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SIAM J. on Matrix Analysis and Applications

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2000

Volume 21, Issue 4, pp. 1051-1393


Look-Ahead Procedures for Lanczos-Type Product Methods Based on Three-Term Lanczos Recurrences

Martin H. Gutknecht and Klaus J. Ressel

SIAM. J. Matrix Anal. & Appl. 21, pp. 1051-1078 (28 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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Lanczos-type product methods for the solution of large sparse non-Hermitian linear systems either square the Lanczos process or combine it with a local minimization of the residual. They inherit from the underlying Lanczos process the danger of breakdown. For various Lanczos-type product methods that are based on the Lanczos three-term recurrence, look-ahead versions are presented which avoid such breakdowns or near-breakdowns at the cost of a small computational overhead. Different look-ahead strategies are discussed and their efficiency is demonstrated by several numerical examples.

Residual-Minimizing Krylov Subspace Methods for Stabilized Discretizations of Convection-Diffusion Equations

Oliver G. Ernst

SIAM. J. Matrix Anal. & Appl. 21, pp. 1079-1101 (23 pages) | Cited 8 times

Online Publication Date: July 31, 2006

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We discuss the behavior of the minimal residual method applied to stabilized discretizations of one- and two-dimensional model problems for the stationary convection-diffusion equation. In the one-dimensional case, it is shown that eigenvalue information for estimating the convergence rate of the minimal residual method is highly misleading due to the strong nonnormality of these operators for large grid Péclet numbers. It is also shown that the field of values is a more reliable tool for assessing the convergence rate. In the two-dimensional model problems considered, we observe two distinct phases in the convergence of the iterative method: the first determined by the field of values and the second by the spectrum. We conjecture that the first phase lasts as long as the longest streamline takes to traverse the grid with the flow.

Convergence of Eigenvalues in State-Discretization of Linear Stochastic Systems

José A. De Doná, Graham C. Goodwin, Richard H. Middleton, and Iain Raeburn

SIAM. J. Matrix Anal. & Appl. 21, pp. 1102-1111 (10 pages)

Online Publication Date: July 31, 2006

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The transition operator that describes the time evolution of the state probability distribution for continuous-state linear systems is given by an integral operator. A state-discretization approach is proposed, which consists of a finite rank approximation of this integral operator. As a result of the state-discretization procedure, a Markov chain is obtained, in which case the transition operator is represented by a transition matrix. Spectral properties of the integral operator for the continuous-state case are presented. The relationships between the integral operator and the finite rank approximation are explored. In particular, the limiting properties of the eigenvalues of the transition matrices of the resulting Markov chains are studied in connection to the eigenvalues of the original continuous-state integral operator.

Implicitly Restarted GMRES and Arnoldi Methods for Nonsymmetric Systems of Equations

Ronald B. Morgan

SIAM. J. Matrix Anal. & Appl. 21, pp. 1112-1135 (24 pages) | Cited 18 times

Online Publication Date: July 31, 2006

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The generalized minimum residual method (GMRES) is well known for solving large nonsymmetric systems of linear equations. It generally uses restarting, which slows the convergence. However, some information can be retained at the time of the restart and used in the next cycle. We present algorithms that use implicit restarting in order to retain this information. Approximate eigenvectors determined from the previous subspace are included in the new subspace. This deflates the smallest eigenvalues and thus improves the convergence. The subspace that contains the approximate eigenvectors is itself a Krylov subspace, but not with the usual starting vector. The implicitly restarted FOM algorithm includes standard Ritz vectors in the subspace. The eigenvalue portion of its calculations is equivalent to Sorensen's IRA algorithm. The implicitly restarted GMRES algorithm uses harmonic Ritz vectors. This algorithm also gives a new approach to computing interior eigenvalues.

Structure of Expansion-Contraction Matrices in the Inclusion Principle for Dynamic Systems

Lubomír Bakule, José Rodellar, and Josep M. Rossell

SIAM. J. Matrix Anal. & Appl. 21, pp. 1136-1155 (20 pages) | Cited 15 times

Online Publication Date: July 31, 2006

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This paper characterizes a general structure of complementary matrices involved in the input-state-output inclusion principle for linear time invariant dynamic systems. Aggregations and restrictions are adopted as two practically important classes within this scheme. Further, contractibility conditions for feedback controllers are presented, including the decentralized control design. The identified structure enables the formulation of conditions on the complementary matrices which can be useful for analysis and synthesis of control problems.

On Shary's Algebraic Approach for Linear Interval Equations

Arnold Neumaier

SIAM. J. Matrix Anal. & Appl. 21, pp. 1156-1162 (7 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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A recent method by Shary for enclosing the solution set of a system of linear interval equations is derived in a new way. It is shown that the method converges to the fixed-point inverse and that it has finite termination with probability 1.

