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On the computation of nonhyperbolic fixed points. (English) Zbl 1162.37316

Summary: In order to deal with nonhyperbolic fixed points of a given real iteration function \(g\), we construct new iteration functions \(C\) which will be called combined. When a nonhyperbolic fixed point of \(g\) becomes a super attractor fixed point of \(C\), the iteration function \(C\) is called flat. Some flat iteration functions are constructed based on Newton’s iteration function. Several numerical examples illustrating the good properties of flat iteration functions are presented.

MSC:

37E05 Dynamical systems involving maps of the interval
37M99 Approximation methods and numerical treatment of dynamical systems
39B12 Iteration theory, iterative and composite equations
65J15 Numerical solutions to equations with nonlinear operators

Software:

Mathematica
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References:

[1] Brezinski C., Extrapolation Methods Theory and Practice. (1991) · Zbl 0744.65004
[2] Burden R. L., Numerical Analysis, (1989) · Zbl 0671.65001
[3] Epureanu B. I., SIAM Rev. 40 pp 102– (1998) · Zbl 0912.65039
[4] Gerlach J., SIAM Rev. 36 pp 272– (1994) · Zbl 0814.65046
[5] Gilbert W. J., Computers and Graphics 18 pp 227– (1994)
[6] Golubitsky M., Singularities and Groups in Bifurcation Theory (1985) · Zbl 0607.35004
[7] Govaerts W., Numerical Methods for Bifurcations of Dynamical Equilibria. (2000) · Zbl 0935.37054
[8] Henrici P., Elements of Numerical Analysis. (1964) · Zbl 0149.10901
[9] Holmgren R. A., A First Course in Discrete Dynamical Systems. (1996) · Zbl 0855.58042
[10] Isaacson E., Analysis of Numerical Methods. (1966) · Zbl 0168.13101
[11] Kress R., Numerical Analysis. (1998) · Zbl 0913.65001
[12] Ostrowski A. M., Solutions of Equations in Euclidean and Banach Spaces., (1973) · Zbl 0304.65002
[13] Sablonnière P., J. Comput. Appl. Math. 19 pp 55– (1987)
[14] Sablonnière P., Numer. Algorithms pp 177– (1991) · Zbl 0749.65002
[15] Traub J. F., Iterative Methods for the Solution of Equations. (1964) · Zbl 0121.11204
[16] Vrscay E. R., Numer. Math. 52 (1988)
[17] Wolfram S., The Mathematica Book, (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.