D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matr. Anal. Appl., 13 (1992), pp. 357--385.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....used to compute an initial iterate x 0 as discussed in Section 2.2. LSTRS is an iterative method that requires the solution of a large scale eigenvalue problem at each step. Unless otherwise indicated, the eigenvalue problems were solved by means of the Implicitly Restarted Lanczos Method (IRLM) [22] as implemented in ARPACK [13] The IRLM is particularly suitable for large scale problems since it has low and fixed storage requirement and relies upon matrix vector products only. In the current implementation of LSTRS, a Mexfile interface was used to access ARPACK. Notice that the capabilities ....

D.C. Sorensen. Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13(1):357-385, 1992.

....(k 1; k) entry is h k 1;k . The columns of V k are computed by a Gram Schmidt orthogonalization process. This process does not guarantee orthogonality of the columns of V k in floating point arithmetic, so reorthogonalization is recommended to improve the numerical stability of the method [32] [134]. Sometimes, B orthogonalization is used, so that V k 1 BV k 1 = I instead of V k 1 V k 1 = I [93] The non Hermitian Lanczos method, also called the two sided Lanczos method, is an oblique projection method. It produces a non Hermitian tridiagonal matrix T k and a pair of matrices V k and ....

D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357--385.

....satisfied. Several restarting strategies for the Arnoldi process have been proposed in the literature with the aim of iteratively determining an initial vector v 1 that gives an Arnoldi decomposition (2.8) with the property (2. 9) The Implicitly Restarted Arnoldi (IRA) method, proposed by Sorensen [13], is one of the most e#ective restarting strategies. This method combines the Arnoldi process with the implicitly shifted QR algorithm to determine an Arnoldi decomposition (2.8) that satisfies (2.9) We outline the IRA method. Let the eigenvalues # j of A satisfy (1.4) and (1.5) and assume for ....

....vector v 1 with normally distributed randomly generated entries to determine the decomposition V 2m = V 2mH 2m # 2m v 2m 1 e 2m . 2.10) Compute the spectrum #(H 2m ) j=1 and assume that . Re( # ) 0 Re( # 1 ) Re( 2m ) Define k = min m, # . Sorensen [13] describes how the decomposition (2.10) can be updated without evaluating any matrix vector products with the matrix A to give the Arnoldi decomposition T V 2m k = V 2m k H 2m k # 2m k v 2m k 1 e 2m k , 2.11) with initial vector v 1 = V 2m k e 1 = 2.12) ....

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D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357--385.

....preconditioner. It can be expressed as: M 1 = V k H 1 k V n k . At each restart, the preconditioner is updated by extracting new eigenvalues which are the smallest in magnitude. The algorithm proposed uses the recursion formulae of the implicitly restarted Arnoldi (IRA) method described in [24], and the determination of the preconditioner does not require the evaluation of any matrix vector products with the matrix A in addition to those needed for the Arnoldi process. Another adaptive procedure to determine a preconditioner during GMRES iterations was introduced in [9] It is based on ....

D. C. Sorensen. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Analysis and Applications, 13:357--385, 1992.

....such as the sparsity of the operators, should be thus taken into account. Whether only the eigenvalues or eigenvectors are required becomes also an important issue. Therefore, the development of new eigensolvers, or the improvement of existing ones, has been the subject of continuous research [8,18,21,22,25]. Krylov subspace based methods such as Lanczos [19] and Arnoldi [21] algorithms have been widely used for treating eigenproblems associated with very large sparse matrices. They can be shown to perform better than vector iteration (inverse or direct) transformation methods (Jacobi, Householder ....

....or those of small real part, a preconditioning (shift invert, polynomials) is generally required [21] The maximum allowable j can also be reached without the convergence of all desired solutions. Then, restarting is an alternative way to proceed and different strategies exist for that purpose [21,22]. 2.4 Packages developed at CERFACS Two main packages which are described below have been developed at CERFACS for the solution of very large eigenproblems: BLZPACK is an implementation of the block Lanczos algorithm for the solution of real symmetric eigenproblems [13] It is intended for the ....

Sorensen, D.C. (1992), "Implicit Application of Polynomial Filters in a k-step Arnoldi Method", SIAM J. Matrix Anal. Appl., 13, pp. 357-385.

.... Lanczos process more robust, by introducing a so called Look ahead strategy [59, 27, 9, 32] ffl The idea of an implicit restart technique for the Arnoldi process, which helps to keep memoryrequirements reasonable, and which makes the Arnoldi process an attractive algorithm for eigencomputations [68, 38]. ffl Further improvements on the Davidson method, culminating in the JacobiDavidson algorithm [66] In this algorithm a major deficiency in the original Davidson method has been removed. The Jacobi Davidson method can be implemented as an accelerated inner outer iteration scheme. Of course, ....

