Computing the Nearest Rank-Deficient Matrix Polynomial

Abstract

Matrix polynomials appear in many areas of computational algebra, control systems theory, differential equations, and mechanics, typically with real or complex coefficients. Because of numerical error and instability, a matrix polynomial may appear of considerably higher rank (generically full rank), while being very close to a rank-deficient matrix. Close is defined naturally under the Frobenius norm on the underlying coefficient matrices of the matrix polynomial. In this paper we consider the problem of finding the nearest rank-deficient matrix polynomial to an input matrix polynomial, that is, the nearest square matrix polynomial which is algebraically singular. We prove that such singular matrices at minimal distance always exist (and we are never in the awkward situation having an infimum but no actual matrix polynomial at minimal distance). We also show that singular matrices at minimal distance are all isolated, and are surrounded by a basin of attraction of non-minimal solutions. Finally, we present an iterative algorithm which, on given input sufficiently close to a rank-deficient matrix, produces that matrix. The algorithm is efficient and is proven to converge quadratically given a sufficiently good starting point. An implementation demonstrates the effectiveness and numerical robustness in practice.