Numerical methods for accurate computation of the eigenvalues of Hermitian matrices and the singular values of general matrices

TL;DR

This paper reviews numerical methods that achieve high accuracy in computing eigenvalues and singular values of Hermitian and structured matrices, even in ill-conditioned cases, by exploiting structured errors and intrinsic matrix parameters.

Contribution

It introduces a structured error analysis framework and an unconventional approach using intrinsic parameters for accurate eigenvalue and singular value computation of challenging matrices.

Findings

Structured error analysis improves sensitivity measurement.

High-accuracy algorithms work despite large condition numbers.

Use of intrinsic parameters enables accurate factorization of structured matrices.

Abstract

This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that guarantee high accuracy even in the cases that are ill-conditioned for the conventional methods. First, it is shown that a particular structure of the errors in a finite precision implementation of an algorithm allows for a much better measure of sensitivity and that computation with high accuracy is possible despite a large classical condition number. Such structured errors incurred by finite precision computation are in some algorithms e.g. entry-wise or column-wise small, which is much better than the usually considered errors that are in general small only when measured in the Frobenius matrix norm. Specially tailored perturbation theory for such structured perturbations of Hermitian matrices guarantees much better bounds for the relative errors in the computed eigenvalues. % Secondly, we review an unconventional approach to accurate computation of the singular values and eigenvalues of some notoriously ill-conditioned structured matrices, such as e.g. Cauchy, Vandermonde and Hankel matrices. The distinctive feature of accurate algorithms is using the intrinsic parameters that define such matrices to obtain a non-orthogonal factorization, such as the \textsf{LDU} factorization, and then computing the singular values of the product of thus computed factors. The state of the art software is discussed as well.