On The Block Decomposition and Spectral Factors of {\lambda}-Matrices

TL;DR

This paper introduces a novel algorithm for factorizing matrix polynomials into spectral factors, extending the Horner method with improved convergence and no need for initial guesses, enhancing computational efficiency.

Contribution

The paper presents a new block Horner algorithm and a combined computational approach for spectral decomposition and factorization of matrix polynomials, avoiding initial guess requirements.

Findings

Faster convergence of the block Horner method.

Complete set of spectral factors obtained without prior knowledge.

Enhanced computational scheme for matrix polynomial factorization.

Abstract

In this paper we factorize matrix polynomials into a complete set of spectral factors using a new design algorithm and we provide a complete set of block roots (solvents). The procedure is an extension of the (scalar) Horner method for the computation of the block roots of matrix polynomials. The Block-Horner method brings an iterative nature, faster convergence, nested programmable scheme, needless of any prior knowledge of the matrix polynomial. In order to avoid the initial guess method we proposed a combination of two computational procedures . First we start giving the right Block-Q. D. (Quotient Difference) algorithm for spectral decomposition and matrix polynomial factorization. Then the construction of new block Horner algorithm for extracting the complete set of spectral factors is given.