โ Scribed by Dietmar Saupe
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We consider the damped Newton's method Nh(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. Nh(z) may be interpreted as an Euler step for the differential equation ~ = -p(z)lp '(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 Nh(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.
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