Ben Petschel

Richard/Nathan, thanks for the suggestions - I've submitted a new update for fr.m and freduce.m. I got around the performance issue in fr.m by using a persistent variable to ensure that the calls to superiorto are done only once.

Nice work, these should be included in the core MATLAB.

Since erf rapidly goes to infinity along the i axis (e.g. erfz(1i*30) = 1i*Inf in floating point), it would be useful to have a function that calculates exp(z^2)*erf(z) or z*exp(z^2)*erfc(z).

Ok, to do that you'd need to define a total ordering on the polynomials by partitioning R[x] into P, -P and {0}, so p(x)>0 if p(x) is in P. See Lang's Algebra chapter 11 (real fields) for examples and details on the theory - e.g. the monomial orderings used for Groebner basis calculation are valid. This way you have sign(p)=1 if p is in P, etc, and similarly abs(p)=sign(p)*p.

Also I forgot to mention that MOD is required, but once this is defined then it's easy to write a GCD function. To define MOD you just need to define the representative elements of the cosets R[x]/p(x), such that MOD(q,p)>=0.

If you're willing to put in the effort implementing this, I'd be keen to see the results, but otherwise you're probably better off with a professional package such as Mathematica (which has an affordable home-use version) or the Symbolic Toolbox.

One more thing, even with my change, I still find that about 60% of the time in fr.m is spent on the line

if exist('vpi', 'class')

at least according to the profiler. So if anyone does not have this toolbox, I'd advise just commenting this whole test out.

I suggest modifying fr.m to check for the existence of the vpi class before calling superiorto like follows:

if exist('vpi', 'class')
try
superiorto('vpi')
catch
% in case VPI toolbox has never been used
end
end

the reason is that if you call fr.m many times and don't have the vpi toolbox, the call to superiorto('vpi') is incredibly slow, and in fact takes up most of the execution time.