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Widely Convergent Method for Finding Multiple Solutions of Simultaneous Nonlinear Equations

IP.com Disclosure Number: IPCOM000078626D
Original Publication Date: 1973-Feb-01
Included in the Prior Art Database: 2005-Feb-26
Document File: 2 page(s) / 47K

Publishing Venue

IBM

Related People

Branin, FH: AUTHOR

Abstract

A method for solving a system of nonlinear equations g(x)=0 is based on solving the related system of differential equations dg/dt+ or -g(x)=0. The sign is changed whenever the corresponding trajectory x(t) encounters a change in sign of the Jacobian determinant absolute value ofag/Dx, and whenever it arrives at a solution of g(x)=0. This procedure endows the method with a wide region of convergence and enables it to find multiple solutions of g(x)=0, one after the other.

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Widely Convergent Method for Finding Multiple Solutions of Simultaneous Nonlinear Equations

A method for solving a system of nonlinear equations g(x)=0 is based on solving the related system of differential equations dg/dt+ or -g(x)=0. The sign is changed whenever the corresponding trajectory x(t) encounters a change in sign of the Jacobian determinant absolute value ofag/Dx, and whenever it arrives at a solution of g(x)=0. This procedure endows the method with a wide region of convergence and enables it to find multiple solutions of g(x)=0, one after the other.

This method is more fully described in an article having the same title by the same author, in IBM J. Res. Develop. 16, 504-522 (September, 1972), and can be implemented by the operation of a properly programmed digital computer such as in IBM 1130 Computing System or an IBM System/370. The flow chart in this publication describes a procedure for carrying out the method.

The following explanation describes the operation of the method in accordance with the flow chart, with the reference numbers in the text keying the description to the reference numbers in the chart:

1) Choose (arbitrary) starting vector Xo and set R=+1 to start search for a solution point. Set Flag=1.

2) Calculate Jacobian determinant, D= absolute value of ag/ax, for the current value of x vector.

3) If D is negative, go to 4; positive, go to 5; zero, go to 6.

4) Set S=-R to flag negative D and go to 7.

5) Set S=+R to flag positive D and...