On the efficient and accurate solution of the skew-symmetric eigenvalue problem: An arrangement of new and already known algorithmic formulations

Abstract

Algorithms are presented to solve the special eigenvalue problem AZ = Zλ , where A is skew-symmetric. The effective use of Householder's method, the bisection method and inverse iteration for solving the complete eigen-value problem are described in some detail. Simultaneous vector iteration is formulated for skew-symmetric matrices. The amount of work for the skew-symmetric Jacobi algorithm and the simultaneous vector iteration may be reduced by using the solution of a simplified eigenvalue problem. For Hermitian matrices also quadratic eigenvalue bounds for groups of eigenvalues and linear bounds for groups of eigenvectors are derived. The case where the set of calculated eigenvectors is not orthonormal is considered in some detail. In principle, the skew-symmetric eigenvalue problem may be easily transformed into a symmetric eigenvalue problem; but such a procedure has the following disadvantages: first, the results are in general less accurate, and, second, the eigenvectors which belong to well separated eigenvalues are not uniquely determined.