[heisig]

Maxima is a surprisingly good alternative to SymPy. We had to learn that the hard way at my job. Both systems offer similar functionality, but it turns out Maxima is typically 50-200x as fast as SymPy for many tasks (I guess because it was written at a time when computers were slow and expensive). For our application, switching from SymPy to Maxima shortened the time-to-solution from hours to minutes.

Also, it shows that Maxima has been in use for a very long time, and maintained by very bright people. I have yet to encounter a bug in Maxima. And I have already encountered plenty of bugs and inconsistent behavior in SymPy.

[defrost]

Maxima sees serious 'more free' versions of computational algorithms that migrate out from less free acedemic systems.

On the computational real actual algebra side (ie. a bit past computing a quadractic at 'x') a number of things that appear in Magma follow into Maxima (but not all the things, although with time they will come).

https://en.wikipedia.org/wiki/Hidden_Field_Equations

https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis

    The first HFE cryptosystem challenge of J. Patarin is solved by Magma in 82.5 seconds, using 1.3GB memory; with 4 threads the time taken is just 29.0 seconds. This involves solving a system of 80 quadratic equations in 80 variables over GF(2) (each input polynomial has about 1600 terms). The challenge was first solved by J.-C. Faugère in 52 hours on an 1GHz Alpha in 2002 using his unpublished F5/2 algorithm. At the time of writing, there is apparently still no publically-available software besides Magma which can solve the challenge.

http://magma.maths.usyd.edu.au/magma/overview/2/27/13/#subsu...

https://maxima.sourceforge.io/docs/manual/maxima_257.html

    Notes on the grobner package: The code is a little bit out of date now. Modern implementation use the fast F4 algorithm described in "A new efficient algorithm for computing Gröbner bases (F4)" Jean-Charles Faugère

( *NB a somewhat old example, I'm not keeping up on the comp.algebra lists and I'm assuming here the online manuals are up to date )