Construction of globally convergent iteration functions for the solution of polynomial equations

Abstract

Iteration functions for the approximation of zeros of a polynomial P are usually given as explicit functions of P and its derivatives. We introduce a class of iteration functions which are themselves constructed according to a certain algorithm given below. The construction of the iteration functions requires only simple polynomial manipulation which may be performed on a computer. Let P be a real monic polynomial of degree n with distinct zeros Pit • • • , Pn and let the dominant zero p be real. The theory may be extended to multiple zeros, dominant complex zeros, and subdominant zeros. Let B{t) be an arbitrary polynomial of degree at most n — 1 with B (pi) j*0. Define a sequence of polynomials of degree n — by