Research Papers

A Fixed-Point Iteration Method With Quadratic Convergence

[+] Author and Article Information
K. P. Walker

 Engineering Science Software, Inc., Smithfield, RI 02917

T.-L. Sham1

 Materials Science and Technology Division,Oak Ridge National Laboratory, Oak Ridge, TN 37831 shamt@ornl.gov


Corresponding author.

The United States Government retains, and the publisher, by accepting this submission for publication, acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this submission, or allow others to do so, for United States Government purposes.

J. Appl. Mech 79(3), 031001 (Apr 05, 2012) (10 pages) doi:10.1115/1.4005878 History: Received May 24, 2011; Revised December 05, 2011; Posted February 06, 2012; Published April 04, 2012; Online April 05, 2012

The fixed-point iteration algorithm is turned into a quadratically convergent scheme for a system of nonlinear equations. Most of the usual methods for obtaining the roots of a system of nonlinear equations rely on expanding the equation system about the roots in a Taylor series, and neglecting the higher order terms. Rearrangement of the resulting truncated system then results in the usual Newton-Raphson and Halley type approximations. In this paper the introduction of unit root functions avoids the direct expansion of the nonlinear system about the root, and relies, instead, on approximations which enable the unit root functions to considerably widen the radius of convergence of the iteration method. Methods for obtaining higher order rates of convergence and larger radii of convergence are discussed.

Copyright © 2012 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 4

Widening of the “convergence plateau” for the quadratic β0 and higher order F(x) approximations

Grahic Jump Location
Figure 1

Exact value of β(x) for the exponential algorithm with f(x) = 12/x and the approximate partial sum steady state solutions β0,β1,β2

Grahic Jump Location
Figure 2

Comparison of the Cesa`ro partial sum with the exact β*, β0, and β5

Grahic Jump Location
Figure 3

Value of β0 in the quadratic exponential algorithm versus the third Bender approximation and the exact solution β*




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