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Fast local convergence with single and multistep methods for nonlinear equations

Published online by Cambridge University Press:  17 February 2009

James P. Abbott  and
Richard P. Brent
Richard P. Brent
Affiliation:
Computer Centre, Australian National University, Canberra 2600, Australia.
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Abstract

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Methods which make use of the differential equation ẋ(t) = −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 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.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 19 , Issue 2 , December 1975 , pp. 173 - 199
Copyright
Copyright © Australian Mathematical Society 1975

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