For some constant alpha_0, alpha(f,z)<alpha_0 implies that z is an approximate zero of f, where


Smale (1986) found a constant alpha approx 0.130707 for the test, and this value was subsequently improved to alpha_0=3-2sqrt(2) approx 0.171573 by Wang and Han (1989), and further improved by Wang and Zhao (Wang and Zhao 1995; Petković et al. 1997, p. 2).

See also

Approximate Zero, Newton's Method, Point Estimation Theory

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Kim, M. Ph.D. thesis. New York: City University of New York, 1985.Petković, M. S.; Herceg, D. D.; and Ilić, S. M. Point Estimation Theory and Its Applications. Novi Sad, Yugoslavia: Institute of Mathematics, 1997.Smale, S. "Newton's Method Estimates from Data at One Point." In The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Ed. R. E. Ewing, K. I. Gross, and C. F. Martin). New York: Springer-Verlag, pp. 185-196, 1986.Wang, X. and Han, D. "On Dominating Sequence Method in the Point Estimate and Smale's Theorem." Scientia Sinica Ser. A, 905-913, 1989.Wang, D. and Zhao, F. "The Theory of Smale's Point Estimation and Its Application." J. Comput. Appl. Math. 60, 253-269, 1995.

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Cite this as:

Weisstein, Eric W. "Alpha-Test." From MathWorld--A Wolfram Web Resource.

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