Computing lower rank approximations of matrix polynomials

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Date

2020-05

Authors

Giesbrecht, Mark
Haraldson, Joseph
Labahn, George

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Publisher

Elsevier

Abstract

Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest matrix polynomial that is algebraically singular with a prescribed lower bound on the dimension given in a previous paper by the authors. In this paper we prove that such lower rank matrices at minimal distance always exist, satisfy regularity conditions, and are all isolated and surrounded by a basin of attraction of non-minimal solutions. In addition, we present an iterative algorithm which, on given input sufficiently close to a rank-at-most 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 of our algorithm in practice.

Description

The final publication is available at Elsevier via https://doi.org/10.1016/j.jsc.2019.07.012. © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

Keywords

matrix polynomials, symbolic-numeric computing, low-rank approximation

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