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SIAM J. on Matrix Analysis and Applications

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2005

Volume 26, Issue 4, pp. 901-1193


Invariant Subspaces of Skew-Adjoint Matrices in Skew-Symmetric Inner Products

Leiba Rodman

SIAM. J. Matrix Anal. & Appl. 26, pp. 901-907 (7 pages)

Online Publication Date: July 31, 2006

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It is proved that, for a real matrix which is skew-adjoint with respect to a skew-symmetric inner product, every given neutral invariant subspace is contained in an invariant subspace which is also maximal semidefinite with respect to an associate symmetric bilinear form. Applications are given to the existence of solutions of continuous and discrete algebraic Riccati equations, with the property that the ranks of skew-symmetric parts of the solutions have a fixed upper bound. As a particular case, a known basic result concerning symmetric solutions of the Riccati equations is recovered.

Exclusion and Inclusion Intervals for the Real Eigenvalues of Positive Matrices

J. M. Peña

SIAM. J. Matrix Anal. & Appl. 26, pp. 908-917 (10 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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Given a real matrix, we analyze an openinterval, called a row exclusion interval, such that the real eigenvalues do not belong to it. We characterize when the row exclusion interval is nonempty. In addition to the exclusion interval, inclusion intervals for the real eigenvalues, alternative to those provided by the Gerschgorin disks, are also considered for matrices whose off-diagonal entries present a restricted dispersion. The results are applied to obtain a sharp upper bound for the real eigenvalues different from 1 of a positive stochastic matrix and a sufficient condition for the stability of a negative matrix, among other applications.

A Rank-Revealing Method with Updating, Downdating, and Applications

T. Y. Li and Zhonggang Zeng

SIAM. J. Matrix Anal. & Appl. 26, pp. 918-946 (29 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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A new rank-revealing method is proposed. For a given matrix and a threshold for near-zero singular values, by employing a globally convergent iterative scheme as well as a deflation technique the method calculates approximate singular values below the threshold one by one and returns the approximate rank of the matrix along with an orthonormal basis for the approximate null space. When a row or column is inserted or deleted, algorithms for updating/downdating the approximate rank and null space are straightforward, stable, and efficient. Numerical results exhibiting the advantages of our code over existing packages based on two-sided orthogonal rank-revealing decompositions are presented. Also presented are applications of the new algorithm in numerical computation of the polynomial GCD as well as identification of nonisolated zeros of polynomial systems.

Perturbation Bounds for Isotropic Invariant Subspaces of Skew-Hamiltonian Matrices

Daniel Kressner

SIAM. J. Matrix Anal. & Appl. 26, pp. 947-961 (15 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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We investigate the behavior of isotropic invariant subspaces of skew-Hamiltonian matrices under structured perturbations. It is shown that finding a nearby subspace is equivalent to solving a certain quadratic matrix equation. This connection is used to derive meaningful error bounds and condition numbers that can be used to judge the quality of invariant subspaces computed by strongly backward stable eigensolvers.

A Technique for Accelerating the Convergence of Restarted GMRES

A. H. Baker, E. R. Jessup, and T. Manteuffel

SIAM. J. Matrix Anal. & Appl. 26, pp. 962-984 (23 pages) | Cited 15 times

Online Publication Date: July 31, 2006

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We have observed that the residual vectors at the end of each restart cycle of restarted GMRES often alternate direction in a cyclic fashion, thereby slowing convergence. We present a new technique for accelerating the convergence of restarted GMRES by disrupting this alternating pattern. The new algorithm resembles a full conjugate gradient method with polynomial preconditioning, and its implementation requires minimal changes to the standard restarted GMRES algorithm.

Numerical Stability of the Parallel Jacobi Method

T. Londre and N. H. Rhee

SIAM. J. Matrix Anal. & Appl. 26, pp. 985-1000 (16 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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In this paper we study numerical stability of the parallel Jacobi method for computing the singular values and singular subspaces of an invertible upper triangular matrix that is obtained from QR decomposition with column pivoting. We show that in this case the parallel Jacobi method locates singular values and singular subspaces to full machine accuracy.

Breakdown-free GMRES for Singular Systems

Lothar Reichel and Qiang Ye

SIAM. J. Matrix Anal. & Appl. 26, pp. 1001-1021 (21 pages) | Cited 13 times

Online Publication Date: July 31, 2006

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GMRES is a popular iterative method for the solution of large linear systems of equations with a square nonsingular matrix. When the matrix is singular, GMRES may break down before an acceptable approximate solution has been determined. This paper discusses properties of GMRES solutions at breakdown and presents a modification of GMRES to overcome the breakdown.

A Bidiagonal Matrix Determines Its Hyperbolic SVD to Varied Relative Accuracy

Beresford N. Parlett

SIAM. J. Matrix Anal. & Appl. 26, pp. 1022-1057 (36 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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Let $T = L \Omega L^t$ be an invertible, unreduced, indefinite tridiagonal symmetric matrix with $\Omega$ a diagonal signature matrix. We provide error bounds on the (relative) change in an eigenvalue and the angular change in its eigenvector when the entries in $L$ suffer small relative changes. Our results extend those of Demmel and Kahan for $\Omega = I$. The relative condition number for an eigenvalue exceeds by 1 its absolutecondition number as an eigenvalue of $\Omega L^t L$. The condition number of an eigenvector is a weighted sum of the relative separations of the eigenvalue from each of the others.
A small example shows that very small eigenpairs can be robust even when the large eigenvalues are extremely sensitive. When $L$ is well conditioned for inversion, then all eigenvalues are robust and the eigenvectors depend only on the relative separations.

