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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

MSC:

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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