....information for nearby eigenvalues. The main problem is that in this kind of restart we try to catch the information for an approximate subspace in one single vector, and apart from this, it is not easy to find the optimal mix. The restart problem has been solved very elegantly by Sorensen [68]. The idea behind his Implicitly Restarted Arnoldi (IRA) method is the following. Suppose we are at step k m, and we want to shrink the current subspace to dimension k again. Obviously we want to maintain the subspace with the best k Ritz vectors, that is, the Ritz vectors corresponding to k Ritz ....

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D. C. Sorensen. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Mat. Anal. Appl., 13(1):357--385, 1992.

....spaces and states some technical lemmata. In Sections 3.2 and 3.3 we derive the polynomial counterparts of the OR and MR residual vectors and express their zeros as Ritz and harmonic Ritz values of A, respectively. Finally, we describe the Implicitly Restarted Arnoldi process of Sorensen [32] for later use as a technique for manipulating Krylov spaces. 3.1 Why Krylov Subspaces One regard in which (38) is a reasonable choice for a correction space is that it enables the successive generation of the sequence fCm g using only matrix vector multiplication by A, an operation which is ....

....v 2 Km Gamma1 (A; v 1 ) generate the Arnoldi factorization associated with K p (A; v) with p as large as possible without performing 29 additional multiplications with A. The technique which accomplishes this task is known as the implicitly restarted Arnoldi (IRA) process and is due to Sorensen [32]. As a member of Km Gamma1 , v has the representation v = q k Gamma1 (A)v 1 with q k Gamma1 of exact degree k Gamma 1, 1 k m. In other words, v 2 K k nK k Gamma1 . We will show that p = m Gamma k is maximal and the resulting Arnoldi factorization has the form A V p = V p H p j p 1;p ....

Danny C. Sorensen. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl., 13(1):357--385, 1992. 52

....of a general large sparse matrix. This problem arises in several important applications including image and signal processing [38] control [6] and matrix pseudospectra [37] The computation of few extremal singular triplets of large sparse matrices has been the focus of many research e#orts, see [3,9,22,21,30,33,35] as well as Preprint submitted to Elsevier Science [2,10,29,36,15] and numerous references therein. Recent needs in applications such as the ones mentioned earlier, however, have motivated research oriented towards the development of algorithms for the computation of the smallest singular ....

....matrices. These improvements are described in the paper, whose structure is as follows. In Section 2 we review Lanczos bidiagonalization and describe its limitations when deployed to compute the smallest singular triplets. In Section 3 we show how to incorporate implicit restarts, introduced in [35], that permit Lanczos bidiagonalization to maintain limited storage and computational requirements per restart. In Section 4 we study the use of Ritz and harmonic Ritz values as implicit shifts. In Section 5 we show how to apply the orthogonal deflation transformation proposed in [34] in the ....

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D.C Sorensen. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl., 13:357--385, 1992. 27

....one may also be forced to restart, even before an eigenpair is found. Restarts have the disadvantage that a subspace, that may contain very useful information, is replaced by one single vector, so that much valuable information is lost. This problem has been solved elegantly for the Arnoldi method [22], and our approach (cf. x2.2 and x3.2) is related to this (see also [20, x5.3] For the Jacobi Davidson method this problem is solved by our new algorithms. In these algorithms the given large system is projected on a suitably filtered subspace, and this leads to a similar, but much smaller, ....

....through v j , and continue the JD algorithm with V = VU( 1 : j min ) 7) It is convenient that VU( 1 : j min ) is already orthogonal. We refer to this reduction strategy as implicit restart. Remark 3 Our restart strategy follows similar ideas as in the Implicitly Restarted Arnoldi (IRA) [22]. However, in [22] implicit shifts are used to delete the unwanted part, instead of explicitly selecting the wanted portion of the Krylov subspace as we do. The situation for IRA is more complicated because the reduced search subspace has to be a Krylov subspace. For further details, see [22] ....

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D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357--385.

....Prior to this discussion, however, the PIES are surveyed and classified according to a few well defined choices. This is not to say, however, that all worthwhile iterative eigensolvers utilize preconditioners and fall into the proposed classification. The implicitly restarted Arnoldi algorithm [101] is an example of a notable nonpreconditioned approach for eigenvalue computations. In this section, it is assumed that the problem to be solved is a standard, symmetric one, i.e. A = A and E = I. Concentrating on this commonly occurring and muchstudied problem allows for a more ....

D. C. Sorensen, "Implicit application of polynomial filters in a k-step Arnoldi method," SIAM J. Matrix Anal. Appl., vol. 13, pp. 357--385, 1992.

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D.C. Sorensen. Implicit application of polynomial filters in a k-step arnoldi method. SIAM J. Matrix. Anal. Appl., 13:357--385, 1992.

....classifications. Primary 65F15, Secondary 65G05 1. Introduction. The implicitly restarted Arnoldi method IRAM is an efficient procedure for approximating a selected subset of the eigenvalues and corresponding eigenvectors of a large sparse or structured n x n matrix A. Implicitly restarting [7] enables the Arnoldi process to compute this selected subset within a pre determined and relatively small amount of storage. This is the underlying algorithm in the large scale eigenvalue package ARPACK [3] The method may be viewed as a truncation of the standard implicitly shifted QR iteration. ....

