R has a larger installed user base than Matlab/Octave?
That is pretty surprising to me... are there user
statistics somewhere? Is this the Xlisp-stat system
underneath it? Can you provide a URL for it?
I'm not surprised that one can come up with a better
design than Matlab .. it was originally pretty much
an ad hoc design. On the other hand, it has grown
especially in terms of add-on packages and popularity.
To me the interest in a matrix (or quadrature...)
package joined to maxima is the possibility of a
relatively smooth transition from the traditional
domain of single/double precision floats to
(a) arbitrary precision floats (e.g. GMP, ARPREC)
(b) exact rationals
(c) symbolic
Thus comparing quadpack to what can be done in
Mathematica, we see
(a) competent numerical quadrature with single/double
vs.
(b) adaptive precision numerical quadrature with
(perhaps) guaranteed error bounds based on (perhaps)
symbolic understanding of analytic functions.
Romberg integration in maxima (which I initially
translated from my numerical analysis course lab
exercise in 1968 or so) was never intended to be
the final word in quadrature; it was certainly never
supposed to be used at or near singular points.
There is also a bigfloat version of it, I think.
Subsequently, other people elaborated on romberg instead
of using a more sophisticated approach.
The bigfloat approach, coupled with an attempt to
understand the nature of the integrand symbolically,
is an interesting (although somewhat research-y)
approach to integration in maxima.. it may
not be faster than quadpack, but addresses the issue
using tools uniquely available to symbolic
computing systems. I think that Mathematica tries
to do the latter.
RJF