From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: Continuous Newton's Method
Date: 18 Nov 1999 11:38:29 GMT
Newsgroups: sci.math.num-analysis

In article <3833241F.80E1E24E@unt.edu>,
 "Matthew T. Brenneman" <mtb0006@unt.edu> writes:
|> Hi,
|> 
|> Can you please tell me of a good introductory text or article on the
|> "continuous" Newton's method?
|> 
|> Thank you very much,
|> 
|> Matt Brenneman
|> 
this is 

   \dot x (t) = -(DF(x(t)))^{-1}F(x(t)),   x(0)=x_0
Papers:
    S Smale: A convergence process in price adjustment and global Newton
             methods. J. Math. Econ. 3 (1976),107--120

    F.H. Branin: A widely convergent method for finding multiple solutions
            of simultaneous nonlinear equations IBM J. Res. Develop. (1972),
            504-522

    H Th Jongen, P. Jonker and F. Twilt: On Newton flows in Optimization. 
            Math. Oper. Res. 31 (1979) 345-359

there is also newer work of Twilt on this process :
Twilt, F.
Some reflections on Newton's method. (English)
[CA] Burgh, Adriaan van der (ed.) et al., Topics in engineering mathematics. Modeling and methods. Dordrecht: Kluwer Academic Publishers, (ISBN 0-7923-2005-0/hbk). Math.
Appl., Dordr. 81, 217-237 (1992).

Jongen, H.Th.; Jonker, P.; Twilt, F.
The continuous, desingularized Newton method for meromorphic functions. (English)
[J] Acta Appl. Math. 13, No.1/2, 81-121 (1988).
Twilt, F.; Jonker, P.; Streng, M.
Gradient Newton flows for complex polynomials. (English)
[CA] Optimization, Proc. 5th French-German Conf., Varetz/Fr. 1988, Lect. Notes Math. 1405, 177-190 (1989).

see also:

Janovsky, Vladimir; Seige, Viktor
Qualitative analysis of Newton's flow. (English)
[J] SIAM J. Numer. Anal. 33, No.5, 2068-2097 (1996). [ISSN 0036-1429] http://epubs.siam.org/sam-bin/dbq/article/24185

there are applications e.g. in solving LP's:
Anstreicher, Kurt M.
Linear programming and the Newton barrier flow. (English)
[J] Math. Program., Ser. A 41, No.3, 367-373 (1988).


hope this helps
peter
==============================================================================

From: Matthew Timothy Brenneman <mtb0006@unt.edu>
Subject: Re: Fractal Artifacts
Date: Sun, 28 Nov 1999 23:19:04 -0600 (Central Standard Time)
Newsgroups: [missing]
To: Dave Rusin <rusin@math.niu.edu>

Hi Dr. Rusin!

The little bit I can tell you is this.

The motivation of the continuous Newton's method is as follows:
Standard Newton's for a complex polynomial:
(i)  z_n+1  =  z_n  -  P(z_n)/P'(z_n)
We can introduce a parameter t, to alter the original eqn as:
(ii)  z_n+1  =  z_n  - t * P(z_n)/P'(z_n)
Rearranging (ii) and taking the limit as t goes to zero gives us:
(iii)  z'(t)  =  - P( z(t) ) / P'( z(t) )
The "improved" continuous Newton's method is used to deal with 
singularities, and it reformulates (iii) as: For z0 in C, a continuous 
function z from all of R to C is sought so that:
(iv) P(z(0)) = z0  and P(z)'(t) = - P(z(t))  t in R
(The "improved" here refers to the fact that it handles singularities 
better.)


The analysis of this problem leads one to be able to prove the 
existence of domains, which are regions in which a point will converge 
to one and only one point in that region. One use of the continuous 
Newton's method has been in the analysis of solns to systems of pde's 
using steepest descent (Sobolev gradient) methods where an insufficient 
number of boundary conditions may not directly guarantee the existence 
of a unique solution. Along this line, this method has also been used 
to exploit the fact that these "mathematical" domains are
manifested in certain physical systems as real domains (like magnetic 
domains) and give insight into problems such as minimizing the 
Ginzberg-Landau functional. It turns out that some nice results are 
being done on the G-L functional that are suprising to the physicists 
(who predicted a different behavior than the math does-a very nice 
example perhaps of the math leading the physics), and it also has some 
applications to understanding the behavior of gravitons. A very 
readable, short account of the continuous Newton's method is in an 
article published by J.W. Neuberger (in the Mathematical Intelligencer, 
vol. 21(3), pgs. 18-23, 1999).

That's about all I know on the subject (as I'm still trying to learn 
more about it).

Sincerely,

Matthew T. Brenneman