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

by   Zlatko Drmač, et al.

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. 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 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.


page 1

page 2

page 3

page 4


Cross product-free matrix pencils for computing generalized singular values

It is well known that the generalized (or quotient) singular values of a...

Global properties of eigenvalues of parametric rank one perturbations for unstructured and structured matrices

General properties of eigenvalues of A+τ uv^* as functions of τ∈ or τ∈ o...

Well-conditioned eigenvalue problems that overflow

In this note we present a parameterized class of lower triangular matric...

On the sensitivity of singular and ill-Conditioned linear systems

Solving a singular linear system for an individual vector solution is an...

Richardson Approach or Direct Methods? What to Apply in the Ill-Conditioned Least Squares Problem

This report shows on real data that the direct methods such as LDL decom...

Graph-Theoretical Based Algorithms for Structural Optimization

Five new algorithms were proposed in order to optimize well conditioning...

Linear Computation Coding: Exponential Search and Reduced-State Algorithms

Linear computation coding is concerned with the compression of multidime...

Please sign up or login with your details

Forgot password? Click here to reset