A defect-correction algorithm for quadratic matrix equations, with applications to quasi-Toeplitz matrices

TL;DR

This paper introduces a defect correction method for quadratic matrix equations, enhancing the Structure-preserving Doubling Algorithm to improve convergence, especially for stochastic models involving quasi-Toeplitz matrices in infinite dimensions.

Contribution

It presents a novel defect correction formula based on invariant subspaces, modifying SDA for faster convergence in solving quadratic matrix equations with applications to quasi-Toeplitz matrices.

Findings

Enhanced convergence speed in solving quadratic matrix equations.

Effective application to stochastic models with infinite quasi-Toeplitz matrices.

Improved solution accuracy through defect correction technique.

Abstract

A defect correction formula for quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=0$ is presented. This formula, expressed by means of an invariant subspace of a suitable pencil, allows us to introduce a modification of the Structure-preserving Doubling Algorithm (SDA), that enables refining an initial approximation to the sought solution. This modification provides substantial advantages, in terms of convergence acceleration, in the solution of equations coming from stochastic models, by choosing a stochastic matrix as the initial approximation. An application to solving random walks in the quarter plane is shown, where the coefficients $A_{-1},A_0,A_1$ are quasi-Toeplitz matrices of infinite size.