## Fast local convergence with single and
multistep methods for nonlinear equations

30. J. P. Abbott and R. P. Brent,
Fast local convergence with single and
multistep methods for nonlinear equations,
* J. Australian Mathematical Society (Series B)*
19 (1975), 173-199.
MR 55\#4677.
Errata: *ibid* 20 (1977), 254 (see below).
MR 58\#13673.
Paper: pdf (2330K).

Errata: pdf (70K), or see below.

## Abstract

Methods which make use of the differential equation
*dx/dt* = -J(*x*)^{-1}*f(x)*,
where J(*x*) is the Jacobian of *f(x)*,
have recently been proposed for solving the system of nonlinear
equations *f(x)* = 0.
These methods are important because of their improved convergence
characteristics. Under general conditions the solution trajectory
*x(t)* of the differential equation converges to a root of
*f* and the problem becomes one of solving a differential equation.
In this paper we note that the special form of the differential equation
can be used to derive single and multistep methods which give improved
rates of local convergence to a root.
## Comments

For a related paper, see Abbott and Brent [44].
## Errata

page 177: An assumption of uniform differentiability is necessary
for Theorem 3.1 and its Corollaries.

page 181, line -6: "4.1" should be "3.1".

page 187, line 6: replace "*G(x*^{*},...,x^{*})"
by "*H*".

page 189, line -1: "*G*" is missing a hat.

page 190, lines 1 and 2: ditto.

page 190, line -13: "*h*_{0}^{3}" should be
"*h*_{0}^{2}".

page 191, line -9: "*h*_{j}" should be
"*h*_{m+j+1}".

page 192, line -6: "3.8" should be "3.9".
Go to next publication

Return to Richard Brent's index page