On the block decomposition and spectral factors of λ-matrices

In this paper we have factorized matrix polynomials into a complete set of spectral factors using a new design algorithm, and with some systematic procedures a complete set of block roots (solvents) have been obtained. The newly developed procedure is just an extension of the (scalar) Horner method to its block form for use in the computation of the block roots of matrix polynomial, the block-Horner method bringing a local iterative nature, faster convergence, nested programmable scheme, needless of any prior knowledge of the matrix polynomial, with the only one inconvenience, which is the strong dependence on the initial guess. In order to avoid this trap, we proposed a combination of two computational procedures, for which the complete program starts with the right block-Q.D. algorithm. It is then followed by a refinement of the right factor by block-Horner’s algorithm. This results in the global nature of the program, which is faster in execution, has well defined initial conditions, and good convergence in much less time.