A composite polynomial zerofinding matrix algorithm

A globally convergent matrix algorithm is presented for finding the real and complex zeros of a (complex) polynomial p(x). A combination of Schmeisserʹs and Fiedlerʹs matrices is used in the algorithm. First, the greatest common divisor, g(x), of p(x) and its derivative, p′(x), is obtained and used to reduce p(x) to a polynomial q(x) = p(x)/q(x) with simple zeros. Second, the zeros of q(x) are computed either by finding the eigenvalues of the first block of Schmeisserʹs matrix or by applying Fiedlerʹs algorithm recursively. In either case, the eigenvalues are obtained by the QR algorithm. Third, the multiplicity of each zero of p(x) is calculated by means of Lagouanelleʹs modified limiting formula. Sample numerical results are given to demonstrate the effectiveness of the algorithm.