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)-1f(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: "h03" should be
"h02".
page 191, line -9: "hj" should be
"hm+j+1".
page 192, line -6: "3.8" should be "3.9".
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