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A NEW ITERATION FOR COMPUTING THE EIGENVALUES OF SEMISEPARABLE (PLUS DIAGONAL) MATRICES

ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2008年 33卷 126-150页

作者： Vandebril, Raf Van Barel, Marc Mastronardi, Nicola Katholieke Univ Leuven Dept Comp Wetenschappen Louvain Belgium Ist Applicaz Calcolo M Picone Sez Bari Bari Italy

This paper proposes a new type of iteration for computing eigenvalues of semiseparable (plus diagonal) matrices based on a structured-rank factorization. Remarks on higher order semiseparability ranks are also made. More precisely, instead of the traditional qr iteration, a QH iteration is used. The QH factorization is characterized by a unitary matrix Q and a Hessenberg-like matrix H in which the lower triangular part is semiseparable (often called a lower semiseparable matrix). The Q factor of this factorization determines the similarity transformation of the QH method. It is shown that this iteration is extremely useful for computing the eigenvalues of structured-rank matrices. Whereas the traditional qr method applied to semiseparable (plus diagonal) and Hessenberg-like matrices uses similarity transformations involving 2p(n-1) Givens transformations (where p denotes the semiseparability rank), the QH iteration only needs p(n-1) Givens transformations, which is comparable to the generalized Hessenberg (symmetric band) situation having p subdiagonals. It is also shown that this method can in some sense be interpreted as an extension of the traditional qr method for Hessenberg matrices, i.e., the traditional case also fits into this framework. It is also shown that this iteration exhibits an extra type of convergence behavior compared to the traditional qr method. The algorithm is implemented in an implicit way, based on the Givens-weight representation of the structured rank matrices. Numerical experiments show the viability of this approach. The new approach yields better complexity and more accurate results than the traditional qr method.

关键词： QH algorithm structured rank matrices implicit computations eigenvalue qr algorithm rational qr iteration

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An atlas for tridiagonal isospectral manifolds

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LINEAR ALGEBRA AND ITS APPLICATIONS 2008年第1期429卷 387-402页

作者： Leite, Ricardo S. Saldanha, Nicolau C. Tomei, Carlos PUC Dept Matemat BR-22453900 Rio De Janeiro Brazil Univ Fed Espirito Santo Dept Matemat BR-29075910 Vitoria ES Brazil

Let T-Lambda be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix Lambda with simple spectrum. We introduce bidiagonal coordinates, charts defined on open dense domains forming an explicit atlas for T-Lambda. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in T-Lambda now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time. (C) 2008 Elsevier Inc. All rights reserved.

关键词： Jacobi matrices tridiagonal matrices norming constants Toda flows qr algorithm

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Zero-preserving iso-spectral flows based on parallel sums

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LINEAR ALGEBRA AND ITS APPLICATIONS 2007年第1期421卷 69-84页

作者： Driessel, Kenneth R. Gerisch, Alf Colorado State Univ Dept Math Ft Collins CO 80523 USA Univ Halle Wittenberg Fachbereich Math & Informat D-06099 Halle Saale Germany

Driessel [K.R. Driessel, Computing canonical forms using flows, Linear Algebra Appl. 379 (2004) 353379] introduced the notion of quasi-projection onto the range of a linear transformation from one inner product space into another inner product space. Here we introduce the notion of quasi-projection onto the intersection of the ranges of two linear transformations from two inner product spaces into a third inner product space. As an application, we design a new family of iso-spectral flows on the space of symmetric matrices that preserves zero patterns. We discuss the equilibrium points of these flows. We conjecture that these flows generically converge to diagonal matrices. We perform some numerical experiments with these flows which support this conjecture. We also compare our zero-preserving flows with the Toda flow. (c) 2006 Elsevier Inc. All rights reserved.

关键词： iso-spectral flow group action orbit eigenvalues sparse matrix dynamical system ordinary differential equation vector field Toda flow double bracket flow qr algorithm differential geometry projection quasi-projection parallel sum harmonic mean

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Comparison of BR and qr eigenvalue algorithms system small signal stability analysis

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IEEE TRANSACTIONS ON POWER SYSTEMS 2006年第4期21卷 1848-1855页

作者： Ma, Jian Dong, Zhao Yang Zhang, Pei Univ Queensland Sch Informat Technol & Elect Engn Brisbane Qld 4072 Australia EPRI Palo Alto CA 94304 USA

The BR algorithm is a novel and efficient method to find all eigenvalues of upper Hessenberg matrices and has never been applied to eigenanalysis for power system small signal stability. This paper analyzes differences between the BR and the qr algorithms with performance comparison in terms of CPU time based on stopping criteria and storage requirement. The BR algorithm utilizes accelerating strategies to improve its performance when computing eigenvalues of narrowly banded, nearly tridiagonal upper Hessenberg matrices. These strategies significantly reduce the computation time at a reasonable level of precision. Compared with the qr algorithm, the BR algorithm requires fewer iteration steps and less storage space without depriving of appropriate precision in solving eigenvalue problems of large-scale power systems. Numerical examples demonstrate the efficiency of the BR algorithm in pursuing eigenanalysis tasks of 39-, 68-, 115-, 300-, and 600-bus systems. Experiment results suggest that the BR algorithm is a more efficient algorithm for large-scale power system small signal stability eigenanalysis.

