Accurate Inverses for Computing Eigenvalues of Extremely Ill-conditioned Matrices and Differential Operators
This work addresses the problem of computing eigenvalues of extremely ill-conditioned matrices, which is important for numerical linear algebra and differential operators, but the approach is limited to diagonally dominant matrices or their products.
The paper shows that smaller eigenvalues of extremely ill-conditioned matrices can be accurately computed by combining standard iterative methods with accurate inversion algorithms for diagonally dominant matrices, with applications to differential operators. Numerical examples demonstrate high accuracy.
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.