A one parameter family of locally quartically convergent zero-finding methods

Authors:
Naoki Osada
Affiliations:
Tokyo Woman's Christian University, Zempukuji, Suginamiku, Tokyo 167-8585, Japan
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

Citing 4
Cited 2

Accelerated convergence in Newton's method

SIAM Review
Accelerated Convergence in Newton's Method

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Rounding Errors in Algebraic Processes

Rounding Errors in Algebraic Processes
Asymptotic error constants of cubically convergent zero finding methods

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Chebyshev-Halley methods for analytic functions

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On Newton-type methods for multiple roots with cubic convergence

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Abstract

A one parameter family of iteration functions for finding simple and multiple zeros of analytic functions is derived. The family includes, as a special case, Traub's quartic square root method and, as limiting cases, the Kiss method of order 4, the Halley and the Newton methods. All the methods of the family are locally quartically convergent for a simple or multiple zero with known multiplicity. The asymptotic error constants for the methods of the family are given. The decreasing ratio at infinity of iteration functions is discussed. The optimum parameter of the family for polynomials is given.