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Journals ACM SIGAPL APL Quote Quad Volume 23 Issue 1

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.

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Globally convergent polynomial iterative zero-finding using APL

Chen, Tien Chi
ACM SIGAPL APL Quote Quad , Volume 23 (1)
Association for Computing Machinery – Jul 15, 1992

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