unknown user
05-20-1991, 10:47 PM
Dear Biomch-L readers,

Yesterday, Paolo de Leva asked about software
for solving overcomplete systems of nonlinear equations. Overcomplete (or
overdetermined) means that there are more equations than unknowns, and an
exact solution does not exist. Instead, the sum of squares of N nonlinear
functions of M variables is minimized. For this type of problem, I have
has some success with LMDIF, a subroutine from the Netlib library. The
application was least squares fitting of a model to experimental data
(N>>M). LMDIF uses the Levenberg-Marquardt algorithm, which is described
by Levenberg as a generalization of the Newton-Raphson method. For my
application, LMDIF performed much better than SVDMIN. SVDMIN is a routine
from ACCULIB (a local numerical library of our computer center), and is
based on singular value decomposition of the NxM Jacobian matrix.
Both routines are written in Fortran.

LMDIF can be obtained by sending a request to the fileserver netlib@ornl.gov.

send index
send index from minpack
send lmdif1 from minpack

Specify sminpack instead of minpack if you need the single precision version.

Literature (LMDIF):
K. Levenberg (1944) A method for the solution of certain non-linear problems
in least squares. Quart.Appl.Math. 2:164-168.
D.W. Marquardt (1963) An algorithm for least squares estimation of nonlinear
parameters. J.Soc.Industr.Appl.Math. 11:431-441.
J.H. Wilkinson and C. Reinsch (1971) Linear Algebra. Springer Verlag, Berlin.
pp. 134-151. (deals only with the singular value decomposition).

I hope this information is useful for Paolo and others.

-- Ton van den Bogert
Dept. of Veterinary Anatomy
University of Utrecht, Netherlands.