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Re: Comparison between Mathematica and other symbolic systems

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92534] Re: Comparison between Mathematica and other symbolic systems
  • From: Albert Retey <awnl at gmx-topmail.de>
  • Date: Fri, 3 Oct 2008 06:41:05 -0400 (EDT)
  • References: <gbt2q3$lcv$1@smc.vnet.net>

Hi,

> As part of a presentation to students, I will have to support the
> claim that "Mathematica is better than other systems when it comes to
> symbolic computations".

isn't this quite a stupid claim? Wouldn't it be necessary to have
information on what exactly people try to achieve to choose a "best"
system? And in many cases, you probably will need more than one system
to get a result: e.g. I know from other posts that there are integrals
that mathematica can't do, but another system can. Of course there are
other integrals (I'm sure a lot more :-) that mathematica can do but the
other system can't. So depending on which one you want to solve, your
rating might vary...

> Some experts in other systems will be
> giving a 15 min presentation to convince the audience of the contrary,
> and then it will be my turn.

If it is about symbolic computations only, I think it will always be
possible to find examples that another, probably more specialiced,
system can do better. If the students work on a specialiced field, e.g.
 all they do is finding Groebner bases of huge polynomial systems or
solving multiloop integrals in quantum field theory by recursion, they
will probably need specialized systems. The strength of mathematica is
that it is an all purpose system which integrates all kind of algorithms
and functionalities - not only symbolic ones - in a very consistent way.
It is very good at almost everything, but probably not the best for
every single specific task.

> At this point I am quite clueless on how to make my point across.
> First of all, I am not at all familiar with any other system's  
> symbolic toolboxes, so I don't know what are the weaknesses. Also, I
> am not sure what kind of demo could really make an impression on
> graduate students and colleagues.
> 
> Has any of you any experience on this? Ideas? Suggestions?

I would concentrate on the strong parts of mathematica, which are in my
opinion:

* It is a (well) integrated all purpose system
* I think it has an extraordinary strong language design
* It has a strong pattern matcher
* The well known symbolic algorithms are probably all
  implemented, and in some areas there might be (much) more than just
  the common standard.
* Notebooks are a very powerful way to store all kind of scientific
  information in one place along with the calculations and can even be
  a starting point for publications. Some, but not all systems have
  something comparable.
* Unlike most other CAS Mathematica is quite strong in Numerics, so you
  won't need to switch to something else when "filling in the numbers"
* Powerful visualization with Graphics(3D) and Dynamic/Manipulate in
  Version 6

When choosing a system, I think one needs to answer these questions:

 1) can the system solve the problem at hand
 2) how much effort is it to feed the problem to the system
 3) how efficient is the system in calculating the solution

when the answer to question one is true for mathematica (which I think
is true for all but some very special cases), I think it will
outperform other systems considering question 2 in almost every case
(assuming some familiarisation with its concepts). Considering question
3 the mileage may vary, but usually the skills of the person
implementing the solution are much more important than the pure
performance of the system...

hth,

albert


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