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This paper is concerned with computations of a few smaller eigenvalues (inabsolute value) of a large extremely ill-conditioned matrix. It is shown thatsmaller eigenvalues can be accurately computed for a diagonally dominant matrixor a product of diagonally dominant matrices by combining a standard iterativemethod with the accurate inversion algorithms that have been developed for suchmatrices. Applications to the finite difference discretization of differentialoperators are discussed. In particular, a new discretization is derived for the1-dimensional biharmonic operator that can be written as a product ofdiagonally dominant matrices. Numerical examples are presented to demonstratethe accuracy achieved by the new algorithms.

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