FriCAS - Wish List - Solvers

I have put my complete 'wish list' on the page here. This page contains the replies to my issue about equation solvers.

Add Ability to Write Equation Solvers for a given Domain.

FriCAS has quite good support for say, systems of polynomial equations in field or OrderedRing, see numsolve.spad.pamphlet for more details.

Would it be possible to generalise this so that someone writing code for an SPAD domain can add their own equation solvers for their domain. With variables that represent elements of that domain.

So, for example, we might want to write equations where the variables are quaternions. So we need to write a solver which can handle a non-commuting algebra.

I suppose someone could write some code in SPAD to do this now? But it really needs a proper framework to do it efficiently?

Reply From Waldek

You can write solvers just now.  Existing solvers are packages
which export 'solve' function.  Of course, you need an algorithm
for given domain.  Given system of linear equations over quaternions
one can convert it to a bigger linear system over base ring
and call existing solver.  Then convert back to quaternions.
In principle similar approach may work for polynomial equations,
but even simple polynomial equations over quaternions will
produce complicated equations over base ring.  Directly working
with quaternions may produce better solver.

Anyway, the main problems is to find a method of solving for
given domain.

Reply From Me

OK thanks, I did not know this, it looks useful I'll have to experiment with it.
Do you know of any tutorials?


Perhaps the Sourceforge feature-request should be changed to the wish to have solvers
for quaternions and what I would really like: solvers for Clifford algebra (and possibly
better documentation/tutorials for writers of solvers?).


By the way, is it helpful to put feature-requests on Sourceforge? It is certainly
useful for me to find out that FriCAS has more capability that I thought. Also It
can't do any harm to have more activity on the project which could also lead to
more people finding FriCAS when searching on Sourceforge. 

Reply From Waldek

Basically, for the interpreter 'solve' is just like another function.
If you add a new one normal overloading resolution will choose
appropriate one.  You need some way of writing equations.
If you want to have '=' sign, then you can use Equation domain.
Eqation can do only a little: '=' sign on command line will
create equation, you can extract left hand side and right
hand side.  

For both sides you need some representation of terms.  For 
polynomial Polynomial is good choice, but for noncommutative
domains you need someting different.  We have 'XPolynomial'
domain which may work OK.  I write 'may work' because
'XPolynomial' assumes that variables commute with coefficients
which strictly speaking is not true if you work with equations
over noncommutative ring.  This is OK if you take coefficients
from the base ring.  And any system of algebraic equations can
be written is such a way that you never multiply variable by
noncommuting coefficient.  The problem is that if you are
not careful writing your equation, then 'XPolynomial' will
change relative order of coefficients and variables which
may lead to inequivalent system.  So, it may be wise to
create a generalization of 'XPolynomial' which preserves
relative order of variables and coefficients.

Anyway, if you have a finite dimensional algebra over a
commutative ring (like Quaternion), then you can
replace each variable from the algebra by a linear
combination of basis elements with coefficients form
base ring.  So, instead of 'x' in quaternion you will
have 'x1 + i*x2 + j*x3 + k*x4'.  Practically, this can
be done by using say 'Quaternion(Polyniomial(Fraction(Integer)))'
and defining appropriate mapping from noncommutative
polynomials (which represent your equations) to the
quaternions with polynomial coefficients.  For polynomials
there is 'PolynomialCategoryLifting' package which
helps building mappings between polynomials and a commitative ring.
Something similar would be needed form noncommutative
polynomials.

Note that the approch above is rather naive and potentially
quite inefficient.

In general case, one can use Expression to represent
equations.  The real difficulty is to have solving
algorithm.  Basically any open problem in mathematics
can be formulated as question about solutions of
aproproate equations, so it unlikely that any general
method exists.

> By the way, is it helpful to put feature-requests on Sourceforge? It is 
> certainly useful for me to find out that FriCAS has more capability that 
> I thought. Also It can't do any harm to have more activity on the 
> project which could also lead to more people finding FriCAS when 
> searching on Sourceforge.

Well, for discussion mailing list is better.  Feature-requests are
better when you have reasonably good ideas how the feature should
look like.  Main advantage of feature-request compared to
mailing list is that thing on mailing list are mixed with all
other traffic, so it is easy to forget them.  Feature-request
will stand out.

I have put some information about the SPAD mechanism for equation solvers on the page here.


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  • I have put some information about the SPAD mechanism for equation solvers on the page here.
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