A Numerical Method for Computing an SVD-like Decomposition

Hongguo Xu

SIAM. J. Matrix Anal. & Appl. 26, pp. 1058-1082 (25 pages)

Online Publication Date: July 31, 2006

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We present a numerical method for computing the SVD-like decomposition B = QDS-1, where Q is orthogonal, S is symplectic, and D is a permuted diagonal matrix. The method can be applied directly to compute the canonical form of the Hamiltonian matrices of the form JBTB, where $J=[{0 \atop -I}{I \atop 0}]$. It can also be applied to solve the related application problems such as the gyroscopic systems and linear Hamiltonian systems. Error analysis and numerical examples show that the eigenvalues of JBTB computed by this method are more accurate than those computed by the methods working on the explicit product JBTB or BJBT.

Block-Toeplitz/Hankel Structured Total Least Squares

Ivan Markovsky, Sabine Van Huffel, and Rik Pintelon

SIAM. J. Matrix Anal. & Appl. 26, pp. 1083-1099 (17 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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A structured total least squares problem is considered in which the extended data matrix is partitioned into blocks and each of the blocks is block-Toeplitz/Hankel structured, unstructured, or exact. An equivalent optimization problem is derived and its properties are established. The special structure of the equivalent problem enables us to improve the computational efficiency of the numerical solution methods. By exploiting the structure, the computational complexity of the algorithms (local optimization methods) per iteration is linear in the sample size. Application of the method for system identification and for model reduction is illustrated by simulation examples.

The Inverse Eigenproblem of Centrosymmetric Matrices with a Submatrix Constraint and Its Approximation

Zheng-Jian Bai

SIAM. J. Matrix Anal. & Appl. 26, pp. 1100-1114 (15 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenproblem defined as follows: given a set of complex $n$-vectors $\{\bx_i\}_{i=1}^{m}$ and a set of complex numbers $\{\la_i\}_{i=1}^m$, and an $s$-by-$s$ real matrix $C_0$, find an $n$-by-$n$ real centrosymmetric matrix $C$ such that the $s$-by-$s$ leading principal submatrix of $C$ is $C_0$, and $\{\bx_i\}_{i=1}^{m}$ and $\{\la_i\}_{i=1}^m$ are the eigenvectors and eigenvalues of $C$, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real $n$-by-$n$ matrix $\tilde{C}$, find a matrix $C$ which is the solution to the constrained inverse problem such that the distance between $C$ and $\tilde{C}$ is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. Some illustrative experiments are also presented.

Necessary and Sufficient Conditions for the Existence of Positive Definite Solutions to the Symmetric Recursive Inverse Eigenvalue Problem

Zhenyue Zhang, Jing Wang, and Min Fang

SIAM. J. Matrix Anal. & Appl. 26, pp. 1115-1131 (17 pages)

Online Publication Date: July 31, 2006

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Necessary and sufficient conditions are completely characterized for the existence of a positive definite or positive semidefinite solution to the symmetric recursive inverse eigenvalue problem (SRIEP). When a prior (indefinite) solution A to the SRIEP is known, positive definite/semidefinite solutions are formulated in terms of A and basis matrices of the column space of the given recursive matrix R and the null space of RT. Taking into account some computational concerns, an algorithm is proposed that can check whether the SRIEP has a positive definite/semidefinite solution and find such a solution if it exists. Several numerical experiments are given to illustrate the performance of the algorithm.

A Method for Generating Infinite Positive Self-adjoint Test Matrices and Riesz Bases

C. V. M. van der Mee and S. Seatzu

SIAM. J. Matrix Anal. & Appl. 26, pp. 1132-1149 (18 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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In this article we propose a method to easily generate infinite multi-index positive definite self-adjoint matrices as well as Riesz bases in suitable subspaces of $L^2({\mathbb{R}}^d)$. The method is then applied to obtain some classes of multi-index Toeplitz matrices which are bounded and strictly positive on $\ell^2({\mathbb{Z}}^d)$. The condition number of some of these matrices is also computed.

An Iterative Method for Solving Complex-Symmetric Systems Arising in Electrical Power Modeling

Victoria E. Howle and Stephen A. Vavasis

SIAM. J. Matrix Anal. & Appl. 26, pp. 1150-1178 (29 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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We propose an iterative method for solving a complex-symmetric linear system arising in electric power networks. Our method extends Gremban, Miller, and Zagha's [in Proceedings of the International Parallel Processing Symposium, IEEE Computer Society, Los Alamitos, CA, 1995] support-tree preconditioner to handle complex weights and vastly different admittances. Our underlying iteration is a modification to transpose-free QMR to enhance accuracy. Computational results are described.

The Scaling and Squaring Method for the Matrix Exponential Revisited

Nicholas J. Higham

SIAM. J. Matrix Anal. & Appl. 26, pp. 1179-1193 (15 pages) | Cited 35 times

Online Publication Date: July 31, 2006

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The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in MATLAB's {\tt expm} function. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Padé approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error analysis of the method (in exact arithmetic) that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency. We also give new rounding error analysis that shows the computed Padé approximant of the scaled matrix to be highly accurate. For IEEE double precision arithmetic the best choice of degree of Padé approximant turns out to be 13, rather than the 6 or 8 used by previous authors. Our implementation of the scaling and squaring method always requires at least two fewer matrix multiplications than {\tt expm} when the matrix norm exceeds 1, which can amount to a 37% saving in the number of multiplications, and it is typically more accurate, owing to the fewer required squarings. We also investigate a different scaling and squaring algorithm proposed by Najfeld and Havel that employs a Padé approximation to the function $x \coth(x)$. This method is found to be essentially a variation of the standard one with weaker supporting error analysis.
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