There is a need for scientists and engineers to have a numerical library that:
There are strengths and weaknesses with existing libraries:
Netlib (http://www.netlib.org/) is probably the most advanced set of numerical algorithms available on the net, maintained by AT&T. Unfortunately most of the software is written in Fortran, with strange calling conventions in many places. It is also not very well collected, so it is a lot of work to get started with netlib.
GAMS (http://gams.nist.gov/) is an extremely well organized set of pointers to scientific software, but like netlib, the individual routines vary in their quality and their level of documentation.
Numerical Recipes (http://www.nr.com, http://cfata2.harvard.edu/nr/) is an excellent book: it explains the algorithms in a very clear way. Unfortunately the authors released the source code under a license which allows you to use it, but prevents you from re-distributing it. Thus Numerical Recipes is not free in the sense of freedom. On top of that, the implementation suffers from fortranitis and other limitations. [http://www.lysator.liu.se/c/num-recipes-in-c.html]
SLATEC is a large public domain collection of numerical routines in Fortran written under a Department of Energy program in the 1970's. The routines are well tested and have a reasonable overall design (given the limitations of that era). GSL should aim to be a modern version of SLATEC.
NAG and IMSL both sell high-quality libraries which are proprietary. The NAG library is more advanced and has wider scope than IMSL. The IMSL library leans more towards ease-of-use and makes extensive use of variable length argument lists to emulate "default arguments".
Forth Scientific Library [see the URL http://www.taygeta.com/fsl/sciforth.html]. Mainly of interest to Forth users.
Numerical Algorithms with C G. Engeln-Mullges, F. Uhlig. A nice numerical library written in ANSI C with an accompanying textbook. Source code is available but the library is not free software.
NUMAL A C version of the NUMAL library has been written by H.T. Lau and is published as a book and disk with the title "A Numerical Library in C for Scientists and Engineers". Source code is available but the library is not free software.
C Mathematical Function Handbook by Louis Baker. A library of function approximations and methods corresponding to those in the "Handbook of Mathematical Functions" by Abramowitz and Stegun. Source code is available but the library is not free software.
CCMATH by Daniel A. Atkinson. A C numerical library covering similar areas to GSL. The code is quite terse. Earlier versions were under the GPL but unfortunately it has changed to the LGPL in recent versions.
CEPHES A useful collection of high-quality special functions written in C. Not GPL'ed.
WNLIB A small collection of numerical routines written in C by Will Naylor. Public domain.
MESHACH A comprehensive matrix-vector linear algebra library written in C. Freely available but not GPL'ed (non-commercial license).
[JT section: written by James Theiler, and not incorporated into the above motivation]
And we furthermore promise to try as hard as possible to document the software: this will ideally involve discussion of why you might want to use it, what precisely it does, how precisely to invoke it, how more-or-less it works, and where we learned about the algorithm, and (unless we wrote it from scratch) where we got the code. We do not plan to write this entire package from scratch, but to cannibalize existing mathematical freeware, just as we expect our own software to be cannibalized.
The long-term goal will be to provide a framework to which the real numerical experts (or their graduate students) will contribute. Such contributors will have to contribute copylefted software and they cannot mind that evil profit-making companies (such as those which sell "environments") might use this software as part of their packages.