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1 Eectronic Transactions on Numerica Anaysis. Voume 7, 1998, pp Copyright 1998,. ISSN ETNA A THEORETICAL COMPARISON BETWEEN INNER PRODUCTS IN THE SHIFT-INVERT ARNOLDI METHOD AND THE SPECTRAL TRANSFORMATION LANCZOS METHOD KARL MEERBERGEN y Abstract. The spectra transformation Lanczos method and the shift-invert Arnodi method are probaby the most popuar methods for the soution of inear generaized eigenvaue probems originating from engineering appications, incuding structura and acoustic anayses and fuid dynamics. The orthogonaization of the Kryov vectors requires inner products. Often, one empoys the standard inner product, but in many engineering appications one uses the inner product using the mass matrix. In this paper, we make a theoretica comparison between these inner products in the framework of the shift-invert Arnodi method. The concusion is that when the square-root of the condition number of the mass matrix is sma, the convergence behavior does not strongy depend on the choice of inner product. The theory is iustrated by numerica exampes arising from structura and acoustic anayses. The theory is extended to the discretized Navier-Stokes equations. Key words. Lanczos method, Arnodi s method, generaized eigenvaue probem, shift-invert. AMS subect cassifications. 65F Introduction. This paper is concerned with the soution of generaized eigenvaue probems of the form (1.1) Ax = Bx; A; B 2 R nn ; x 6= 0; where A may be symmetric or non-symmetric, and B is symmetric positive (semi) definite, by the spectra transformation Lanczos method [6, 18] and the shift-invert Arnodi method [17]. Appications incude the moda anaysis of structures without damping, which eads to (1.2) Ku =! 2 Mu; where K and M are symmetric matrices and often positive definite [9]. Typicay, the number of wanted eigenmodes for representing the structura properties for ow and mid frequencies ranges from a few tens to a few thousands. The moda extraction of acoustic finite eement modes aso eads to a probem of the form (1.2). The required number of eigenmodes is often sma, since the modes are usuay empoyed for a ow frequency anaysis. For the (Navier) Stokes probem, we have K (1.3) C C T 0 u p M 0 = 0 0 u p where M is symmetric positive definite, C is of fu rank and K is symmetric (Stokes [14]) or nonsymmetric (Navier-Stokes). This eigenvaue probem arises in the determination of the stabiity of a steady state soution. Here ony the rightmost eigenvaue is wanted [14, 3]. This paper concentrates on the soution of (1.2), but (1.3) wi aso be touched on. In both appications, M is a discretization of the continuous identity operator, i.e., the continuous inner product hx; yi is repaced by the discrete x T My. As a resut, the condition number of M is usuay sma. We study this specific case. The theory is iustrated by numerica exampes arising from rea appications. ; Received November 11, Accepted for pubication August 4, Recommended by R. Lehoucq. y LMS Internationa, Intereuvenaan 70, 3001 Leuven, Begium. Current address: Rutherford Appeton Laboratory, Chiton, Didcot, OX11 0QX, UK. (K.Meerbergen@r.ac.uk) 90

2 Kar Meerbergen 91 One approach to the soution of generaized eigenvaue probems is the shift-invert Arnodi method [17, 20, 15]. Instead of soving (1.1) directy, one soves the shifted and inverted probem (1.4) (A, B),1 Bx = x by the Arnodi method. The scaar is caed the shift, which expains the name shiftinvert. If (; x) is an eigenpair of (A, B),1 B,then( +,1 ;x) is an eigenpair of Ax = Bx. This reation demonstrates that s can be computed from s. Without oss of generaity, we assume a shift = 0 is used. In genera, A,1 B is a nonsymmetric matrix, even when A and B are symmetric, and this is the reason why the Arnodi method is used. However, when A is symmetric and B is symmetric positive definite, A,1 B is sef-adoint with respect to the B-inner product. This impies that the Lanczos method can be used, when the B-inner product x T By is empoyed instead of the standard inner product x T y. This idea was proposed by Ericsson [5] and Nour-Omid, Parett, Ericsson and Jensen [18]. A bock version was proposed by Grimes, Lewis and Simon [10]. In the case where B is positive semi-definite, which, e.g. arises in appications of the form (1.3), the B-semi-inner product can be used in the Lanczos method or the Arnodi method. This is suggested by Ericsson [5], Nour-Omid, Parett, Ericsson and Jensen [18], and Meerbergen and Spence [16] and appied to inearized and discretized Navier-Stokes equations by Lehoucq and Scott [12]. In this paper, we show by both anaysis and numerica exampes that if the square root of the condition number of the mass matrix B is sma, the choice of inner product does not infuence the convergence speed. The choice of inner product shoud be based on other criteria than rate of convergence. We iustrate this for two casses of appications. When A is symmetric, the use of the B-inner product reduces the Arnodi method to the Lanczos method. The Lanczos method has two advantages over the Arnodi method. First, the eigenvaues have quadratic