On algorithmization of Janashia-Lagvilava matrix spectral factorization method

TL;DR

This paper explores three algorithmic approaches to the Janashia-Lagvilava spectral factorization method, analyzing their efficiency, applicability to different matrix sizes, and implementation speed, with numerical comparisons to existing algorithms.

Contribution

It introduces three distinct algorithms for spectral factorization, optimizing for speed and applicability, and provides a comparative analysis with numerical simulation results.

Findings

The first algorithm is faster but limited to low-dimensional matrices.

The second algorithm handles larger matrices effectively.

The third algorithm is superfast but restricted to polynomial cases with zeros not near the boundary.

Abstract

We consider three different ways of algorithmization of the Janashia-Lagvilava spectral factorization method. The first algorithm is faster than the second one, however, it is only suitable for matrices of low dimension. The second algorithm, on the other hand, can be applied to matrices of substantially larger dimension. The third algorithm is a superfast implementation of the method, but only works in the polynomial case under the additional restriction that the zeros of the determinant are not too close to the boundary. All three algorithms fully utilize the advantage of the method which carries out spectral factorization of leading principal submatrices step-by-step. The corresponding results of numerical simulations are reported in order to describe the characteristic features of each algorithm and compare them to other existing algorithms.