Partial Fraction Expansions for Newton`s and Halley`s Iterations for Square Roots

- Journal title : Kyungpook mathematical journal
- Volume 52, Issue 3, 2012, pp.347-357
- Publisher : Department of Mathematics, Kyungpook National University
- DOI : 10.5666/KMJ.2012.52.3.347

Title & Authors

Partial Fraction Expansions for Newton`s and Halley`s Iterations for Square Roots

Kouba, Omran;

Kouba, Omran;

Abstract

When Newton`s method, or Halley`s method is used to approximate the pth root of 1-z, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev`s polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).

Keywords

Newton`s method;Halley`s method;Series expansion;Square roots;Chebyshev`s Polynomials;

Language

English

Cited by

References

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