Numerical stability of the Halley-iteration for the solution of a system of nonlinear equations
Annie A. M. Cuyt
Math. Comp. 38 (1982), 171-179
Primary 65H10; Secondary 65G05, 65J15
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Abstract: Let and a simple root in of the system of nonlinear equations .
Abstract Padé approximants (APA) and abstract Rational approximants (ARA) for the operator F have been introduced in  and . The adjective ``abstract'' refers to the use of abstract polynomials  for the construction of the rational operators.
The APA and ARA have been used for the solution of a system of nonlinear equations in . Of particular interest was the following third order iterative procedure: with the 1st Fréchet-derivative of F in the Newton-correction where the 2nd Fréchet-derivative of F in where is the bilinear operator evaluated in , and componentwise multiplication and division in . For this technique is known as the Halley-iteration [6, p. 91]. In this paper the numerical stability  of the Halley-iteration for the case is investigated and illustrated by a numerical example.
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