Re: Symbolic inverse of a 12x12 matrix

Madhusudan Singh wrote:
> Thanks for your response.
>
> > A 12x12 matrix would have 144 entries so at the naive level an
> > expression for the inverse would involve the 144 unknowns.
> >
>
> No. Nearly 2/3 of the entries are zero, and the terms that remain are
all
> functions of 4-6 variables.
>
> Unfortunately, the matrix is not tridiagonal, or UT, or LT or any
simple
> form like that.
>
> > To cover all the cases the will be 12! (12 factorial) terms in
> > the expression. Each term will be the product of 12 variables.
> > Remember this will just be a determinant expansion.
>
> Please read above. I should have made this clearer earlier.
>
> >
> > The long execution time for the symbolic inversion is just
> > the computers subtle way of trying to point out that it
> > just might no be the most sensible thing to do.
> >
>
> I think it is, in light of the above. Awaiting further suggestions.


Unless the structure is in some way VERY nice, a 12x12 symbolic
inversion is not to be in your near future. In Mathematica or
otherwise.

Things you might try:

(1) Inverse[matrix, Method->OneStepRowReduction]

(2) Have a look at

http://groups-beta.google.com/group/sci.math.symbolic/browse_thread/thread/774f4a4a83bc0cf5/a3290d56d39612a1?q=matrix+Inverse+Lichtblau&rnum=4#a3290d56d39612a1

The method indicated there, using interpolation, will be difficult to
adapt because you have multivariate entities. In general this will
involve need for a sparse polynomial interpolator, which is not
terribly easy to implement. And i'm not certain offhand if you can
compute the needed information for a determinant computation in the way
it is found for e.g. Zippel's sparse multivariate gcd algorithm.

(3) Work with the numeric matrices. I put this last because I figured
only after you saw (1) and (2) would you change your mind about the
numeric approach. It is almost certainly the appropriate method. Note
that you can check conditioning for each system you solve, hence get a
fair assessment of numerical problems. In contrast, a huge symbolic
result involving rational functions might have horrible numeric
instabilities in evaluating at finite precision, either from
cancellation error or from a denominator expression with vanishing set
that comes close to many of the points you might find of interest.


Daniel Lichtblau
Wolfram Research

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