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

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.