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SOLVING SPLINE COLLOCATION APPROXIMATION TO NONLINEAR TWO-POINT 13OUNDARY VALUE PROBLEMS BY A HOMOTOPY METHOD

IP.com Disclosure Number: IPCOM000128713D
Original Publication Date: 1984-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 18 page(s) / 48K

Publishing Venue

Software Patent Institute

Related People

Layne T. Watson: AUTHOR [+4]

Abstract

The Chow-Yorke algorithm is a homotopy method that has been proved globally convergent for Brouwer fixed point problems, certain classes of zero finding and nonlinear programming problems, and two-point boundary value approximations based on shooting and finite differences. The method is numerically stable and has been successfully applied to a wide range of practical engineering problems. Here the Chow-Yorke algorithm is proved globally convergent foe a class of spline collocation approximations to nonlinear two-point boandary value problems. Several numerical implementations of the algorithm are briefly described, and computational results are presented for a fairly difficult fluid dynamics boundary value problem. Key words. homotopy method, Chow-Yorke algorithm, globally convergent, two-point boundary value problem. spline collocation, nonlinear equations

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

SOLVING SPLINE COLLOCATION APPROXIMATION TO NONLINEAR TWO-POINT 13OUNDARY VALUE PROBLEMS BY A HOMOTOPY METHOD

by Layne T. Watson Melvin R. Scott CS84015-R <> SOLVING SPLINE COLLOCATION APPROXIMATIONS TO NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS BY A HOMOTOPY METHOD

LAYNE T. WATSONfi and MELVIN R. SCOTT:

Abstract.

The Chow-Yorke algorithm is a homotopy method that has been proved globally convergent for Brouwer fixed point problems, certain classes of zero finding and nonlinear programming problems, and two-point boundary value approximations based on shooting and finite differences. The method is numerically stable and has been successfully applied to a wide range of practical engineering problems. Here the Chow-Yorke algorithm is proved globally convergent foe a class of spline collocation approximations to nonlinear two-point boandary value problems. Several numerical implementations of the algorithm are briefly described, and computational results are presented for a fairly difficult fluid dynamics boundary value problem.

Key words. homotopy method, Chow-Yorke algorithm, globally convergent, two-point boundary value problem. spline collocation, nonlinear equations

1. Introduction.

The foundation of the Chow-Yorke algorithm was laid in 1976, and since that time both the theory and the scope of its practical applicability have been greatly extended. This homotopy algorithm is accurately described as a globally convergent probability one algorithm. It is truly globally convergent in the sense that it will converge to a solution of the problem from an arbitrary starting point. The phrase 'probability one' refers to the rigorous theoretical results which guarantee convergence for almost all choices of some parameter, i.e., with probability one.

Homotopy methods (both continuous (1) and simpIicial (9,23j) were once believed to be hope- lessly inefficient, and dismissed by some as inherently inferior to quasi-Newton algorithms. Another prevalent point of view was that homotopy algorithms were just continuation, and nothing new. These beliefs have been somewhat dispelled by a series of problems, successfully solved by homo-topy methods, on which continuation and quasi-Newton methods either totally failed or experienced great difficulty (391. Current implementations of these globally convergent probability one homotopy algorithms are reasonably efficient, and their robustness, stability, and accuracy have never been in doubt. A reasonable attitude toward homotopy methods is that they area method of last resort, but a very powerful and realistic method of last resort.

Virginia Polytechnic Institute and State University Page 1 Dec 31, 1984

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SOLVING SPLINE COLLOCATION APPROXIMATION TO NONLINEAR TWO-POINT 13OUNDARY VALUE PROBLEMS BY A

HOMOTOPY METHOD

There are three distinct, but interrelated, aspects of homotopy methods: 1) construction of the right homotopy map...