Generic factorizations and alternating direction method for fixed-rank approximation of matrix polynomials

Abstract

We present an algorithm for approximating a given matrix polynomial of degree $d$ by a matrix polynomial of prescribed rank and degree at most $d$. The method builds on recent advances in the theory of generic eigenstructures and factorizations of matrix polynomials with bounded rank and degree. We also give a specialized treatment of the case of matrix pencils (degree one matrix polynomials), motivated by its particular theoretical and practical importance. Lastly, numerical experiments are reported to assess the performance of the proposed approach and to compare it with existing methods.