ISSN:
1572-9036
Keywords:
65H05
;
30D05
;
58F12
;
Relaxed Newton's method
;
basins of attraction
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00047502
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