error bounds and their convergence is we understood [19, 20]. Second, the cost per iteration consists of the action of A,1 B on a vector and the orthogonaization of the new iteration vector against the previous ones. The cost for the construction of the Kryov basis is smaer than for the Arnodi method, since ony the ast two basis vectors are used in the orthogonaization process. The Arnodi method uses a vectors. The Lanczos method uses the B-inner product which can be quite expensive compared to the standard inner product. The overa orthogonaization cost, however, can be much smaer than for the Arnodi method with standard inner product, when the number of iteration vectors is arge. This is often the case for a structura anaysis for ow and mid frequencies since a arge number of eigenmodes is wanted. The use of the standard inner product instead of the B-inner product may be preferred when A is nonsymmetric and the Arnodi method needs to be used anyway, so fu orthogonaization against a previous basis vectors cannot be avoided. Lehoucq and Scott [12] demonstrate for discretized Navier-Stokes appications that the B-inner product is more expensive than the standard inner product, but eads to a more reiabe Arnodi method. A side effect of the use of B-orthogonaization is that the approximate eigenvectors are B-orthogona, when A is symmetric. This is very natura since the exact eigenvectors corresponding to different eigenvaues are B-orthogona. In finite eement appications, it is assumed that the computed eigenmodes satisfy this property. This is automaticay satisfied by the Lanczos method with B-orthogonaization, but not by the Arnodi method. The pan of this paper is as foows. In x2, a theoretica comparison between standard and B orthogonaization is estabished. In x3, we present an easy way of obtaining B-orthogona eigenvectors from the Arnodi method when A is symmetric. In x4, we iustrate the theory by numerica exampes. Section 5 generaizes the ideas from x2 to the Navier-Stokes probem. Finay, we summarize the main concusions in x6. We assume computations in exact arithmetic.

3 92 Theoretica comparison between inner products 2. A reation between standard and B-orthogonaization. In this section, a theoretica study of the Arnodi method with standard orthogonaization and B-orthogonaization is estabished for the eigenvaue probem (1.2). The goa is to reate the residua norms as we as the Hessenberg matrix (which is tridiagona for the Lanczos method) and the computed eigenvaues for both types of inner product. The anaysis assumes exact arithmetic. First, in x2.1, some preiminaries and notation are presented. Second, x2.2 puts both types of orthogonaization into a singe theoretica framework: we present the agorithm and some properties. The reation between standard and B-inner products in the Arnodi method wi be formuated and derived in x Notation and preiminaries. This section is devoted to some notation and matrix properties. In genera, we use the Eucidean norm for vectors and matrices, denoted by kk 2 or kk. The matrix Frobenius norm is denoted by kk F. Let (C) denote the condition number of the matrix C. First, since B is a positive definite matrix, there exists L 2 R nn such that B = L T L. LEMMA 2.1. Consider V;W 2 R nk.letv T V = I, W T BW = I and et the coumns of V span the same space as the coumns of W. Then there is an S such that V = WS. Moreover, (S) =(W ) p (B). Proof. It is cear that there is an S such that V = WS. Hence V T BV = S T S and W T W = S,T S,1. Since kv k = 1, wehaveksk 2 = kv T BV k kbk, and since W T BW =(LW ) T (LW )=I, wehaveks,1 k 2 = kw T W k = k(lw ) T B,1 (LW )k kb,1 k. This competes the proof. We wi compare two agorithms that differ primariy in their choice of inner product or norm. We wi use the notation hx; yi to stand for a generic inner product, such as x T y or x T By. The notation is aso generaized to matrices V =[v 1 ;:::;v k ] and W =[w 1 ;:::;w ], as foows : hv;xi =[hv ;xi] k =1 2 Rk hw;vi =[hw i ;v i] (;k) (i;)=(1;1) 2 Rk : Since h; i is an inner product, it foows that hws;vzi = S T hw;viz. The B norm of a vector x is defined by kxk B = p xt Bx. TheB norm of a matrix C is defined by kck B = klck 2,whereB = L T L. Obviousy, for two matrices, V 2 R nk and W 2 R n,wehavekv T BWk kv k B kw k B. For a matrix C, the Kryov space K k (C; v 1 ) of order k with starting vector v 1 is defined by K k (C; v 1 ) = spanfv 1 ;Cv 1 ;C 2 v 1 ;:::;C k,1 v 1 g : We assume that a Kryov spaces of order k have dimension k. In practice, a space is represented by a basis. The foowing emma gives a reation between two different bases. LEMMA 2.2. Let V k ;W k 2 R nk be such that the first coumns of V k and the first coumns of W k form two bases for K (C; v 1 ) for =1; ::; k. Then there is a fu rank upper trianguar matrix S k 2 R kk such that V k = W k S k. The foowing emma shows the uniqueness of a normaized Kryov basis. LEMMA 2.3 (Impicit Q Theorem). ([8, Theorem 7.4.2]) Let the first coumns of V k and W k 2 R nk form two bases for K (C; v 1 ) for =1;:::;k and hv k ;V k i = I = hw k ;W k i. Then v i = w i i with i = 1 for i =1;:::;k.

