Construction of a fundamental series of solutions of a pencil of matrices

Abstract

Solution of spectral problems for a singular polynomial pencil of matrices D (λ) of degree s⩾1 and sizem×n is considered. Two algorithms for constructing polynomials solutions of pencils D (λ) are considered: the first is a modification of an algorithm proposed earlier by one of the authors for determining polynomial solutions of a linear pencil; the second algorithm is based on other ideas and consists of two steps. At the first step a finite sequence of auxiliary pencils is constructed for each of which a basis of polynomial solutions of degree zero is found. At the second step the basis so constructed are rearranged into polynomial solutions of the original polynomial pencil D(λ). Both algorithms make it possible to find solutions of the original pencil in order of increasing degrees. For constructing a fundamental series of solutions of the pencil D(λ) two new algorithms are proposed which work independently with either of the algorithms mentioned above for constructing polynomial solutions by rearranging them into linearly independent solutions of the pencil.