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SIAM J. on Matrix Analysis and Applications

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Issue 4 | 1999 | pp. 839-1098 †
Issue 3 | 1999 | pp. 563-837
Issue 2 | 1998 | pp. 279-561
Issue 1 | 1998 | pp. 1-278

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1999

Volume 20, Issue 3, pp. 563-837


On Some Eigenvector-Eigenvalue Relations

S. Elhay, G. M. L. Gladwell, G. H. Golub, and Y. M. Ram

SIAM. J. Matrix Anal. & Appl. 20, pp. 563-574 (12 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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This paper generalizes the well-known identity which relates the last components of the eigenvectors of a symmetric matrix A to the eigenvalues of A and of the matrix An-1, obtained by deleting the last row and column of A. The generalizations relate to matrices and to Sturm--Liouville equations.

Nonlinear Eigenproblems

Philippe Guillaume

SIAM. J. Matrix Anal. & Appl. 20, pp. 575-595 (21 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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Let $A(\lambda )$ be a holomorphic matrix-valued function defined on a domain $\Omega \subset {\Bbb C}.$ The nonlinear eigenproblem of finding generalized eigenpairs $\lambda ,\ v$ such that $A(\lambda )v=0$ is considered. The method which is exposed for solving this problem is based on the derivatives of the function $x(\lambda )=A(\lambda )^{-1}b$, where $b$ is a given vector. Theoretical convergence results are established, and an algorithm is proposed for computing a few eigenvalues close to a given complex number together with some corresponding eigenvectors.

Absolute Schur Algebras and Unbounded Matrices

Pachara Chaisuriya and Sing-Cheong Ong

SIAM. J. Matrix Anal. & Appl. 20, pp. 596-605 (10 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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Let p, q, r be real numbers such that $p, q, r \ge 1$, and let $\B$ be a Banach algebra. Let $\Bl$ denote the set of all matrices which define bounded linear transformations from $\ell^p$ into $\ell^q$. The set $$ \SB = \Big\{A = \left[a_{jk}\right]: a_{jk} \in \B \ \mbox{ and} \ A^{[r]} = \left[\N{a_{jk}}^r\right] \in \mathcal{B}(\ell^p, \ell^q)\Big\} $$ of infinite matrices over $\B,$ is shown to be a Banach algebra under the Schur product operation, and the norm $\rn{A} = \|A^{[r]}\|^{1/r}$. For $r \ge 2$ and $\B = \mathbb{C}$, the complex field, $\mathcal{S}^p = \mathcal{S}^p(\mathbb{C})$ contains the set $\Bl$. For $r= 2$, $\mathcal{S}^2$ contains the bounded matrices $\mathcal{B}(\ell^p, \ell^q)$ as an ideal.

Modifying a Sparse Cholesky Factorization

Timothy A. Davis and William W. Hager

SIAM. J. Matrix Anal. & Appl. 20, pp. 606-627 (22 pages) | Cited 18 times

Online Publication Date: July 31, 2006

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Given a sparse symmetric positive definite matrix ${\bf AA}^{\sf T}$ and an associated sparse Cholesky factorization ${\bf LDL}^{\sf T}$ or ${\bf LL}^{\sf T}$, we develop sparse techniques for obtaining the new factorization associated with either adding a column to ${\bf A}$ or deleting a column from ${\bf A}$. Our techniques are based on an analysis and manipulation of the underlying graph structure and on ideas of Gill et al.\ [ Math. Comp., 28 (1974), pp. 505--535] for modifying a dense Cholesky factorization. We show that our methods extend to the general case where an arbitrary sparse symmetric positive definite matrix is modified. Our methods are optimal in the sense that they take time proportional to the number of nonzero entries in ${\bf L}$ and ${\bf D}$ that change.

A Nonstandard Cyclic Reduction Method, Its Variants and Stability

Tuomo Rossi and Jari Toivanen

SIAM. J. Matrix Anal. & Appl. 20, pp. 628-645 (18 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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A nonstandard cyclic reduction method is introduced for solving the Poisson equation in rectangular domains. Different ways of solving the arising reduced systems are considered. The partial solution approach leads to the so-called partial solution variant of the cyclic reduction (PSCR) method, while the other variants are obtained by using the matrix rational polynomial factorization technique, including the partial fraction expansions, the fast Fourier transform (FFT) approach, and the combination of Fourier analysis and cyclic reduction (FACR) techniques. Such techniques have originally been considered in the standard cyclic reduction framework. The equivalence of the partial solution and the partial fraction techniques is shown. The computational cost of the considered variants is ${\mathcal O} (N\log N)$ operations, except for the FACR techniques for which it is ${\mathcal O} (N\log\log N)$. The stability estimate for the considered method is constructed, and the stability is demonstrated by numerical experiments.

Logarithmic Norms for Matrix Pencils

Inmaculada Higueras and Berta Garcia-Celayeta

SIAM. J. Matrix Anal. & Appl. 20, pp. 646-666 (21 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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We extend the usual concepts of least upper bound norm and logarithmic norm of a matrix to matrix pencils. Properties of these seminorms and logarithmic norms are derived. This logarithmic norm can be used to study the growth of the solutions of linear variable coefficient differential algebraic systems.

A Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils. Part II: A Stratification-Enhanced Staircase Algorithm

Alan Edelman, Erik Elmroth, and Bo Kågström

SIAM. J. Matrix Anal. & Appl. 20, pp. 667-699 (33 pages) | Cited 13 times

Online Publication Date: July 31, 2006

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Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a well-known ill-posed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm. Here we discuss and complete the mathematical theory of these relationships and show how they may be applied to the staircase algorithm. This paper is a continuation of our Part I paper on versal deformations, but it may also be read independently.

Effective Methods for Solving Banded Toeplitz Systems

Dario Andrea Bini and Beatrice Meini

SIAM. J. Matrix Anal. & Appl. 20, pp. 700-719 (20 pages) | Cited 13 times

Online Publication Date: July 31, 2006

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We propose new algorithms for solving n x n banded Toeplitz systems with bandwidth m. If the function associated with the Toeplitz matrix has no zero in the unit circle, then $O(n\log m + m\log ^2 m\log\log \epsilon^{-1})$ arithmetic operations (ops) are sufficient to approximate the solution of the system up to within the error $\epsilon$; otherwise the cost becomes $O(n\log m +m\log^2 m\log {n\over m})$ ops. Here $m=o(n)$ and $n>\log \epsilon^{-1}$. Some applications are presented. The methods can be applied to infinite and bi-infinite systems and to block matrices.

A Supernodal Approach to Sparse Partial Pivoting

James W. Demmel, Stanley C. Eisenstat, John R. Gilbert, Xiaoye S. Li, and Joseph W. H. Liu

SIAM. J. Matrix Anal. & Appl. 20, pp. 720-755 (36 pages) | Cited 138 times

Online Publication Date: July 31, 2006

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We investigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. We introduce the notion of unsymmetric supernodes to perform most of the numerical computation in dense matrix kernels. We introduce unsymmetric supernode-panel updates and two-dimensional data partitioning to better exploit the memory hierarchy. We use Gilbert and Peierls's depth-first search with Eisenstat and Liu's symmetric structural reductions to speed up symbolic factorization.
We have developed a sparse LU code using all these ideas. We present experiments demonstrating that it is significantly faster than earlier partial pivoting codes. We also compare its performance with UMFPACK, which uses a multifrontal approach; our code is very competitive in time and storage requirements, especially for large problems.

An Algorithm for Checking Regularity of Interval Matrices

Christian Jansson and Jiri Rohn

SIAM. J. Matrix Anal. & Appl. 20, pp. 756-776 (21 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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Checking regularity (or singularity) of interval matrices is a known NP-hard problem. In this paper a general algorithm for checking regularity/singularity is presented which is not a priori exponential. The algorithm is based on a theoretical result according to which regularity may be judged from any single component of the solution set of an associated system of linear interval equations. Numerical experiments (with interval matrices up to the size n = 50) confirm that this approach brings an essential decrease in the amount of operations needed.

Cauchy-like Preconditioners for Two-Dimensional Ill-Posed Problems

Misha E. Kilmer

SIAM. J. Matrix Anal. & Appl. 20, pp. 777-799 (23 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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Ill-conditioned matrices with block Toeplitz, Toeplitz block (BTTB) structure arise from the discretization of certain ill-posed problems in signal and image processing. We use a preconditioned conjugate gradient algorithm to compute a regularized solution to this linear system given noisy data. Our preconditioner is a Cauchy-like block diagonal approximation to a unitary transformation of the BTTB matrix. We show that the preconditioner has desirable properties when the kernel of the ill-posed problem is smooth: the largest singular values of the preconditioned matrix are clustered around one, the smallest singular values remain small, and the subspaces corresponding to the largest and smallest singular values, respectively, remain unmixed. For a system involving np variables, the preconditioned algorithm costs only O(np (lgn + lg p)) operations per iteration. We demonstrate the effectiveness of the preconditioner on three examples.

On Smooth Decompositions of Matrices

Luca Dieci and Timo Eirola

SIAM. J. Matrix Anal. & Appl. 20, pp. 800-819 (20 pages) | Cited 22 times

Online Publication Date: July 31, 2006

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In this paper we consider smooth orthonormal decompositions of smooth time varying matrices. Among others, we consider QR-, Schur-, and singular value decompositions, and their block-analogues. Sufficient conditions for existence of such decompositions are given and differential equations for the factors are derived. Also generic smoothness of these factors is discussed.

Decay Rates of the Inverse of Nonsymmetric Tridiagonal and Band Matrices

Reinhard Nabben

SIAM. J. Matrix Anal. & Appl. 20, pp. 820-837 (18 pages) | Cited 13 times

Online Publication Date: July 31, 2006

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It is well known that the inverse C = [ci,j] of an irreducible nonsingular symmetric tridiagonal matrix is given by two sequences of real numbers, {ui} and {vi}, such that ci,j = u i vj for $i \leq j$. A similar result holds for nonsymmetric matrices A. There the inverse can be described by four sequences {ui},{vi}, {xi},$ and {vi} with u ivi = xiyi. Here we characterize certain properties of A, i.e., being an M-matrix or positive definite, in terms of the ui, vi,xi, and yi. We also establish a relation of zero row sums and zero column sums of A and pairwise constant ui,vi, xi, and yi. Moreover, we consider decay rates for the entries of the inverse of tridiagonal and block tridiagonal (banded) matrices. For diagonally dominant matrices we show that the entries of the inverse strictly decay along a row or column. We give a sharp decay result for tridiagonal irreducible M-matrices and tridiagonal positive definite matrices. We also give a decay rate for arbitrary banded M-matrices.
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