Starlink provides access to computer algebra by supporting
the Maple package. You might have access to Maple on your
own Starlink node, but if not, you may use it on the machine
star.rl.ac.uk, if you have an account there.
If you are a Starlink user, you should apply for an account
by mailing star@star.rl.ac.uk.
Maple allows you to enter mathematical expressions using a fairly natural syntax, substitute into them, simplify them, differentiate and (with limits) integrate them, and finally go on to graph them. As an added bonus, you can also produce output in C, Fortran and LaTeX.
For example, consider the following example.
star:nxg> maple
|\^/| Maple V Release 4 (Rutherford Appleton Laboratory)
._|\| |/|_. Copyright (c) 1981-1996 by Waterloo Maple Inc. All rights
\ MAPLE / reserved. Maple and Maple V are registered trademarks of
<____ ____> Waterloo Maple Inc.
| Type ? for help.
> gi := amp * exp(-(xparam^2/sa^2)/2);
2
xparam
gi := amp exp(- 1/2 -------)
2
sa
exp is the exponential
function, so that it knows, for example, how to
differentiate it.Now define the variable xparam, and redisplay
gi.
> xparam:= cos(theta)*(xc-x0);
xparam := cos(theta) (xc - x0)
> gi;
2 2
cos(theta) (xc - x0)
amp exp(- 1/2 ----------------------)
2
sa
Then differentiate the gaussian, and assign the result to
gid. The result is something you're happy not
to have had to work out yourself.
> gid := diff (gi, theta);
2 2
2 cos(theta) (xc - x0)
amp cos(theta) (xc - x0) sin(theta) exp(- 1/2 ----------------------)
2
sa
gid := ----------------------------------------------------------------------
2
sa
If that the purpose of this was to do a calculation somewhere, you might want to code this expression in Fortran. Doing this by hand would be error-prone, but Maple can produce output in Fortran as well as this `prettyprinted' style.
> fortran (gid,optimized);
t1 = cos(theta)
t4 = (xc-x0)**2
t6 = sa**2
t7 = 1/t6
t10 = t1**2
t15 = amp*t1*t4*t7*sin(theta)*exp(-t10*t4*t7/2)optimized argument tells Maple to try to
produce Fortran code without repeated subexpressions. You
can save this to a file with the expression fortran
(gid, filename=`gaussian.f`, optimized); You can
produce output in C as well, though because the
identifierC is potentially such a common one,
you must explicity load the C library first.
> readlib(C):
> C([gf=gid],optimized);
t1 = cos(theta);
t4 = pow(xc-x0,2.0);
t6 = sa*sa;
t7 = 1/t6;
t10 = t1*t1;
gf = amp*t1*t4*t7*sin(theta)*exp(-t10*t4*t7/2);gid on the fly.
The second is that the expression for t4 is not
the most efficient -- it is very bad to use the
pow() function for raising expressions to small
integer powers: much better would be t4a=xc-x0;
t4=t4a*t4a;, as has happened automatically for
t10.You can also produce results in LaTeX
> latex(gid);
{\it amp}\,\cos(\theta)\left ({\it xc}-{\it x0}\right )^{2}\sin(\theta
){e^{-1/2\,{\frac {\left (\cos(\theta)\right )^{2}\left ({\it xc}-{
\it x0}\right )^{2}}{{{\it sa}}^{2}}}}}{{\it sa}}^{-2}
variable, and it has got all the braces matching
correctly, but it has expressed the exponential as a simple
e-to-the-power which will look rather ugly (as well, the
exponential should be written with \mathrm{e}).Leave Maple by giving the command quit.
SUN/107 provides an
introduction to Maple, and SGP/47 is a comparison of Maple and Mathematica.
Also, the Maple manual and tutorial are very clear. There
is help within Maple (type ?intro), and this
gives enough information to get you going. Maple's web
pages are at <http://www.maplesoft.com/>, but they
don't currently (December 1998) give a lot of tutorial help.
See also the example Maple program in Appendix A.7.
There's also a GUI for maple, which you can invoke with
xmaple.
As a final point, don't fall into the common trap of thinking that because you've produced your result using computer algebra, it must be right. This is as false of computer algebra as it is of numerical programming -- be inventive in thinking of cross-checks.