ACCURATE INVERSES FOR COMPUTING EIGENVALUES OF EXTREMELY ILL-CONDITIONED MATRICES AND DIFFERENTIAL OPERATORS

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.