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AN ADAPTIVE CONTINUATION PROCESS FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS

IP.com Disclosure Number: IPCOM000147996D
Original Publication Date: 1975-Jul-31
Included in the Prior Art Database: 2007-Mar-28
Document File: 25 page(s) / 1M

Publishing Venue

Software Patent Institute

Related People

Rheinboldt, Werner C.: AUTHOR [+2]

Abstract

July 1975 Technical Report TR-393 NSF-OCA-GJ-35568X-393

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Page 1 of 25

July 1975

Technical Report TR-393
NSF-OCA-GJ-35568X-393

AN ADAFTIVE CONTINUATION PROCESS FOR SOLVING
SYSTEMS OF NONLINEAR EQUATIONS

      by
Werner C. Rheinboldt

Abstract

The so-called iterative continuation approach for solving nonlinear
equations in R~ uses locally convergent iterations to proceed along the
continuation curve. The main delimiter of the parameter steps are then the
sizes of the convergence domains of the local methods. In this reyort, the
details of an earlier-announced adaptive iterative continuation process are
presented in which, in particular, the steps are adjusted on the basis of
estimates of the local attraction domains. Since effective, computable
estimates for such convergence radii are rarely available, it is natural
that here heuristic procedures play an important role. The overall process
has been applied to a variety of problems with excellent success; some
results have been given before; others are summarized here.

Invited Lecture, Semester or, ''Mathematical Models
and Numerical Methods", Stefan Banach International
blathematical Center, Polish Academy of Science,
Warsaw, Poland; to be published in the Proceedings,
of that Semester.

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   AN ADAPTIVE CONTINUATION PROCESS
FOR SOLVING SIST~IS OF NONLINEAR E~UATIONS')
by
Werner C. ~heinboldt')

1. Introduction. In general, iterative methods for solving a nonlinear

equation in Rn depend strongly on the selection of the initial data. In

order to reduce this dependence, the continuation processes use a family

of equations

(1.1) H(x,t) = 0, 0 5 t 5 1, x € R"

which for t = 1 contains the given equation. If for each t c [O, 11 a

solution x(t) of (1.1) exists that varies continuously with t, then the

function x : [0,1]

-

             R" constitutes a curve in R" between the- -assumed
to be given--point xo = x(0) and the unknown solution x* = x(1) of the

original equation. Hence iterative processes may be considered which use

the curve as a guide and channel their iterates in its proximity from

xO to the intended limit x*.

While the continuation idea itself has a long history (see, e.g.,

[4]), its first use as a numerical tool is often attributed to E. Lahaye

([ 6 ] ,

     [ 7 I) and D. F. Davidenko (see, e. g. , [ll] for some translations and
a lengthy bibliography). The Davidenko methods are based on the observa-
tion that--under suitable differentiability conditions--the unknown contin-

uation curve is a solution of the initial value problem

* -,

his work was in part supported by the National Science Fjoundation under

Grant GJ-35568X. I

')computer Science Center, University of Maryland, College Park, Id. 20742.

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Accordingly, we may approximate this curve by applying some discrete varia- ble method to (1.2). On the other hand, Lahaye's iterative continu...