Matrix Functions And Matrix Equations

Zhaojun Bai;Weiguo Gao;Yangfeng Su, Zhaojun Bai, Weiguo Gao;Yangfeng Su

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814675789
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Matrix functions and matrix equations are widely used in science, engineering and social sciences due to the succinct and insightful way in which they allow problems to be formulated and solutions to be expressed. This book covers materials relevant to advanced undergraduate and graduate courses in numerical linear algebra and scientific computing. It is also well-suited for self-study. The broad content makes it convenient as a general reference to the subjects.Matrix functions and matrix equations are widely used in science, engineering and social sciences due to the succinct and insightful way in which they allow problems to be formulated and solutions to be expressed. This book covers materials relevant to advanced undergraduate and graduate courses in numerical linear algebra and scientific computing. It is also well-suited for self-study. The broad content makes it convenient as a general reference to the subjects.Readership: Researchers and graduate students in numerical and computational mathematics.Key Features:The book covers underlying theory and a variety of algorithms for matrix functions and matrix equations. The book also covers high performance linear system solvers and eigenvalue computations which are computational kernels to matrix functions and matrix equationsThe book provides the current developments and applications beyond the material found in regular university courses and textbooks. It includes a comprehensive list of latest referencesThe authors of the chapters are leading experts who are also well-known for their expository skills

Rayleigh Quotient Based Optimization
Methods for Eigenvalue Problems

Ren-Cang Li^∗

Abstract

Four classes of eigenvalue problems that admit similar min-max principles and the Cauchy interlacing inequalities as the symmetric eigenvalue problem famously does are investigated. These min-max principles pave ways for efficient numerical solutions for extreme eigenpairs by optimizing the so-called Rayleigh quotient functions. In fact, scientists and engineers have already been doing that for computing the eigenvalues and eigenvectors of Hermitian matrix pencils A − λB with B being positive definite, the first class of our eigenvalue problems. But little attention has gone to the other three classes: positive semidefinite pencils, linear response eigenvalue problems, and hyperbolic eigenvalue problems, in part because most min-max principles for the latter were discovered only very recently and some more are being discovered. It is expected that they will drive the effort to design better optimization based numerical methods for years to come.

1Introduction

Eigenvalue problems are ubiquitous. Eigenvalues explain many physical phenomena well such as vibrations and frequencies, (in)stabilities of dynamical systems, and energy levels in molecules or atoms. This chapter focuses on classes of eigenvalue problems that admit various min-max principles and the Cauchy interlacing inequalities as the symmetric eigenvalue problem famously does [4, 38, 47]. These results make it possible to efficiently calculate extreme eigenpairs of the eigenvalue problems by optimizing associated Rayleigh quotients.

Consider the generalized eigenvalue problem

where both A and B are Hermitian. The first class of eigenvalue problems are those for which B is also positive definite. Such an eigenvalue problem is equivalent to a symmetric eigenvalue problem B^−1/2AB^−1/2y = λx and thus, not surprisingly, all min-max principles (Courant-Fischer, Ky Fan trace min/max, Wielandt-Lidskii) and the Cauchy interlacing inequalities have their counterparts in this eigenvalue problem. The associated Rayleigh quotient is

When B is indefinite and even singular, (1.1) is no longer equivalent to a symmetric eigenvalue problem in general and it may even have complex eigenvalues which clearly admit no min-max representations. But if there is a real scalar λ₀ such that A − λ₀B is positive semidefinite, then the eigenvalue problem (1.1) has only real eigenvalues and they admit similar min-max principles and the Cauchy interlacing inequalities [25, 27, 29]. This is the second class of eigenvalue problems and it shares the same Rayleigh quotient (1.2) as the first class. We call a matrix pencil in this class a positive semidefinite pencil. Opposite to the concept of a positive semidefinite matrix pencil, naturally, is that of a negative semidefinite matrix pencil A − λB by which we mean that A and B are Hermitian and t...

Table of contents

• Cover
• Halftitle
• Series Editors
• Title
• Copyright
• Preface
• Contents
• Matrix Functions: A Short Course
• A Short Course on Exponential Integrators
• Matrix Equations and Model Reduction
• Rayleigh Quotient Based Optimization Methods for Eigenvalue Problems
• Factorization-Based Sparse Solvers and Preconditioners