## 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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### References:

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