....Pair. When working with real nonsymmetric matrices, it is desirable to compute in real arithmetic and this requires the ability to work with complex conjugate pairs of eigenvalues as a unit. This theme is standard for the double implicit shift both in implicit QR and in implicit restarting [7]. Suppose H(x iy) x iy) 0 iF) with xTx q yTy = 1 and I ler (x, y)11 = eigenvalues 0 q itt and we may express the relationship in real arithmetic as H(x,Y) x,Y) 0 P) tt 0 As shown in [1, 2] it may not be possible to lock this pair using a set of orthogonal vectors. Here, we ....

D.C. Sorensen. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Analysis and Applications, 13(1):357-385, January 1992.

.... 8000 6000 ,000 2000 2000 real axis FIG. 2. The full spectrum of a 2500 x 2500 A . one must construct a starting vector v I such that the subspace spanned by columns of Vk contains the desired eigencomponents. The construction of Vl is not trivial. The Implicitly Restatted Arnoldi Method (IRAM) [7] provides an efficient scheme to repeat edly modify an arbitrary starting vector Vl so that the unwanted eigencomponents Vl are annihilated by a polynomial in A. The analysis and some of the implementation issues of IRAM are also contained in [4] The basic theory is outlined below. 6 Given a ....

D.C. Sorensen. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM Journal on Matrix Analysis and Applications, 13(1):357-385, January 1992.

....selected which corresponds to the new starting vectors = A = 6) 7) In Section 2, a new and inexpensive technique, implicitly restarting the Lanczos algorithm, is developed for directly generating this modified projector, from r. Analogous to the implicitly restarted Arnoldi method of [31], this approach incorporates shifted HR steps [6, 7] into the nonsymmetric Lanczos method to produce . In Section 3, the relationship between the modification of the projector and the resulting reduced order model is explored. A strategy is developed for choosing the parameters, i, in (6,7) to ....

....method. These statements are offered at this point only to motivate the need for implicit restarts. A more complete discussion of these two observations is postponed until Section 4. The approach taken for implicitly restarting the Lanczos method is completely analogous to one developed in [31] for the Arnoldi method. In [31] QR steps (see [17] are combined with the Arnoldi method to yield an implicitly restarted approach. In this section, a process denoted as the HR step is incorporated into the nonsymmetric Lanczos method in order to yield Lanczos restarts. As opposed to the ....

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D.C. Sorensen, "Implicit application of polynomial filters in a K-step Arnoldi method," SIAM J. Matrix Anal. Appl., vol. 13, pp. 357-385, 1992.

.... The numerically stable generation of the Arnoldi factorization AV = VH fe where A, n x n matrix Hk, k x k projected matrix ( Upper Hessenberg ) Vk, n x k matrix, k n Set of Arnoldi vectors T f, residual vector, length n, V f = 0 coupled with an implicit restarting mechanism [1] is the basis of the ARPACK codes. The simple parMlelization scheme used for P ARPACK is as follows. Arnoldi Factorization A V V H fe H replicated on every processor Vk is distributed across a 1 D processor grid. Blocked by rows) f and workspace distributed accordingly The SPMD code ....

....among the processes. 1 go(11i( 11 ) o ) i ) 3) J)T W (j) gsum (4) H v )0 ) Table 1. The exphcit steps of the process responsible for the j block. Since H is replicated on each processor, the parallelization of the implicit restart mechanism described in [1, 2] remains untouched. The only difference is that the local block V(j) is in place of the full matrix V. All operations on the matrix H are replicated on each processor. Thus there is no communication overhead but there is a serial bottleneck here due to the redundant work. If k is small relative ....

D.C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM Journal on Matrix Analysis and Apphcations, 13(1):357-385, Jan- uary 1992.

....of largest real part for the block upper triangular matrix in (24) are the desired eigenvalues. When k is small, it is possible to compute the eigenvalues of H in advance of the computation of the partial Schur decomposition in (24) Within this framework, the implicitly restarted Arnoldi method [29] (implemented in ARPACK [31] can be used effectively to compute this partial Schur decomposition. If there is a reasonable gap between the eigenvalues of H and the imaginary axis, then IRA will be successful in computing the k eigenvalues 13 of largest real part using only matrix vector ....

D.C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Applic., 13:357-385 (1992).

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D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matr. Anal. Appl., 13 (1992), pp. 357--385.

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D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357 -- 385.

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D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357--385.

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D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357--385.

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D. C. Sorensen, "Implicit Application of Polynomial Filters in a k--step Arnoldi Method," SIAM J. Matrix Anal. Appl., 13, n. 1, pp. 357--385 (1992).

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D.C. SORENSEN. Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl., 13(1):357--385, 1992.

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D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357--385.

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D.C. Sorensen, "Implicit application of polynomial filters in a k-step Arnoldi method", SIAM J. Matrix Anal. Appl., 13(1), pp. 357--385, 1992.

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D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method,SIAM Journal on Matrix Analysis and Applications, 13 (1992), pp. 357--385.

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