Summary
The existence of attractive cycles constitutes a serious impediment to the solution of nonlinear equations by iterative methods. This problem is illustrated in the case of the solution of the equationz tanz=c, for complex values ofc, by Newton's method. Relevant results from the theory of the iteration of rational functions are cited and extended to the analysis of this case, in which a meromorphic function is iterated. Extensive numerical results, including many attractive cycles, are summarized.
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This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grants A3028 and A7691
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Howland, J.L., Vaillancourt, R. Attractive cycles in the iteration of meromorphic functions. Numer. Math. 46, 323–337 (1985). https://doi.org/10.1007/BF01389489
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DOI: https://doi.org/10.1007/BF01389489