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).
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.
,
implies that 
is an approximate
zero of 
,
where
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).