4 Kar Meerbergen A genera (theoretica) framework for Kryov methods. The Arnodi and Lanczos methods for the soution of (1.2) are Kryov subspace methods, i.e., the eigenvaues and eigenvectors are computed from the proection of A,1 B on a Kryov space. The foowing agorithm covers both methods. Reca that hx; yi denotes the inner product, e.g. the standard inner product hx; yi = x T y or the B-inner product hx; yi = x T By. ALGORITHM 1. Genera framework for the Lanczos and Arnodi methods. 0. Given v 1 with hv 1 ;v 1 i =1. 1. For =1to k do 1.1. Form p = A,1 Bv Compute the Gram Schmidt coefficients h i = hv i ;p i;i =1;:::; Update q = p, P v i=1 ih i Compute norm h +1; = hq ;q i 1= Normaize : v +1 = q =h +1;. 2. Let H k =[h i ] (k+1;k) (i;)=(1;1) 2 Rk+1k where h i =0whenever i>+1. Let H k be the first k rows of H k. Let V k =[v 1 ;:::;v k ]. 3. Compute eigenpairs (; z) of H k, with z 2 R k, by the QR method. 4. Compute the Ritz vector x = V k z 2 R n. 5. Compute the residua norm = h k+1;k e T k z. Step 1.1 is performed by a matrix vector mutipication with B and the soution of a inear system with A. The soution of the inear system is usuay performed by a direct method, since one can take advantage of the fact that A needs to be factored ony once. Moreover, direct methods usuay give a soution with a sma backward error. This is required for the Arnodi method to find the eigenpairs of (1.1) [15]. Steps 1.1 to 1.5 compute a basis V k+1 =[v 1 ;:::;v k+1 ] of the Kryov space K k+1 (A,1 B;v 1 ) = spanfv 1 ;A,1 Bv 1 ;:::;(A,1 B) k v 1 g ; normaized such that hv k+1 ;V k+1 i = I. The normaization is performed by Gram-Schmidt orthogonaization. For reasons of numerica stabiity, practica impementations use reorthogonaization [4] in the Arnodi method and modified Gram-Schmidt with partia reorthogonaization in the Lanczos method [10]. The Gram-Schmidt coefficients are coected in the upper Hessenberg matrix H k. Note that H k is a k +1by k matrix. We denote the k k upper submatrix of H k by H k. By the eimination of p and q from Steps 1.1 to 1.5, it foows that (2.1a) (2.1b) A,1 BV k = V k+1 H k = V k H k + v k+1 h k+1;k e T : k This is the we known recurrence reation for the Arnodi and Lanczos methods. Usuay, k n, sothath k has much smaer dimensions than A and B. In Steps 3-4, an eigenpair (; x) is computed by the Gaerkin proection of A,1 B on K k, i.e., x 2K k and the residua r = A,1 Bx, x is orthogona to the Kryov space : (2.2) x = V k z with H k z = z : The s are the eigenvaues of H k and are caed Ritz vaues and the x s are the corresponding Ritz vectors. They form an approximate eigenpair of A,1 B. Reca that H k is an upper Hessenberg matrix, so the eigenpairs (; z) are efficienty computed by the QR method [8, x7.5]. The residua r = A,1 Bx, x aso foows from (2.1b) : (2.3) r = A,1 Bx, x = v k+1 h k+1;k e T k z

5 94 Theoretica comparison between inner products and the induced norm is very cheapy computed as (2.4) = hr;ri 1=2 = h k+1;k e T k z : This expains Step 5 in the agorithm. The residua norm is a measure of the accuracy of the eigenvaues. (See eigenvaue perturbation theory, e.g., [20, Ch. III] and [1, Ch. 2].) In the foowing, we aso use a bock formuation of the recurrence reation for the Ritz vectors. Let X = [x 1 ;:::;x ] = V k Z and D = diag( 1 ;:::; ) represent k Ritz pairs and R =[r 1 ;:::;r ] the corresponding residua terms of the form (2.3). Then it foows that (2.5) A,1 BX = X D + R ; R = h k+1;k v k+1 e T k Z : In the rest of this paper, we use the foowing notation to distinguish between the standard and B-orthogonaization. For standard orthogonaization, the Kryov vectors are denoted by V k+1 and the Hessenberg matrix is H k. They satisfy A,1 BV k = V k+1 H k ; H k = V T k+1a,1 BV k ; V T k+1v k+1 = I: For B-orthogonaization, the Kryov vectors are denoted by W k+1 and the Hessenberg matrix by T k. They satisfy A,1 BW k = W k+1 T k ; T k = W T k+1ba,1 BW k ; W T k+1bw k+1 = I: It is cear that if A is symmetric, the Hessenberg matrix T k = V T k BA,1 BV k is