关键词： BR algorithm CPU time eigenvalue calculation power systems qr algorithm small signal stability stopping criteria upper Hessenberg matrix

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Reachable matrices by a qr step with shift

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2006年第1期5卷 91-107页

作者： Chu, DL Chu, M Natl Univ Singapore Dept Math Singapore 117543 Singapore N Carolina State Univ Dept Math Raleigh NC 27695 USA

One of the most interesting dynamical systems used in numerical analysis is the qr algorithm. An added maneuver to improve the convergence behavior is the qr iteration with shift which is of fundamental importance in eigenvalue computation. This paper is a theoretical study of the set of all isospectral matrices "reachable" by the dynamics of the qr algorithm with shift. A matrix B is said to be reachable by A if B = RQ + mu I, where A - mu I = qr is the qr decomposition for some mu is an element of R. It is proved that in general the qr algorithm with shift is neither reflexive nor symmetric. Examples are given to demonstrate that this relation is neither transitive nor antisymmetric. It is further discovered that the reachable set from a given n x n matrix A forms 2(n-1) disjoint open loops if n is even and 2(n-2) disjoint components each of which is no longer a loop when n is odd.

关键词： qr algorithm shift analytic qr decomposition reachable matrices reflexivity symmetry transitivity

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A quadratically convergent qr-like method without shifts for the Hermitian eigenvalue problem

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LINEAR ALGEBRA AND ITS APPLICATIONS 2006年第2-3期417卷 478-495页

作者： Zha, Hongyuan Zhang, Zhenyue Ying, Wenlong Zhejiang Univ Dept Math Hangzhou 310027 Peoples R China Penn State Univ Dept Comp Sci & Engn University Pk PA 16802 USA

We propose a new qr-like algorithm, symmetric squared qr (SSqr) method, that can be readily parallelized using commonly available parallel computational primitives such as matrix-matrix multiplication and qr decomposition. The algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts. We provide a rigorous convergence analysis of SSqr and derive structures for several of the important quantities generated by the algorithm. We also discuss various practical implementation issues such as stopping criteria and deflation techniques. We demonstrate the convergence behavior of SSqr using several numerical examples. (c) 2005 Elsevier Inc. All rights reserved.

关键词： eigenvalue problem qr algorithm Hermitian matrix convergence analysis

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Sobre um método assemelhado ao de Francis para a determinação de autovalores de matrizes

Sobre um método assemelhado ao de Francis para a determina...

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作者： Oliveira, Danilo Elias de Universidade Estadual Paulista

O principal objetivo deste trabalho é apresentar, discutir as qualidades e desempenho e provar a convergência de um método iterativo para a solução numérica do problema de autovalores de uma matriz, que chamamos de Método Assemelhado ao de Francis (MAF). O método em questão distingue-se do qr de Francis pela maneira, mais simples e rápida, de se obter as matrizes ortogonais Qk, k = 1; 2. Apresentamos, também, uma comparação entre o MAF e os algoritmos qr de Francis e LR de *** main purpose of this work is to presente, to discuss the qualities and performance and to prove the convergence of an iterative method for the numerical solution of the eigenvalue problem, that we have called the Método Assemelhado ao de Francis (MAF)þþ. This method di ers from the qr method of Francis by providing a simpler and faster technique of getting the unitary matrices Qk; k = 1; 2; We present, also, a comparison analises between the MAF and the qr of Francis and LR of Rutishauser algorithms.

关键词： Álgebra linear Matrizes (Matematica) Eigenvalue qr algorithm LR algorithm Cholesky decomposition Dissertação de mestrado

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Product eigenvalue problems

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SIAM REVIEW 2005年第1期47卷 3-40页

作者： Watkins, DS Washington State Univ Dept Math Pullman WA 99164 USA

Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix A are wanted, but A is not given explicitly. Instead it is presented as a product of several factors: A = A(k)A(k-1)... A(1). Usually more accurate results are obtained by working with the factors rather than forming A explicitly. For example, if we want eigenvalues/vectors of B-T B, it is better to work directly with B and not compute the product. The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept. Diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems. For all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic GR algorithm applied to a related cyclic matrix.

关键词： eigenvalue qr algorithm GR algorithm generalized eigenvalue problem SVD symmetric skew symmetric positive definite totally positive pseudosymmetric Hamiltonian symplectic unitary

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Transformations to rank structures by unitary similarity

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LINEAR ALGEBRA AND ITS APPLICATIONS 2005年第1-3期402卷 126-134页

作者： Bevilacqua, RB Bozzo, EB Del Corso, GM Univ Pisa Dipartimento Informat I-56127 Pisa Italy

We study transformations by unitary similarity of a nonderogatory matrix to certain rank structured matrices. This kind of transformations can be described in a unified framework that involves Krylov matrices. The rank structures here addressed are preserved by qr iterations, and every iterate can be associated with a suitable Krylov matrix. (c) 2005 Elsevier Inc. All rights reserved.

关键词： banded matrices low rank submatrices Krylov matrices qr algorithm

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A numerical method for computing an SVD-like decomposition

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2005年第4期26卷 1058-1082页

作者： Xu, HG Univ Kansas Dept Math Lawrence KS 66045 USA

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 JB(T)B, where J = [(-I 0) (0 I)]. 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 JB(T)B computed by this method are more accurate than those computed by the methods working on the explicit product JB(T)B or BJB(T)

关键词： skew-symmetric matrix Hamiltonian matrix symplectic matrix orthogonal symplectic matrix eigenvalue problem SVD SVD-like decomposition Schur form Jordan canonical form qr algorithm Jacobi algorithm

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