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Globally convergent polynomial iterative zero-finding using APL

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Tien Chi ChenAuthors Info & Claims

ACM SIGAPL APL Quote Quad, Volume 23, Issue 1

Pages 52 - 59

https://doi.org/10.1145/144052.144079

Published: 15 July 1992 Publication History

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Abstract

We report the APL study of a global iteration theory for polynomial zero-finding, with the potential to converge from almost any point in the complex plane, as distinguished from classical, “neighborhood” theory requiring the starting guess to be close to the zero being sought. The iteration functions studied involve the function and its first two derivatives, evaluated at the guess Z_k.

Using the symmetric cluster as a reference, we discovered Newton's method to be unsuitable, and the multiplicity-adjusted Laguerre formula deficient. But a new formula converges to symetric cluster zeros in just one iteration excepting for roundoff error, possesses cubic neighborhood convergence, and appears reliable for general polynomials. Roundoff errors are reduced through reasonable starting guesses. Reboundingby subclusters, a major cause of nonconvergence is resolved by a local polynomial approach. Most polynomials tested converge within a dozen iterations.

References

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Kendall E. Atkinson, An Introduction to Numerical Analysis, Wiley (New York 1978). P. 87.

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Tien Chi Chen, "Global convergence to zeros in the presence of clusters," Proc. Inter-national __~_m_~., Taipei, Taiwan, pp. 270-276, 1988.

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Tien Chi Chen, "Iterative zero-finding revisited," pp. 583-590 in W.L. Hogarth and B.J. Noye (Eds.), Computational Techniques and Ap- plications: CTAC-89 (Proc. Computational Techniques and Applications Conference, Brisbane, Australia, July 1989), Hemisphere Pub. Corp. (New York 1990).

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E. Hansen and M. Patrick, "Estimating the multiplicity of a root," Numerische Math., vol. 27, pp. 121-131 (1976).

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E. Hansen and M. Patrick, "A family of root finding methods," Numerische Math., vol. 27, pp. 257-269 (1977).

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A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, 2nd. Ed., McGraw-Hill (New York 1978). Pp.391, 395.

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Cited By

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(2007)Newton's and Related MethodsNumerical Methods for Roots of Polynomials, Part I10.1016/S1570-579X(07)80008-5(131-206)Online publication date: 2007
https://doi.org/10.1016/S1570-579X(07)80008-5
Chen TLuk W(1994)Aberth's method for the parallel iterative finding of polynomial zerosACM SIGAPL APL Quote Quad10.1145/190468.19028125:1(40-49)Online publication date: 1-Aug-1994
https://dl.acm.org/doi/10.1145/190468.190281
Chen TLuk W(1994)Aberth's method for the parallel iterative finding of polynomial zerosProceedings of the international conference on APL : the language and its applications: the language and its applications10.1145/190271.190281(40-49)Online publication date: 1-Aug-1994
https://dl.acm.org/doi/10.1145/190271.190281
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Globally convergent polynomial iterative zero-finding using APL

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Information

Published In

ACM SIGAPL APL Quote Quad Volume 23, Issue 1

July 1992

309 pages

ISSN:0163-6006

DOI:10.1145/144052

Editor:
Lynne C. Shaw

Issue’s Table of Contents

APL '92: Proceedings of the international conference on APL
July 1992
326 pages
ISBN:0897914775
DOI:10.1145/144045
Editor:
Lynne C. Shaw

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 15 July 1992

Published in SIGAPL Volume 23, Issue 1

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Cited By

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(2007)Newton's and Related MethodsNumerical Methods for Roots of Polynomials, Part I10.1016/S1570-579X(07)80008-5(131-206)Online publication date: 2007
https://doi.org/10.1016/S1570-579X(07)80008-5
Chen TLuk W(1994)Aberth's method for the parallel iterative finding of polynomial zerosACM SIGAPL APL Quote Quad10.1145/190468.19028125:1(40-49)Online publication date: 1-Aug-1994
https://dl.acm.org/doi/10.1145/190468.190281
Chen TLuk W(1994)Aberth's method for the parallel iterative finding of polynomial zerosProceedings of the international conference on APL : the language and its applications: the language and its applications10.1145/190271.190281(40-49)Online publication date: 1-Aug-1994
https://dl.acm.org/doi/10.1145/190271.190281
Chen T(1993)SCARFS, an efficient polynomial zero-finder systemACM SIGAPL APL Quote Quad10.1145/166198.16620424:1(47-54)Online publication date: 1-Sep-1993
https://dl.acm.org/doi/10.1145/166198.166204
Chen T(1993)SCARFS, an efficient polynomial zero-finder systemProceedings of the international conference on APL10.1145/166197.166204(47-54)Online publication date: 1-Sep-1993
https://dl.acm.org/doi/10.1145/166197.166204

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