symmetric, hence tridiagona Comparison between both orthogonaization schemes. In this section, a reationship between standard and B-orthogonaization is estabished. Ceary, the computed Kryov spaces are the same, but the proections differ and this may ead to different eigenvaue approximations. The theoretica resut in this section uses Lemma 2.2, which makes the ink between two Kryov bases by use of an upper trianguar matrix. First, in Theorem 2.4, we reate h k+1;k and t k+1;k and H k and T k, and in Theorem 2.5, we reate the eigenvaues of T k and H k. THEOREM 2.4. Let v 1 = w 1 =kw 1 k. Then there is a matrix S k 2 R kk, such that (2.6) T k = S k H k S,1 k + E kek 2 h k+1;k (W k+1 ) (W k+1 ),1 t k+1;k h k+1;k (W k+1 ) ; and (S k );(W k+1 ) p (B) : Proof. Reca that V k+1 and H k satisfy the Arnodi recurrence reation (2.1a). Let S k+1 2 R k+1k+1 be the upper trianguar Choesky factor of V T k+1 BV k+1, i.e., V T k+1bv k+1 = S T k+1s k+1 : As a consequence, V k+1 S,1 k+1 forms a B-orthogona Arnodi basis for the Kryov space. Since Kryov bases are unique (see Lemma 2.3), W k+1 = V k+1 S,1 k+1. (Eventuay, the coumns

6 Kar Meerbergen 95 S k+1 must be mutipied by,1, see i in Lemma 2.3.) The Arnodi reation (2.1a) can be rewritten as A,1 B(V k S,1 k )=(V k+1s,1 k+1 )(S k+1h k S,1 k ); where S k is the k k principe submatrix of S k+1. Consequenty, T k = S k+1 H k S,1. k Decompose S k+1 H k S,1 Sk s H k = S,1 k 0 s k+1;k+1 h k+1;k e T k k T T k k = : t k+1;k e T k Then, we have where (2.7) and T k = S k H k S,1 k + E E = h k+1;k s,1 k;k set k t k+1;k = h k+1;k s k+1;k+1 s,1 k;k : The proof now foows from Lemma 2.1. This theorem says that when the square root of the condition number of B is sma, the residua terms are of the same order. It aso says that T k is a rank-one update of H k of the order of h k+1;k. The foowing theorem estabishes a reationship between eigenpairs of T k and H k. THEOREM 2.5. Let (; V k z) be a Ritz pair of the Arnodi method and et be the corresponding residua norm. Assume that v 1 = w 1 =kw 1 k. Then there exists an eigenvaue of T k such that, (Y k )(W k+1 ); where Y k is the matrix whose coumns contain the eigenvectors of T k. Proof. From (2.6) and the fact that H k z = z,wehave S k H k z, T k S k z =,ES k z T k S k z, S k z = ES k z: With y = S k z=ks k zk and = h k+1;k e T k z, it foows from (2.7) and et k S k = s k;k e T k that (2.8) kt k y, yk h k+1;ks,1 k;k ksket k S kz ks k zk h k+1;kkske T z ksk k = ks k zk ks k zk h k+1;ke T k z (S k+1 ) = (W k+1 ): Foowing the Bauer-Fike Theorem [20, Theorem 3.6], it foows that T k has an eigenvaue for which, (Y k )kt k y, yk ;

7 96 Theoretica comparison between inner products from which the proof foows. The eigenvaues of H k can simiary be reated to the eigenvaues of T k. When A is symmetric, so is T k. Therefore (Y k )=1, and the error bound becomes sharper. The most important concusion is that the Ritz vaues with a sma residua norm are amost the same for both the standard and the B-inner products. 3. Computation of B-orthogona eigenvectors in the Arnodi method. When the B- inner product is used and A and B are symmetric, the tridiagona matrix T k is symmetric, so T k has an orthogona set of eigenvectors z ; =1;:::;k. Therefore, the Ritz vectors W k z, =1;:::;kform a B-orthogona set of k vectors. When the standard inner product is used, the Ritz vectors are not necessariy B- orthogona. The foowing theorem shows the dependence of the B-orthogonaity on the separation between the eigenvaues and on the norms of the residuas. THEOREM 3.1. Let ( ;x ), =1;:::; < k be Ritz pairs and r the corresponding residuas computed by the Arnodi method with the standard inner product. Then, for 1 i;, we have x T i Bx kx i k B kx k B kr k B =kx k B + kr i k B =kx i k B i, provided that i 6=. Proof. Denote by D the diagona matrix containing 1 ;:::;, and et the coumns of X be the corresponding Ritz vectors. By mutipying (2.5) on the eft by X T B,wehave ; (3.1) X T BA,1 BX = X T BX D + X T BR : Since the eft-hand side