For some constant ,  implies that  is an approximate zero of , where

Smale (1986) found a constant  for the test, and this value was subsequently improved to  by Wang and Han (1989), and further improved by Wang and Zhao (Wang and Zhao 1995; Petković et al. 1997, p. 2).

REFERENCES:

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.