Computing the Nearest Rank-Deficient Matrix Polynomial

Matrix polynomials appear in many areas of computational algebra, control systems theory, di‚erential equations, and mechanics, typically with real or complex coecients. 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-de€cient matrix. “Close” is de€ned naturally under the Frobenius norm on the underlying coecient matrices of the matrix polynomial. In this paper we consider the problem of €nding the nearest rank-de€cient 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 in€mum 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 aŠraction of non-minimal solutions. Finally, we present an iterative algorithm which, on given input suciently close to a rank-de€cient matrix, produces that matrix. Œe algorithm is ecient and is proven to converge quadratically given a suciently good starting point. An implementation demonstrates the e‚ectiveness and numerical robustness in practice. ACM Reference format: Mark Giesbrecht, Joseph Haraldson, and George Labahn. 2017. Computing the Nearest Rank-De€cient Matrix Polynomial. In Proceedings of ISSAC ’17, Kaiserslautern, Germany, July 25-28, 2017, 8 pages. DOI: hŠp://dx.doi.org/10.1145/3087604.3087648