Changes in nonlinear least squares module

[Patrick Alken <patrick.alken@Colorado.EDU>]

Hello all,

   I've been working a lot on the nonlinear least squares module lately, 
and wanted to inform everyone of some changes.

1) There is a brand new Levenberg-Marquardt solver called 'lmniel' based 
on a smoother updating procedure for the damping parameter proposed by 
Nielsen 1999. This is the algorithm implemented in the popular 'levmar' 
library. This algorithm is not as robust as the current 'lmsder' 
algorithm, since it does not use a trust region approach and has a 
simpler method of rescaling to deal with badly scaled problems. However 
there are two significant advantages in lmniel:

     a) The algorithm is much simpler than the lmsder/MINPACK algorithm 
and will be very easy to maintain going forward. Despite its simplicity, 
it has proven very effective on a large class of problems.

     b) The most important reason I wanted to include this is that in 
each iteration, the method solves the normal equation system:
             (J^T J + \mu I) dx = -J^T f
and so it does a QR decomposition of the p-by-p matrix J^T J instead of 
the full n-by-p matrix J, which lmsder operates on. This means that for 
problems with a huge number of residuals (ie: n >> p), the method is far 
more efficient than lmsder, which can sit there for hours trying to QR 
decompose the n-by-p Jacobian which could have millions of rows. The 
drawback of this approach is that the normal equations solution could be 
significantly error-prone if the Jacobian matrix is near-singular, since 
forming J^T J would make the matrix much closer to singular. In these 
cases lmsder would be preferable.

In principle, lmsder could be modified to solve the normal equations 
also, however the lmsder algorithm is rather complex and I haven't taken 
the time to fully understand it. And of course for smaller systems where 
efficiency is not a big concern lmsder is more accurate and will 
converge more quickly in many cases.

2) There is now direct support for weighted nonlinear least squares, via 
the _wset functions. Previously, the user had to take care of weighting 
themselves in their f and df evaluation functions.

3) There is a new interface fdfridge for adding ridge regularization to 
nonlinear least squares problems.

4) There is now an extensive test suite for the nonlinear least squares 
module. Previously there existed 3 tests and now there are around 40. 
These tests helped to identify a bug in the scaling procedure of lmsder, 
and they also demonstrate that lmsder is an extremely robust 
implementation. There were some questions about this on the mailing list 
and bug tracker (ie: switching to the ALGLIB or levmar algorithms). 
Levmar is now implemented as the lmniel solver for big systems, but I 
can't find any reason to doubt the quality of the lmsder solver; it 
converges on many of the test suite problems which levmar/lmniel cannot.

Patrick