Maxima and IMaxima on Emacs
Sometimes (actually less frequently than I should) I try to do some maths. I tend to do more of it in teaching than in my research, as my research is mostly experimental and usually I’m using someone else’s maths. Most of my research calculations are numerical, and J works really well and efficiently for these sorts of things. However sometimes you need to use analytical maths, and when that happens and you forget the calculus you learned as an undergraduate, then computational algebra systems (CAS) can be very useful.
Probably the most well known CAS is mathematica, which became popular at the time when Macintosh GUI-based applications were coming out, but mathematica was at least in part the outgrowth of existing primarily text-based CASs such as Macsyma, Derive and others. Many of these started their lives as commercial packages, but were either discontinued because of mathematica’s dominance of the commercial space (like MathCad, which I used during my PhD) or were open-sourced. There are several open-source CASs around now, with the most common being maxima, SageMath and Fricas. Here I’m going to be talking about maxima.
Maxima is a descendent of Macsyma, a lisp-based CAS first written in the 1960s and released into the public domain. I have chosen this CAS, as it has good hooks into emacs, in particular a nice plugin that is part of the maxima package called imaxima.
There is another package called wxmaxima that contains a GUI interface to maxima, and many people use this because it has lots of nice icons and is easy to use. However, as an emacs user, I prefer to use the maxima REPL within an emacs buffer, and fortunately maxima comes with a file imaxima.el that allows all the power of maxima to be available from an emacs buffer.
Installation
First, install maxima itself. On linux systems, use your favourite package manager (eg apt on Debian systems). In my case, I use Manjaro, so I use pacman as the package manager. You can see the maxima-based packages using
pacman -Ss maxima
This produces
extra/maxima 5.46.0-2 [installed]
A sophisticated computer algebra system
extra/maxima-ecl 5.46.0-2 [installed]
ECL backend for Maxima
extra/maxima-fas 5.46.0-2
Maxima FAS module for ECL
extra/maxima-sbcl 5.46.0-2 [installed]
SBCL backend for Maxima
on my system. As I use sbcl, maxima-sbcl provides a back-end for that Lisp interpreter. Installation on my system is done typing
pacman -S maxima maxima-ecl maxima-sbcl
on the command line.
If you have the locate command installed (it can also be installed via pacman on Manjaro) then after typing
sudo updatedb
on the command line, followed by
locate imaxima.el
produces
/usr/share/emacs/site-lisp/maxima/imaxima.el
as an output. Then, to install imaxima, I put the following lines in my init.el file
;----------------------------------------------------------------------- ; imaxima - interactive computational algebra system ;----------------------------------------------------------------------- (push "/usr/share/emacs/site-lisp" load-path) (autoload 'imaxima "imaxima" "Maxima frontend" t) (autoload 'imath "imath" "Interactive Math mode" t) (setq imaxima-fnt-size "Large")
You may need to change the path based upon your locate command. These commands run the imaxima and imath functions that allow interactive use of imaxima and inline insertion of latex-quality equations in the output.
Examples
Once installed, typing M-x imaxima starts the interactive shell for imaxima. I’m not even going to try to make a maxima tutorial here, as there are already several available. Figure 1 contains an example of calculation of normal unit vectors for a vector function at a given location. Note that the equations are nicely typeset.
Figure 1: Maxima session showing vector calculations
And of course no demo would be complete without the arbitrary plotting of a pretty 3D function, in this case a hyperboloid.
Figure 2: Maxima with pretty plot
The M-x maxima command also works if the maxima package is installed (see next section), but the output is in text form, which is not as aesthetically appealing. The text below is the equivalent maxima output to that in Figure 1 for imaxima.
(%i92) r(t) := [t, cos(t), sin(t)];
(%o92) r(t) := [t, cos(t), sin(t)]
(%i93) limit(r(t), t, 2, plus);
(%o93) [2, cos(2), sin(2)]
(%i94) limit(r(t), t, 3, minus);
(%o94) [3, cos(3), sin(3)]
(%i95) diff(r(t), t);
(%o95) [1, - sin(t), cos(t)]
(%i96) define(rp(t), diff(r(t), t));
(%o96) rp(t) := [1, - sin(t), cos(t)]
(%i97) load(eigen);
(%o97) /usr/share/maxima/5.46.0/share/matrix/eigen.mac
(%i98) uvect(rp(t));
1 sin(t)
(%o98) [---------------------------, - ---------------------------,
2 2 2 2
sqrt(sin (t) + cos (t) + 1) sqrt(sin (t) + cos (t) + 1)
cos(t)
---------------------------]
2 2
sqrt(sin (t) + cos (t) + 1)
(%i99) trigsimp(%);
1 sin(t) cos(t)
(%o99) [-------, - -------, -------]
sqrt(2) sqrt(2) sqrt(2)
It does the job, but the text is harder to read, at least for me, than the nicely typeset output.
Other Computer Algebra Systems
In addition to imaxima, there is a package on ELPA called maxima, which installs a major mode for editing maxima files. Also there is an interface to the Fricas CAS, called Frimacs, which I have not used, but which is worth investigating. Fricas is descended from Axiom, another commercial CAS, and is apparently strong at automatic integration of functions.
Conclusion
There is no question, these CASs can be very useful, either for learning algebra and calculus, or for using them in mathematics, physics and engineering applications. They can be used for anything from a simple calculator alternative (although M-x calc is much better for this) to a solver of integrals, derivatives and linear algebra problems, and for plotting of functions. Often the output of calculations is not simplified in the way one might expect when doing it by hand, but there are simplifying commands that can get around those limitations. It’s also a great tool when I can’t remember my integration techniques. I’ll be trying to use it more in my maths calculations in future now that it’s set up.
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