Chapter 50 - Newton's method for multiple roots

Abstract

Newton's method for finding a real or complex root of a function is very efficient near a simple root because the algorithm converges quadratically in the neighborhood of such a root. At a multiple root––that is, a root of order greater than one–– Newton's method only converges linearly. Various modifications of Newton's method have been proposed that converge quadratically at multiple roots. This chapter discusses one standard method that finds the roots of the function g(z) by applying Newton's method to the function g(z)/g'(z). Newton's method applied to this quotient always converges quadratically near all the roots of g(z). This method introduces extraneous fixed points into Newton's method, but they are always repelling. The chapter plots the basins of attraction for the roots in the complex plane and illustrates what happens to these basins as two simple roots coalesce to form a multiple root.