is symmetric, (3.2) X T BX D + X T BR = D X T BX + R T BX and, so X T BX D, D X T BX = R T BX, X T BR : The (i; ) eement of this matrix eads to the resut of the theorem. This theorem shows that eigenvectors corresponding to different distinct eigenvaues are amost B-orthogona, but the eigenvectors corresponding to custered eigenvaues can ose B-orthogonaity. So, an additiona B-orthogonaization of the eigenvectors is desirabe. In the foowing, we use the notations of (2.5), i.e., D denotes the diagona matrix containing k Ritz vaues 1 ;:::; and the coumns of X denote the corresponding Ritz vectors. The most robust way to obtain B-orthogona Ritz vectors, is to compute the B- orthogona proection of A,1 B onto the range of X and compute new eigenpairs from this proection. This is estabished as foows. Let S be the upper trianguar Choesky factor of X T BX = S T S. Then, with Y = X S,1,wehaveY T BY = I. A Ritz pair obtained by B-orthogonaization is of the form (; y = Y z) with (; z) satisfying (3.3) Y T BA,1 BY z = z : This additiona proection eads to quadratic error bounds for the eigenvaues. (See e.g. the Kato-Tempe theorem [20, Theorem 3.8].) The matrix on the eft-hand side of (3.3) can be computed expicity, but can as we easiy be computed without the action of A,1 by the use of (3.1) : (3.4) Y T BA,1 BY = S D S,1 + Y T BR S,1

8 Kar Meerbergen 97 Box car interior exhaust pipe FIG.4.1.Meshes for the three eigenvaue probems in x4 (by courtesy of Bosa) In practice, we coud compute (; z) from (3.5) (S D S,1 + Y T BR S,1 )z = z instead of (3.3). However, there is an aternative to performing an additiona proection. The coumns of Y can be taken as the B-orthogona Ritz vectors. From the foowing theorem, it foows that if the residua norms of ( ;x ) for = 1;:::; are sma and p (B) is sma, the residuas for ( ;Y e ) for = 1;:::; are aso sma. Of course, the Ritz vaues do not satisfy a quadratic error bound. THEOREM 3.2. Reca the definition of D and X from Eq. (2.5). Let S be the upper trianguar Choesky factor of X T BX and et Y = X S,1.Then ka,1 BY, Y D k B (p +1) p (B)kR k : Proof. From (3.4), we have Y T BA,1 BY = S D S,1 + E with E = Y T BR S,1.SinceS is upper trianguar, and D diagona, S D S,1 = D + U where U is stricty upper trianguar. Note that S D S,1 + E is symmetric and so is U + E : U + E = U T + E T : This impies U, U T = E, E T 2kUk 2 F 2kEk 2 F and so kuk 2 kek F p kek 2. From (2.5), we have from which the proof foows. A,1 BY, Y D = Y (S D S,1, D )+R S,1 = Y U + R S,1 ka,1 BY, Y D k B kuk 2 + p (B)kR k 2 (p +1) p (B)kR k 2 4. Numerica exampes. A exampes have been generated using SYSNOISE, a software too for vibro-acoustic simuation [22]. The matrices A and B arise from acoustic and structura finite eement modes and are symmetric. We present resuts for the foowing probems.

9 98 Theoretica comparison between inner products PROBLEM 1. The first appication concerns the structura moda anaysis of a square box with Young moduus 2: Pa, Poisson coefficient 0:3 and voume density 7800 kg=m 3. The thickness of the materia is 0:01m. One of the six faces of the box is fixed. For the finite eement anaysis, we used 150 she eements. This anaysis eads to an eigenvaue probem (1.2). For this probem, A = K and B = M. The dimension of the probem is n = 852 and p (B) 1: We used a shift =,10 in the shift-invert transformation of (1.4). PROBLEM 2. The acoustic moda anaysis of a Vovo 480 3D car interior discretized with 744 cubic eements and 68 pentaedra finite eements eads to a probem of the form (1.2) with K positive semi-definite. The fuid has voume density 1:225 kg=m 3 and sound has a speed of 340 m/s (air). For this probem, A = K and B = M. The dimension of the probem is n = 1176 and p (B) 21:2. We used a shift =,10 in the shift-invert transformation of (1.4). PROBLEM 3. This concerns the couping of a structura and an acoustic probem. We consider a square box of 1 m 3 with Young moduus 2: Pa, Poisson coefficient 0:3 and voume density 7800 kg=m 3. The thickness of the materia