Minpack added for solving non-linear equations and non-linear least-squares

Hi!

Nice to see minimisation in maxima finally. I would like to stress  
the importance of proper maxima-level interface to minpack. In the  
past Andrej V. tried to do a simple interface to quadpack routines,  
but somehow mailing list couldn't agree on the name, so maxima still  
has no interface to numerical integration functions.

Hopefully this will not be the case this time.

In my opinion the underlying method names and routine names should be  
hidden to the user. Maxima level function should be named by "what  
they do", not "what method they use". So I'm in favour of functions  
"find_minimum" rather than "minpack_minimize" or something along  
those lines. The user should call an appropriately named function  
(this goes for quadpack also), which should then decide which method  
will be called. Ofcourse user has also the option to choose the  
method and set options if he wants to, but an usable soultion should  
be found with a simple function call.

Sometime ago, some debated, what's the purpose of numerical  
subroutines in a CAS like maxima. for non-perfomnace cricitcal  
operations (like model fitting, numerical ode integration, numerical  
quadrature ... - single calls of course) a CAS like maxima enables  
the user to formulate the problem in a straightforward way, while the  
CAS abilities can then transform the problem into a form, suitable  
for numerical solving. One could compare nonlinear fitting in  
Mathematica and MATLAB to compare the two. In Mathematica, one simply  
writes the model in its mathematical form, while in MATLAB one has to  
provide a function/program for the fitting routine to evaluate. Also  
ODE integration - numerical integrators integrate a system of first  
order ODEs. In matlab, one has to provide the rhs of such a system as  
is usual with numerical solvers. If Maxima had a numerical ode  
solver, one could state ode system as usual equations, while the  
symbolic abilities of Maxima would take care of converting the user's  
(mathematically looking) ode to a system of 1st order odes, compute  
the solution using some standard algorithm and return the appropriate  
solution (interpolating function preferably). Using a CAS for some  
numerical tasks is much better than fortran-in-disguise a.k.a matlab/ 
octave. With todays computers, performance is often not an issue.

I suggest the following naming:
- maxima (user) level functions with names according to what they do:  
"find_minimum" (for iterative numerical minimum searching, where  
local minimums can be found, consistent with find_root)

- lisp level functions with names suggesting the underlying algorithm/ 
fortran name: "minpack-lmderl", "lapack-deesgv", "quadpack-qagi" (at  
least for fortran numerical packages, where it's nice to know which  
code is in maxima) etc.

Another important thing is to make use of maxima sybolic abilities,  
to make the formulation of the problem as mathematically simple as  
possible (jacobian can be calculated automaticly as suggested, or  
optionallly user defined).

I'm still slowly working on a fitting package - looks like i won't  
have to implement levenberg myself if minpack works.

I would also suggest agains naming packages via their fortran names.  
A package "load(optimization)$" should include minimization, perhaps  
fitting, linearprogramming/simplex functions imo. I think the user  
has a right of not getting to know the names of fortran numerical  
libraries, when he wants a simple numerical approximation of an  
integral/ fit a function. Maxima should provide that.

Hope someone writes nice wrapping functions for minpack.

Ziga