Accurate Inverses for Computing Eigenvalues of Extremely Ill-conditioned Matrices and Differential Operators

TL;DR

This paper introduces a method to accurately compute small eigenvalues of extremely ill-conditioned matrices by combining iterative methods with accurate inversion algorithms, especially for diagonally dominant matrices and differential operators.

Contribution

It presents a novel approach that enables precise eigenvalue computation for ill-conditioned matrices using accurate inversion techniques and applies this to discretized differential operators.

Findings

Accurate eigenvalues computed for ill-conditioned matrices.

New discretization for 1D biharmonic operator as a product of diagonally dominant matrices.

Numerical results demonstrate high accuracy of the proposed algorithms.

Abstract

This paper is concerned with computations of a few smaller eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that smaller 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.