The mathematical basis and a prototype implementation of a new polynomial rootfinder with quadratic convergence

Abstract

Formulas developed originally by Weierstrass have been used since the 1960s by many others for the simultaneous determination of all the roots of a polynomial. Convergence to simple roots is quadratic, but individual approximations to a multiple root converge only linearly. However, it is shown here that the mean of such individual approximations converges quadratically to that root. This result, along with some detail about the behavior of such approximations in the neighborhood of the multiple root, suggests a new approach to the design of polynomial rootfinders. It should also be noted that the technique is well suited to take advantage of a parallel environment. This article first provides the relevant mathematical results: a short derivation of the formulas, convergence proofs, an indication of the behavior near a multiple root, and some error bounds. It then provides the outline of an algorithm based on these results, along with some graphical and numerical results to illustrate the major theoretical points. Finally, a new program based on this algorithm, but with a more efficient way of choosing starting values, is described and then compared with corresponding programs from IMSL and NAG with good results. This program is available from Mathon (combin@cs.utoronto.ca).