Discrete versus continuous Newton’s method: A case study. (English) Zbl 0669.65037

Author’s summary: We consider the damped Newton’s method \(N_ h(z)=z- hp(z)/p'(z),\) \(0<h<1\) for polynomials p(z) with complex coefficients. For the usual Newton’s method \((h=1)\) and polynomials p(z), it is known that the method may fail to converge to a root of p and rather leads to an attractive periodic cycle. \(N_ h(z)\) may be interpreted as an Euler step for the differential equation \(\dot z=-p(z)/p'(z)\) with step size h. 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 of p. We show that this property generally carries over to Newton’s method \(N_ h(z)\) only for certain nondegenerate polynomials and for sufficiently small step sizes \(h>0\). Further we discuss the damped Newton’s method applied to the family of polynomials of degree 3.
Reviewer: E.Allgower


65H05 Numerical computation of solutions to single equations
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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