# 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.

## Key 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.