A Continuation Method for a Right Definite Two-Parameter Eigenvalue Problem

Bor Plestenjak

SIAM. J. Matrix Anal. & Appl. 21, pp. 1163-1184 (22 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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The continuation method has been successfully applied to the classical $Ax=\lambda x$ and to the generalized $Ax=\lambda Bx$ eigenvalue problems. Shimasaki applied the continuation method to the right definite two-parameter problem, which resulted in a discretization of a two-parameter Sturm--Liouville problem. We show that the continuation method can be used for a general right definite two-parameter problem and wegive a sketch of the algorithm. For a local convergent method we use the tensor Rayleigh quotient iteration (TRQI), which is a generalization of the Rayleigh iterative method to two-parameter problems. We show its convergence and compare it with Newton's method and with the generalized Rayleigh quotient iteration (GRQI), studied by Ji, Jiang, and Lee.

A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra

Nicholas J. Higham and Françoise Tisseur

SIAM. J. Matrix Anal. & Appl. 21, pp. 1185-1201 (17 pages) | Cited 13 times

Online Publication Date: July 31, 2006

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The matrix 1-norm estimation algorithm used in LAPACK and various other software libraries and packages has proved to be a valuable tool. However, it has the limitations that it offers the user no control over the accuracy and reliability of the estimate and that it is based on level 2 BLAS operations. A block generalization of the 1-norm power method underlying the estimator is derived here and developed into a practical algorithm applicable to both real and complex matrices. The algorithm works with n × t matrices, where t is a parameter. For t=1 the original algorithm is recovered, but with two improvements (one for real matrices and one for complex matrices). The accuracy and reliability of the estimates generally increase with t and the computational kernels are level 3 BLAS operations for t > 1. The last t-1 columns of the starting matrix are randomly chosen, giving the algorithm a statistical flavor. As a by-product of our investigations we identify a matrix for which the 1-norm power method takes the maximum number of iterations. As an application of the new estimator we show how it can be used to efficiently approximate 1-norm pseudospectra.

An Efficient and Stable Algorithm for the Symmetric-Definite Generalized Eigenvalue Problem

S. Chandrasekaran

SIAM. J. Matrix Anal. & Appl. 21, pp. 1202-1228 (27 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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A new, efficient, and stable algorithm for computing all the eigenvalues and eigenvectors of the problem $Ax = \lambda Bx$, where A is symmetric indefinite and B is symmetric positive definite, is proposed.

On the Solution Sets of Linear Complementarity Problems

G. S. R. Murthy, T. Parthasarathy, and B. Sriparna

SIAM. J. Matrix Anal. & Appl. 21, pp. 1229-1235 (7 pages)

Online Publication Date: July 31, 2006

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In this article we consider two problems related to the solution sets of linear complementarity problems (LCPs)---one on the connectedness and the other on the convexity. In Jones and Gowda [ Linear Algebra Appl., 246 (1996), pp. 299--312], it was shown that the solution sets of LCPs arising out of $P_0 \cap Q$-matrices are connected, and they conjectured that this is true even in the case of $P_0\ \cap\ Q_0$-matrices. We verify this, at least in the case of nonnegative matrices. Our second problem is related to the class of fully copositive $(C_0^f)$-matrices introduced in Murthy and Parthasarathy [ Math. Programming, 82 (1998), pp. 401--411]. The class $C_0^f\cap Q_0,$ which contains the class of positive semidefinite matrices, has several properties that positive semidefinite matrices have. This article further supplements this by showing that the solution sets arising from LCPs with $C_0^f\cap Q_0$-matrices and their transposes are convex. This means that $C_0^f\cap Q_0$-matrices are sufficient matrices, another well known class in the theory of linear complementarity problem introduced by Cottle, Pang, and Venkateswaran [ Linear Algebra Appl., 114/115 (1989), pp. 231--249].

Sparse Approximate Inverse Smoother for Multigrid

Wei-Pai Tang and Wing Lok Wan

SIAM. J. Matrix Anal. & Appl. 21, pp. 1236-1252 (17 pages) | Cited 15 times

Online Publication Date: July 31, 2006

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Various forms of sparse approximate inverses (SAI) have been shown to be useful for preconditioning. Their potential usefulness in a parallel environment has motivated much interest in recent years. However, the capability of an approximate inverse in eliminating the local error has not yet been fully exploited in multigrid algorithms. A careful examination of the iteration matrices of these approximate inverses indicates their superiority in smoothing the high-frequency error in addition to their inherent parallelism. We propose a new class of SAI smoothers in this paper and present their analytic smoothing factors for constant coefficient PDEs. The following are several distinctive features that make this technique special: By adjusting the quality of the approximate inverse, the smoothing factor can be improved accordingly. For hard problems, this is useful. In contrast to the ordering sensitivity of other smoothing techniques, this technique is ordering independent. In general, the sequential performance of many superior parallel algorithms is not very competitive. This technique is useful in both parallel and sequential computations. Our theoretical and numerical results have demonstrated the effectiveness of this new technique.