is 0:01 m. One of the six faces of the box is fixed. The box is fied with air (speed of sound 340 m/s and voume density 1:29 kg=m 3 ). The box is discretized by 294 she eements, which represents the structura mode. The acoustic behavior of the fuid is modeed by a boundary eement mesh of 294 quadranguar eements. The matrix A is the structura stiffness matrix, whie B is the sum of the structura mass matrix and the added mass matrix coming from the acoustic mode. The dimension of the probem is n = 1692 and p (B) 1: Ashift =0was used. PROBLEM 4. The acoustic moda anaysis of a Jaguar X100MS exhaust pipe discretized with cubic finite eements eads to a probem of the form (1.2) with K positive semidefinite. The fuid has voume density 1:225kg=m 3 and the sound has a speed of 340 m/s (air). The dimension of the probem is n = and p (B) 42:8. For this probem, A = K and B = M. Theshift is chosen such that the 11 dominant eigenvaues of (A, B),1 B correspond to the eigenfrequencies between 0 and 200Hz Iustration for Theorem 3.1. We compare the Ritz vaues of (A +10B),1 B for Probems 1 and 2 and their residua norms after k = 10 Arnodi/Lanczos steps, started with a random initia vector. Theorem 3.1 states that the Ritz vaues obtained by B- orthogonaization and standard orthogonaization ie within a distance of p (B) where is the residua norm (2.4). For Probem 1, we have h k+1;k = 3: ,7, t k+1;k = 4: ,6 and (W k+1 )=24:755 and for Probem 2, we have h k+1;k =4: ,7, t k+1;k =3: ,7 and (W k+1 )=2:7. These vaues are consistent with Theorem 2.4. The matching significant digits of the respective Ritz vaues and the residua norms are shown in Tabes 4.1 and 4.2, except for the mutipe eigenvaue at 0:1 ( 1 ;:::; 4 ) for Probem 1, which is not dispayed. Ritz vaues that do not share any digits are not shown. A Ritz vaues satisfy Theorem Iustration of the B-orthogonaization of the Ritz vectors. When Ritz vectors are computed by Arnodi s method with standard orthogonaization, they are not B- orthogona. We coud perform an expicit B-orthogona proection by soving (3.5). Instead, we use the coumns of Y as Ritz vectors, as suggested in x3. We iustrate Theorem 3.2, which states that the coumns of Y are sufficienty accurate Ritz vectors. The are unchanged. The resuts reported come from the Arnodi computations from x4.1. The Ritz vectors before and after B-orthogonaization are denoted by x and y respectivey. Tabes 4.3 and 4.4 show the expicity computed residua norms before and after B-orthogonaization. (Note that the (2) in Tabes 4.3 and 4.4 corresponds we to the in Tabes 4.1 and 4.2 respectivey. The difference is that is computed from (2.4) and (2) is the expicity computed residua norm.)

10 Kar Meerbergen 99 TABLE 4.1 Matching the eigenvaues of H k and T k after 10 steps for Probem 1. The vaue is defined by (2.4). hx; yi = x T y hx; yi = x T By 5 5:94110,6 710,9 410,9 6 3: ,6 510,8 410, ,6 910,7 610, ,7 210,7 210, ,7 310, ,7 210,8 TABLE 4.2 Matching the eigenvaues of H k and T k after 10 steps for Probem 2. The vaue is defined by (2.4). hx; yi = x T y hx; yi = x T By 1 9: ,2 0: 0: 2 6: ,6 210,13 210,13 3 2:20810,6 210,9 210,9 4 1:810,6 210,7 210,7 5 1:36410,6 310,8 310, ,7 210,7 310, ,7 310,7 210, ,7 110,7 110, ,8 810, ,8 210,8 The resuts are consistent with Theorem 3.2. In Tabe 4.3, (B) 4 10,6, which is much arger than (2). This is possibe since y 4 4 is a inear combination of x 1 ;:::;x 4 and (B) 4 depends on kr 4 k (2) 1 10, Iustration for the impicity restarted Arnodi/Lanczos methods. In practica cacuations, if sma residua norms need to be obtained for the desired eigenvaues, the Kryov subspaces often become very arge. In the iterature, some remedies against the growth of this subspace are proposed. One idea is to restart the Kryov method with a new poe in (1.4), as suggested in [10, 7]. In this paper, we use the impicity restarted Lanczos and Arnodi methods with exact shifts [21, 11]. Roughy speaking, the impicity restarted Arnodi method is mathematicay equivaent to the foowing scheme.

