue Problems: A Practcal Gude, SIAM, Phladelpha, Z. Ba, R. Barrett, D. Day, J. Demmel and J. Dongarra, Test matrx collecton (non- Hermtan egenvalue problems), 4. L. Bao, Y. Ln and Y. We, A new projecton method for solvng large Sylvester equatons, Appl. Numer. Math., 57 (2007), M. Bellalj, K. Jblou and H. Sadok, New convergence results on the global GMRES method for dagonalzable matrces, Techncal Report, L. M. P. A., F. Chateln and D. Ho, Arnold-Tchebychev procedure for large scale nonsymmetrc matrces, Math. Model. Numer. Anal., 24 (1990), C. C. Chu, M. H. La and W. S. Feng, MIMO nterconnects order reductons by usng the multple pont adaptve-order ratonal global Arnold algorthm, IEICE Trans. Elect., E89-C (2006), C. C. Chu, M. H. La and W. S. Feng, The multple pont global Lanczos method for mutple-nputs multple-outputs nterconnect order reducton, IEICE Trans. Elect., E89-A (2006), C. C. Chu, M. H. La and W. S. Feng, Model-order reductons for MIMO systems usng global Krylov subspace methods, Math. Comput. Smulat., to appear. 10. J. K. Cullum and W. E. Donath, A block Lanczos algorthm for computng the q algebracally largest egenvalues and a correspondng egenspace for large, sparse symmetrc matrces, n: Proceedngs of the 1994 IEEE Conference on Decson and Control, IEEE Press, Pscataway, NJ, 1974, pp J. W. Demmel, Appled Numercal Lnear Algebra, SIAM, Phladelpha, G. Golub and R. Underwood, The block Lanczos method for computng egenvalues, n: Mathematcal Software III, J. Rce (ed.), Academc Press, New York, 1977, pp C. Gu and Z. Yang, Global SCD algorthm for real postve defnte lnear systems wth multple rght-hand sdes, Appl. Math. Comput., 189 (2007), M. Heyoun, The global Hessenberg and CMRH methods for lnear systems wth multple rght-hand sdes, Numer. Algor., 26 (2001), M. Heyoun and K. Jblou, Matrx Krylov subspace methods for large scale model reducton problems, Appl. Math. Comput., 181 (2006),

28 1524 Congyng Duan and Zhongxao Ja 16. K. Jblou, A. Messaoud and H. Sadok, Global FOM and GMRES algorthms for matrx equatons, Appl. Numer. Math., 31 (1999), K. Jblou, An Arnold based algorthm for large algebrac Rccat equatons, Appl. Math. Lett., 19 (2006), K. Jblou, Low rank approxmate solutons to large Sylvester matrx equatons, Appl. Math. Comput., 177 (2006), Z. Ja, The convergence of generalzed Lanczos methods for large unsymmetrc egenproblems, SIAM J. Matrx Anal. Appl., 16 (1995), Z. Ja, Generalzed block Lanczos methods for large unsymmetrc egenproblems, Numer. Math., 80 (1998), Z. Ja, Arnold type algorthms for large unsymmetrc multple egenvalue problems, J. Comput. Math., 17 (1999), W. Karush, An teratve method for fndng characterstcs vectors of a symmetrc matrx, Pacfc J. Math., 1 (1951), R. B. Lehoucq and K. J. Maschhoff, Implementaton of an mplctly restarted block Arnold method, Preprnt MCS-P , Argonne Natonal Laboratory, Argonne, IL, Y. Ln, Implctly restarted global FOM and GMRES for nonsymmetrc matrx equatons and Sylvester equatons, Appl. Math. Comput., 167 (2005), C. C. Page, The computaton of egenvalues and egenvectors of very large sparse matrces, Ph. D. thess, London Unversty, London, England, Y. Saad, Varatons on Arnold's method for computng egenelements of large unsymmetrc matrces, Lnear Algebra Appl., 34 (1980), Y. Saad, Projecton methods for solvng large sparse egenvalue problems, n: Matrx Pencls, B. Kagström, A. Ruhe (eds.), Lecture Notes n Mathematcs 973, Sprnger, Berln, 1983, pp Y. Saad, Chebyshev acceleraton technques for solvng nonsymmetrc egenvalue problems, Math. Comput., 42 (1984), Y. Saad, Numercal Methods for Large Egenvalue Problems, Manchester Unversty Press, 1992, UK. 30. D. K. Salkuyeh, CG-type algorthms to solve symmetrc matrx equatons, Appl. Math. Comput., 172 (2006), D. K. Salkuyeh and F. Toutounan, New approaches for solvng large Sylvester equatons, Appl. Math. Comput., 173 (2006), V. Smoncn and E. Gallopoulos, Convergence propertes of block GMRES and matrx polynomals, Lnear Algebra Appl., 247 (1996), D. C. Sorensen, Implct applcaton of polynomal flters n a k-step Arnold method, SIAM J. Matrx Anal. Appl., 13 (1992),

29 A Global Arnold Method for Large Non-Hermtan Egenproblems G. W. Stewart, Matrx Algorthms II: Egensystems, SIAM, Phladelpha, Congyng Duan and Zhongxao Ja Department of Mathematcal Scences Tsnghua Unversty Bejng P. R. Chna E-mal:

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