A Multilinear Singular Value Decomposition

Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle

SIAM. J. Matrix Anal. & Appl. 21, pp. 1253-1278 (26 pages) | Cited 216 times

Online Publication Date: July 31, 2006

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We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors.

An Augmented Conjugate Gradient Method for Solving Consecutive Symmetric Positive Definite Linear Systems

Jocelyne Erhel and Frédéric Guyomarc'h

SIAM. J. Matrix Anal. & Appl. 21, pp. 1279-1299 (21 pages) | Cited 14 times

Online Publication Date: July 31, 2006

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Many scientific applications require one to solve successively linear systems Ax=b with different right-hand sides $b$ and a symmetric positive definite matrix A. The conjugate gradient method applied to the first system generates a Krylov subspace which can be efficiently recycled thanks to orthogonal projections in subsequent systems. A modified conjugate gradient method is then applied with a specific initial guess and initial descent direction and a modified descent direction during the iterations. This paper gives new theoretical results for this method and proposes a new version. Numerical experiments show the efficacy of our method even for quite different right-hand sides.

Constraint Preconditioning for Indefinite Linear Systems

Carsten Keller, Nicholas I. M. Gould, and Andrew J. Wathen

SIAM. J. Matrix Anal. & Appl. 21, pp. 1300-1317 (18 pages) | Cited 9 times

Online Publication Date: July 31, 2006

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The problem of finding good preconditioners for the numerical solution of indefinite linear systems is considered. Special emphasis is put on preconditioners that have a 2 × 2 block structure and that incorporate the (1,2) and (2,1) blocks of the original matrix. Results concerning the spectrum and form of the eigenvectors of the preconditioned matrix and its minimum polynomial are given. The consequences of these results are considered for a variety of Krylov subspace methods. Numerical experiments validate these conclusions.

Some Properties of Symmetric Quasi-Definite Matrices

Alan George, KH. Ikramov, and A. B. Kucherov

SIAM. J. Matrix Anal. & Appl. 21, pp. 1318-1323 (6 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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Symmetric quasi-definite matrices arise in numerous applications, notably in interior point methods in mathematical programming. Several authors have derived various properties of these matrices. This article provides a list of some previously known properties and adds a number of others that are believed to be new.

On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors

Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle

SIAM. J. Matrix Anal. & Appl. 21, pp. 1324-1342 (19 pages) | Cited 128 times

Online Publication Date: July 31, 2006

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In this paper we discuss a multilinear generalization of the best rank-R approximation problem for matrices, namely, the approximation of a given higher-order tensor, in an optimal least-squares sense, by a tensor that has prespecified column rank value, row rank value, etc. For matrices, the solution is conceptually obtained by truncation of the singular value decomposition (SVD); however, this approach does not have a straightforward multilinear counterpart. We discuss higher-order generalizations of the power method and the orthogonal iteration method.

Nonlinear Matrix Iterative Processes and Generalized Coefficients of Ergodicity

Marc Artzrouni and Olivier Gavart

SIAM. J. Matrix Anal. & Appl. 21, pp. 1343-1353 (11 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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A nonlinear matrix iterative process is a linear dynamical system for which there is a nonlinear feedback of the current vector on the entries of the matrix. Stability conditions for an asymptotically exponential solution are studied for such a process in the positive quadrant of $\mathbb{R}^{n}.$ The results hinge on a generalization of the coefficient of ergodicity of a nonnegative matrix.

Locally X Matrices, Spectral Distributions, Preconditioning, and Applications

Stefano Serra Capizzano

SIAM. J. Matrix Anal. & Appl. 21, pp. 1354-1388 (35 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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Starting from a set X= {Xn}n of Hermitian positive definite $n\times n$ matrices, we constructively define a class ${\cal LL}(X)$ of "locally X" matrices which can be viewed as the range of a special sequence of linear normally positiveoperators. Regarding the spectra of these matrices and of the related preconditioned matrices, we prove some Szego-style ergodic formulas. These results allow one to define a very general procedure for devising optimal and superlinear preconditioners. As special cases, we can deal with matrices coming from the discretization of elliptic and semi-elliptic differential equations defined on multidimensional domains as well as matrices coming from optimization problems connected with graph theory.

A Counterexample of the Convexity of Sums of Singular Values

Liu Hongwei and Liu Sanyang

SIAM. J. Matrix Anal. & Appl. 21, pp. 1389-1391 (3 pages)

Online Publication Date: July 31, 2006

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In this note we present a counterexample to Seeger's conjecture on the convexity of sums of singular values of rectangular matrices. Thus we answer the related open question in the negative.

A Note on P-Regular Splitting of Hermitian Matrix

Zhi-Hao Cao

SIAM. J. Matrix Anal. & Appl. 21, pp. 1392-1393 (2 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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In this note we examine the spectral property of the iteration matrix induced by the P-regular splitting of an invertible Hermitian matrix and point out that a theorem given by Weissler (cf. Theorem A2 in [ SIAM J. Matrix Anal. Appl., 16 (1995), pp. 448--461]) is incorrect.
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