11 100 Theoretica comparison between inner products TABLE 4.3 Residua norms for Probem 1 before and after B-orthogonaization. We denote (2) = k(a+10b),1 Bx, x k and (B) = k(a +10B),1 By, y k B. (2) (B) 1 310,5 510, ,9 810, ,8 210, ,9 210, ,9 110, ,8 110, ,7 310, ,7 810, ,7 310, ,7 410,8 TABLE 4.4 Residua norms for Probem 2 before and after B-orthogonaization. We denote (2) = k(a+10b),1 Bx, x k and (B) = k(a +10B),1 By, y k B. (2) (B) 1 610,17 410, ,13 410, ,9 510, ,8 310, ,7 110, ,7 210, ,7 210, ,7 110, ,8 110, ,8 910,8 ALGORITHM 2. (impicity) restarted Arnodi method 0. Given is an initia vector v Iterate : 1.1. Form V k+1 and H k by k steps of Arnodi Compute Ritz pairs (; x) and residua norm Get a new initia vector v 1 = V k z. Unti TOL The impicity restarted Arnodi method is an efficient and reiabe impementation of this agorithm. For the probems that are soved in this paper, the new initia vector v 1 is a inear combination of the wanted Ritz vectors, so that unwanted eigenvaues do not show up in foowing iterations. (For a precise definition and practica agorithms, see [21, 11].) In genera, the impicity restarted Arnodi and Lanczos methods do not have the same subspaces after two iterations, since the Ritz vectors are different and so is the inear combination. This can ead to different subspaces and therefore to different rates of convergence. The numerica resuts are obtained using the ARPACK routines dsaupd (Lanczos) and dnaupd (Arnodi) [13]. Note that ARPACK uses the Arnodi process, i.e., fu (re)orthogonaization,

12 Kar Meerbergen 101 TABLE 4.5 Comparison between impicity restarted Lanczos (x T By) and Arnodi (x T y) for Probem 3 hx; yi inear soves matrix vector iterations time products with B (sec.) x T By : x T y : TABLE 4.6 Comparison between impicity restarted Lanczos (x T By) and Arnodi (x T y) for Probem 4 hx; yi inear soves matrix vector iterations time products with B (sec.) x T By x T y for symmetric probems, rather than the more efficient Lanczos process with partia reorthogonaization. This impies that timings reductions are possibe for the resuts of the Lanczos method with B-orthogonaization. The matrixb was representedby a compressed sparse row matrix storage format, which aows for efficient storage and matrix vector products. The inear system sovers where soved by a bock skyine (or profie) sover. A computations were carried out within the SYSNOISE [22] environment on an HP PA 7100 C110 workstation. Tabe 4.5 contains the numerica resuts for Probem 3. The 20 eigenmodes nearest 0 are required. The Impicity Restarted Arnodi/Lanczos methods were run with the same initia vector, Kryov subspace dimension k = 30, and reative residua toerance TOL = 10,5. The number of iterations are equa for both methods. The Lanczos method with B-orthogonaization is more expensive due to the additiona matrix vector products in the reorthogonaization of the B-orthogonaization. For the Arnodi method with standard orthogonaization, the number of matrix vector products with B is equa to the number of soves with A, B pus the number needed for the B-orthogonaization of the Ritz vectors (x3). Tabe 4.6 contains the numerica resuts for Probem 4. The eigenfrequencies between 0 and 250 Hz are sought. The impicity restarted Arnodi/Lanczos methods were run with the same initia vector, Kryov subspace dimension k = 21, and reative residua toerance TOL =10,5. A matrix vector product by B is cheap compared to a back transformation. The concusions are very simiar to Probem 3, but the differences between the CPU times are ess pronounced. 5. Extension to (Navier) Stokes probems. Reca the (Navier) Stokes probem (1.3). For this probem, B is typicay positive semi-definite. The use of the B-semi-inner product in the Lanczos method was suggested by Ericsson [5], Nour-Omid, Parett, Ericsson and Jensen [5]. The use of this semi-inner product was ustified by Ericsson [5] and Meerbergen and Spence [16]. Moreover, it purifies the Ritz vaues from possibe contamination arising from the singuarity of B. For the detais, see [5, 18, 16]. Ciffe, Garratt, Goding and Spence [2] suggested the use of the matrix P = I instead of M in the inner product. This P -inner product has the same purification property as the B-inner product (see [16, Theorem 2] with M = I). Thus, an aternative to the Arnodi method with

13 102 Theoretica comparison between inner products B-orthogonaization is the Arnodi method with P -orthogonaization. In fact, the situation is very simiar to acoustic and structura eigenvaue probems. Indeed, decompose Uk+1 U ~ V k+1 = and W k+1 k+1 = ~Q k+1 Q k+1 and normaize V k+1 such that V T PV k+1 k+1 = U T U k+1 k+1 = I and W k+1 such that W T BW k+1 k+1 = U ~ T M ~ k+1 U k+1 = I. Consequenty, Theorem 2.4 is sti vaid with the ony difference that (B) shoud be repaced by (M ), V k+1 by U k+1 and W k+1 by U ~ k Concusions and fina remarks. In this paper, we estabished a theoretica comparison between standard orthogonaization and B-orthogonaization for the soution of eigenvaue probems, for which the square root of the condition number of the mass matrix is sma. This is often the case in acoustic and structura eigenvaue probems and for the discretized (Navier) Stokes equations. Roughy speaking, the orthogonaization seems to pay a minor roe in the