A Krylov Subspace Method for Quadratic Matrix Polynomials with Application to Constrained Least Squares Problems

Abstract

We present a Krylov subspace-type projection method for a quadratic matrix polynomial $\lambda^2 I -\lambda A - B$ that works directly with A and B without going through any linearization. We discuss a special case when one matrix is a low rank perturbation of the other matrix. We also apply the method to solve quadratically constrained linear least squares problem through a reformulation of Gander, Golub, and von Matt as a quadratic eigenvalue problem, and we demonstrate the effectiveness of this approach. Numerical examples are given to illustrate the efficiency of the algorithms.