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

作者：Ye Qiang^*

来源：Mathematics of Computation, 2018, 87(309): 237-259.

DOI：10.1090/mcom/3223

• Eigenvalue
• ill-conditioned matrix
• accuracy
• Lanczos method
• differential eigenvalue problem
• biharmonic operator

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.

• 出版日期2018-1