A Fixed-Point Iteration Method with Quadratic Convergence

Journal Article · Sun Jan 01 00:00:00 EST 2012 · Journal of Applied Mechanics

DOI:https://doi.org/10.1115/1.4005878· OSTI ID:1051423

Walker, Kevin P. ^[1]; Sham, Sam ^[2]

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1. Engineering Science Software, Inc.
2. ORNL

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.

• Research Organization: Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
• Sponsoring Organization: Work for Others (WFO)
• DOE Contract Number: DE-AC05-00OR22725
• OSTI ID: 1051423
• Journal Information: Journal of Applied Mechanics, Vol. 79, Issue 3; ISSN 0021-8936
• Country of Publication: United States
• Language: English