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## 3.1 Computer algebra

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

We start up Maple, and enter an expression for a gaussian. Maple makes an attempt to display the result intelligibly. If we had ended the expression with a colon rather than a semicolon, Maple would have suppressed the display. Note that undefined variables represent themselves, and that Maple knows that 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)
The 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);
There are two things to note here. The first is that we have renamed the expression 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}
Maple has done the correct thing with the cosine and sine functions, and with the 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.

Next: 3.2 Data visualisation
Up: 3 Theory support
Previous: 3 Theory support
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