Finding polynomial roots: A fast algorithm convergent on the complex plane

Abstract

AbstractAn algorithm is suggested which performs fast calculations of all the roots of a polynomial with maximal computer accuracy using, as the only primary information, the coefficients and the degree of the polynomial. The algorithm combines global as well as local convergences, i.e. it ensures a rapid hit into a small neighbourhood of a root for 2–3 iterations from any random initial approximation and cubic convergence within the neighbourhood. A modification of the method is given which allows the roots of entire functions to be found. A numerical comparison of the method with commonly used methods (Newton–Raphson, Bairstow–Newton, steepest descent) shows its advantages in speed, accuracy and stability both for real and complex polynomials.