Accurate Inverses for Computing Eigenvalues of Extremely Ill-conditioned Matrices and Differential Operators
Numerical Analysis
2017-05-16 v2
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.
Cite
@article{arxiv.1512.05292,
title = {Accurate Inverses for Computing Eigenvalues of Extremely Ill-conditioned Matrices and Differential Operators},
author = {Qiang Ye},
journal= {arXiv preprint arXiv:1512.05292},
year = {2017}
}