convergence of the eigenvectors in the Kryov space. The theoretica resuts are extended to the Navier-Stokes probem as we, where the Arnodi method with B-semiorthogonaization and the Arnodi method with P -semi-orthogonaization behave simiary. Foowing the theory and our numerica experiments the type of inner product shoud not be chosen on the basis of rates of convergence. The decision shoud be based on cost per iteration, or reiabiity. For exampe, it foows from the numerica resuts for Probem 4 that there is no advantage in performance for the Arnodi method with standard orthogonaization with respect to the Lanczos method with B-orthogonaization, though the inner product is more expensive for the Lanczos method. Moreover, when the number of iteration vectors is high, it is expected that the Gram-Schmidt process with reorthogonaization becomes prohibitive, so that the Lanczos method becomes much cheaper than the Arnodi method. We aso noticed that the Lanczos method with B-orthogonaization is more reiabe than the Arnodi method with standard orthogonaization: sometimes, sought-after eigenvaues are missed by the Arnodi method. For Navier-Stokes appications, where the Arnodi method shoud be used anyway since A is nonsymmetric, the use of the P -inner product rather than the B-inner product may ead to a more efficient code. Lehoucq and Scott [12] report that the B-inner product seems to be more reiabe than the standard inner product. They do not report resuts for the P -inner product. 7. Acknowedgment. The author is gratefu to John Lewis who made usefu comments on the impementation and appication of the spectra transformation Lanczos method and suggested some other materia for further discussion, and to Jean-Pierre Coyette and Christophe Lecomte for hepfu comments on the appications in the numerica exampes. The author aso thanks the referees for the many hepfu suggestions that improved the presentation of the paper. REFERENCES [1] F. CHATELIN, Eigenvaues of matrices, John Wiey and Sons, [2] K. CLIFFE, T.GARRATT, J.GOLDING, AND A. SPENCE, On the equivaence of bock matrices under Arnodi s method, Tech. Rep. Preprint 94/08, Schoo of Mathematica Sciences, University of Bath, Bath, UK, [3] K. CLIFFE, T. GARRATT, AND A. SPENCE, Eigenvaues of bock matrices arising from probems in fuid mechanics, SIAM J. Matrix Ana. App., 15 (1994), pp [4] J. DANIEL, W. GRAGG, L. KAUFMAN, AND G. STEWART, Reorthogonaization and stabe agorithms for updating the Gram-Schmidt QR factorization, Math. Comp., 30 (1976), pp [5] T. ERICSSON, A generaised eigenvaue probem and the Lanczos agorithm, in Large Scae Eigenvaue Probems, J. Cuum and R. Wioughby, eds., Esevier Science Pubishers BV, 1986, pp

14 Kar Meerbergen 103 [6] T. ERICSSON AND A. RUHE, The spectra transformation Lanczos method for the numerica soution of arge sparse generaized symmetric eigenvaue probems, Math. Comp., 35 (1980), pp [7] T. GARRATT, G. MOORE, AND A. SPENCE, Two methods for the numerica detection of Hopf bifurcations, in Bifurcation and Chaos: Anaysis, Agorithms, Appications, R. Seyde, F. Schneider, and H. Troger, eds., Birkhäuser, 1991, pp [8] G. GOLUB AND C. VAN LOAN, Matrix computations, The Johns Hopkins University Press, 3rd ed., [9] R. GRIMES, J. LEWIS, AND H. SIMON, Eigenvaue probems and agorithms in structura engineering, in Large Scae Eigenvaue Probems, J. Cuum and R. Wioughby, eds., Esevier Science Pubishers B.V., 1986, pp [10], A shifted bock Lanczos agorithm for soving sparse symmetric generaized eigenprobems, SIAMJ. Matrix Ana. App., 15 (1994), pp [11] R. B. LEHOUCQ, Restarting an Arnodi reduction, Preprint MCS-P , Argonne Nationa Laboratory, Argonne, IL 60439, [12] R. B. LEHOUCQ AND J. A. SCOTT, Impicity restarted Arnodi methods and eigenvaues of the discretized Navier-Stokes equations, Technica Report SAND J, Sandia Nationa Laboratory, Abuquerque, New Mexico, [13] R. B. LEHOUCQ, D.C.SORENSEN, AND C. YANG, ARPACK USERS GUIDE: Soution of Large Scae Eigenvaue Probems with Impicity Restarted Arnodi Methods, SIAM, Phiadephia, PA, [14] D. MALKUS, Eigenprobems associated with the discrete LBB condition for incompressibe finite eements, Internat. J. Engrg. Sci., 19 (1981), pp [15] K. MEERBERGEN AND D. ROOSE, Matrix transformations for computing rightmost eigenvaues of rea nonsymmetric matrices, IMA J. Numer. Ana., 16 (1996), pp [16] K. MEERBERGEN AND A. SPENCE, Impicity restarted Arnodi and purification for the shift-invert transformation, Math. Comp., 66 (1997). [17] R. NATARAJAN, An Arnodi-based iterative scheme for nonsymmetric matrix pencis arising in finite eement stabiity probems, J. Comput. Phys., 100 (1992), pp [18] B. NOUR-OMID, B. PARLETT, T. ERICSSON, AND P. JENSEN, How to impement the spectra transformation, Math. Comp., 48 (1987), pp [19] B. PARLETT, The symmetric eigenvaue probem, Prentice Ha series in computationa mathematics, Prentice- Ha, [20] Y. SAAD, Numerica methods for arge eigenvaue probems, Agorithms and Architectures for Advanced Scientific Computing, Manchester University Press, Manchester, UK, [21] D. C. SORENSEN, Impicit appication of poynomia fiters in a k-step Arnodi method, SIAM J. Matrix Ana. App., 13 (1992), pp [22] SYSNOISE REV 5.4, A software too for vibro-acoustic simuation. LMS Internationa, Intereuvenaan 70, 3001 Leuven, Begium, 1999.