## 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).

## Abstract

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
erf(*x*).
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 2*n*. For example, there is a method
of order four which uses only one evaluation of *f*
and two of *f'* per iteration.

## Comments

For a related paper, see Brent
[26].
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