Eigenvalue problem for a regular linear pencil of matrices close to singular ones

Abstract

The solution is examined of the eigenvalue problem

(1) for a regular linear pencil of matrices A and B of which at least one is close to being singular. Two groups of algorithms are proposed for solving (1). Both groups of algorithms work in the situation when the eigenvalues of the original pencil can be separated into groups of eigenvalues “large” and “small” in absolute value. The algorithms reveal this situation. The algorithms of the first group permit the passage from the original pencil to a pencil strictly equivalent to it, which in form is close to a quasitriangular pencil (or coincides with a quasitriangular one in case at least one of the pencil's matrices is singular). The eigenvalues of the diagonal blocks of the pencil constructed yield approximations to the eigenvalues of problem (1). If the approximations obtained are refined by the Newton method, using the normalized decomposition of auxiliary constructed matrices, then both the eigenvalues of (1) as well as all the linearly independent eigenvectors corresponding to them can be found. The algorithms of the second group permit the passage from the original pencil to a strictly equivalent pencil representable as a sum of two singular pencils whose null spaces are mutually perpendicular; next, with the aid of an iteration process based on the use of perturbation theory, these algorithms permit the finding of the eigenvalues of pencil (1), small (large) in absolute value, and the eigenvectors corresponding to them. Ill-conditioned regular pencils close to singular ones also are examined. For them an algorithm is suggested which permits the ill conditioning to be revealed and permits approximations to the stable (to perturbations in the original data) eigenvalues of the pencil to be obtained.