A class of optimal-order zero-finding methods
using derivative evaluations
27. R. P. Brent,
A class of optimal-order zero-finding methods using derivative evaluations,
in Analytic Computational Complexity
(edited by J. F. Traub),
Academic Press, 1975, 59-73.
MR 52#15938, 54#9073.
Paper: pdf (898K).
It is often necessary to find an approximation to a simple zero
of a function f, using evaluations of f and f'.
We consider some methods which are efficient if f' is easier to
evaluate than f. Examples of such functions are given; they include
functions defined by indefinite integrals, such as the error function
The main result is that there exist methods which
use one evaluation of f and n evaluations of f'
per iteration, and (under suitable smoothness conditions)
have order of convergence 2n. For example, there is a method
of order four which uses only one evaluation of f
and two of f' per iteration.
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