Polynomial zerofinding iterative matrix algorithms

Newbery’s method is completed to a method for the construction of a (complex) symmetric or nonsymmetric matrix with a given characteristic polynomial. The methods of Fiedler, Schmeisser, and Dörfler and Schmeisser for similar constructions of symmetric matrices are reviewed. Polynomials found in the literature are solved iteratively by one of Fiedler’s methods with initial values supplied either by Schmeisse’s method, or taken on a large circle or randomly in a region of the complex plane. The determinental equations are solved by the QR algorithm. Fielder’s method used iteratively exhibits fast convergence to simple roots, even in the presence of multiple roots. If, at some iteration step, the values of the iterates, which are converging to a multiple root, are averaged according to the Hull-Mathon procedure, then fast convergence is also attained for multiple roots. This combination appears to have nice features for polynomials of small to moderate degree.