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

Solution of spectral problems for a singular polynomial pencil of matrices $D(\lambda)$ of degree $s\geqslant1$ and size $m\times n$ is considered. Two algorithms for constructing polynomials solutions of pencils $D(\lambda)$ 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 $D(\lambda)$. 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 $D(\lambda)$ 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.