Wiener-Hopf matrix factorization using ordinary differential equations in the commutative case

TL;DR

This paper introduces a novel method for factorizing 2x2 algebraic matrices in Moiseev's class by reducing the problem to solving two ordinary differential equations, improving computational efficiency.

Contribution

The paper develops a new approach that transforms the matrix factorization into solving a nonlinear ODE and a linear ODE, offering a practical numerical solution.

Findings

Method effectively computes matrix factorization in examples

Reduces problem to solving two ODEs on a half-line

Demonstrates improved efficiency over existing methods

Abstract

A matrix factorization problem is considered. The matrix to be factorized is algebraic, has dimension 2 X 2 and belongs to Moiseev's class. A new method of factorization is proposed. First, the matrix factorization problem is reduced to a Riemann-Hilbert problem using the Hurd's method. Secondly, the Riemann-Hilbert problem is embedded into a family of Riemann-Hilbert problems indexed by a variable b taking values on a half-line. A linear ordinary differential equation (ODE1) with respect to b is derived. The coefficient of this equation remains unknown at this step. Finally, the coefficient of the ODE1 is computed. For this, it is proven that it obeys a non-linear ordinary differential equation (ODE2) on a half-line. Thus, the numerical procedure of matrix factorization becomes reduced to two runs of solving of ordinary differential equations on a half-line: first ODE2 for the coefficient of ODE1, and then ODE1 for the unknown function. The efficiency of the new method is demonstrated on some examples.