Accurate inverses for computing eigenvalues of extremely ill-conditioned matrices and differential operators

Abstract

This paper is concerned with computations of a few smallest eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that a few smallest eigenvalues can be accurately computed for a diagonally dominant matrix or a product of diagonally dominant matrices by combining a standard iterative method with the accurate inversion algorithms that have been developed for such matrices. Applications to the finite difference discretization of differential operators are discussed. In particular, a new discretization is derived for the 1-dimensional biharmonic operator that can be written as a product of diagonally dominant matrices. Numerical examples are presented to demonstrate the accuracy achieved by the new algorithms. * 2010 Mathematics Subject Classification: 65F15, 65F35, 65N06, 65N25.

Preconditioning for accurate solutions of ill‐conditioned linear systems

“…We list the computed eigenvalues µ chol 1 and µ aldu 1 and their relative errors. As explained in [29], the error |µ chol…”

“…is a modest number; see [29]. In particular, we may also consider a preconditioner that is a product of diagonally dominant matrices; see Examples 2 and 4 in §4.…”

“…Noting that the largest eigenvalue (in absolute value) can be computed accurately independent of κ 2 (A), one way to compute the smallest eigenvalue accurately is by computing the largest eigenvalue of A −1 using the Lanczos algorithm/the power method. For this to succeed, we need to be able to compute the matrix-vector product A −1 v accurately; see [29]. An inverse-equivalent algorithm for solving Au = v would yield A −1 v as accurately as the one obtained by multiplying the exact A −1 with v, which is sufficient to compute the largest eigenvalue of A −1 accurately.…”

Preconditioning for Accurate Solutions of Linear Systems and Eigenvalue Problems

“…is a modest number; see [36]. In particular, we may also consider a preconditioner that is a product of diagonally dominant matrices; see Examples 2 and 4 in §5.…”

“…An error analysis together with numerical examples will be presented to demonstrate the stability gained. While the present paper is focused on linear systems, we will also use this accurate preconditioning method to accurately compute a few smallest eigenvalues of an ill-conditioned matrix through the accurate inverse approach presented in [36].…”