From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: solving A x = f(x)
Date: 28 Nov 2001 15:13:32 GMT
Newsgroups: sci.math.num-analysis
Summary: SOR-Newton method for some specific non-linear systems of equations

In article <5618d4bd.0111262134.73fafb0a@posting.google.com>,
 sermsak@eng.cmu.ac.th (sermsak) writes:
|> I have the following system of equations
|> 
|>   a_11 x_1 + a_12 x_2 + ... + a_1n x_n = f(x_1)
|>   a_21 x_1 + a_22 x_2 + ... + a_2n x_n = f(x_2)
|>   ...
|>   a_n1 x_1 + a_n2 x_2 + ... + a_nn x_n = f(x_n)
|> 
|> where f(.) is a nonlinear function. And n is large.
|> 
|> Since f(.) depends only on x_i, I woule like to know
|> if there may be some techniques other than newton 
|> which can solve it.
|> 
|> Thank you very much.

this is the classical case for SOR-Newton. look up 
the classical text of Ortega&Rheinboldt : solution of nonlinear 
equations in several variables which has several sections exactly on this
case. If A is positive definite and f monotonic decreasing then you might 
consider this as a gradient system of a uniformly convex function
and this offers you the possibility of using
a wealth of optimization methods suitable for large scale applications

hope this helps
peter