Discrete versus continuous Newton's method: A case study

Abstract

We consider the damped Newton's method N _h (z) = z − hp(z)/p′(z), 0<h<1 for polynomialsp(z) with complex coefficients. For the usual Newton's method (h=1) and polynomialsp(z), it is known that the method may fail to converge to a root ofp and rather leads to an attractive periodic cycle.N _h(z) may be interpreted as an Euler step for the differential equation ż=−p(z)/p′(z) with step sizeh. In contrast to the possible failure of Newton's method, we have that for almost all initial conditions to the differential equation that the solutions converge to a root ofp. We show that this property generally carries over to Newton's methodN _h(z) only for certain nondegenerate polynomials and for sufficiently small step sizesh>0. Further we discuss the damped Newton's method applied to the family of polynomials of degree 3.