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Algorithm for Solving Systems of Nonlinear Equations

IP.com Disclosure Number: IPCOM000077729D
Original Publication Date: 1972-Sep-01
Included in the Prior Art Database: 2005-Feb-25
Document File: 2 page(s) / 22K

Publishing Venue

IBM

Related People

Brent, RP: AUTHOR

Abstract

An algorithm is given for finding the solution x* = (x*(1),...,x*(n)) of a system of n simultaneous nonlinear equations (f(i)(x = 0 (j = 1,...,n) in n real variables. Two distinguishing novel features are (i) an efficient numerical procedure for approximating by finite differences the Jacobian of the system of equations and (ii) the optimal reuse of the information from (i) in the course of the calculation. The algorithm for the procedure is given below. For a complete derivation and results of tests comparing the method to standard methods, reference is made to the IBM publication RC-3725 which is an IBM Research Report available to the public. Copies may be requested from: IBM Thomas J. Watson Research Center, P. O. Box 218, Yorktown Heights, New York, 10598.

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Algorithm for Solving Systems of Nonlinear Equations

An algorithm is given for finding the solution x* = (x*(1),...,x*(n)) of a system of n simultaneous nonlinear equations (f(i)(x = 0 (j = 1,...,n) in n real variables. Two distinguishing novel features are (i) an efficient numerical procedure for approximating by finite differences the Jacobian of the system of equations and
(ii) the optimal reuse of the information from (i) in the course of the calculation. The algorithm for the procedure is given below. For a complete derivation and results of tests comparing the method to standard methods, reference is made to the IBM publication RC-3725 which is an IBM Research Report available to the public. Copies may be requested from: IBM Thomas J. Watson Research Center, P. O. Box 218, Yorktown Heights, New York, 10598.

Given n, the parameter k is chosen so as to maximize the quantity 2(log(k+1))/n+2k+1, which can be shown to be the (Ostrowski) efficiency of the procedure. Initially there are given a trial solution x(0), a positive number h(0), the n x n matrix Q(0) = I, and the termination parameter t.

Suppose that, after the iteration, we have an approximation x(i), an orthogonal matrix Q(i), and a positive step size h(i). The method generates x(i+1), Q(i+1) and h(i+1) in the following way:

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