An improvement of the region of accessibility of Chebyshev’s method from Newton’s method

Ezquerro, J.; Hernández, M.

doi:10.1090/S0025-5718-09-02193-0

An improvement of the region of accessibility of Chebyshev’s method from Newton’s method
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by J. A. Ezquerro and M. A. Hernández;

Math. Comp. 78 (2009), 1613-1627

DOI: https://doi.org/10.1090/S0025-5718-09-02193-0

Published electronically: January 12, 2009

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Abstract:

A simple modification of Chebyshev’s method is presented, so that the region of accessibility is extended to the one of Newton’s method.

References

I. K. Argyros and Dong Chen, Results on the Chebyshev method in Banach spaces, Proyecciones 12 (1993), no. 2, 119–128. MR 1254863, DOI 10.22199/s07160917.1993.0002.00002
Ioannis K. Argyros and Ferenc Szidarovszky, The theory and applications of iteration methods, Systems Engineering Series, CRC Press, Boca Raton, FL, 1993. MR 1272012
Ioannis K. Argyros, Newton methods, Nova Science Publishers, Inc., Hauppauge, NY, 2005. MR 2128628
D. D. Bruns and J. E. Bailey, Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32 (1977) 257-264.
V. Candela and A. Marquina, Recurrence relations for rational cubic methods. II. The Chebyshev method, Computing 45 (1990), no. 4, 355–367 (English, with German summary). MR 1088077, DOI 10.1007/BF02238803
M. A. Hernández and M. A. Salanova, Indices of convexity and concavity. Application to Halley method, Appl. Math. Comput. 103 (1999), no. 1, 27–49. MR 1686356, DOI 10.1016/S0096-3003(98)10047-4
M. A. Hernández and M. A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method, J. Comput. Appl. Math. 126 (2000), no. 1-2, 131–143. MR 1806112, DOI 10.1016/S0377-0427(99)00347-7
M. A. Hernández, The Newton method for operators with Hölder continuous first derivative, J. Optim. Theory Appl. 109 (2001), no. 3, 631–648. MR 1835077, DOI 10.1023/A:1017571906739
L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
Kyle Kneisl, Julia sets for the super-Newton method, Cauchy’s method, and Halley’s method, Chaos 11 (2001), no. 2, 359–370. MR 1843721, DOI 10.1063/1.1368137
F. A. Potra and V. Pták, Nondiscrete induction and iterative processes, Research Notes in Mathematics, vol. 103, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 754338
Juan L. Varona, Graphic and numerical comparison between iterative methods, Math. Intelligencer 24 (2002), no. 1, 37–46. MR 1889920, DOI 10.1007/BF03025310
Stephen Wolfram, The Mathematica$^\circledR$ book, 4th ed., Wolfram Media, Inc., Champaign, IL; Cambridge University Press, Cambridge, 1999. MR 1721106