Help with very very slow determinant calcuation

If the matrix entries are all polynomials in a single variable, say z, 
then you can use modular arithmetic and the "Chinese Remainder Theorem"
to find the symbolic determinant, if it is a polynomial of modest degree.

You compute the determinant several times modulo prime numbers p1, p2, 
... , pn.
Then you use the CRT to determine the polynomial of degree n-1 that 
assumes those values.

This can also work with polys in several variables.   You have managed 
to not describe your problem
sufficiently to actually discuss it from a symbolic standpoint.  From 
this standpoint what matters is the
structural complexity of the entries:  how many variables, polynomial or 
rational, degree, perhaps sparsity.
It might be easier if you just put the matrix on a file and gave people 
a link to it.  Or perhaps just posted
it if it is small enough.
RJF
On 8/7/2011 6:28 AM, Nghia Ho wrote:
> I think you might be right, I'm probably trying to calculate something 
> very impractical mathematically. I was following an algorithm from a 
> computer vision paper and was hoping to keep everything symbolic so I 
> could write a function that calculates the result in "one shot" 
> (albeit some monstrous equation). The 10x10 matrix that they have ends 
> up having mostly numerical values mixed with a single polynomial 
> symbol (z or z^2), which is probably practical to calculate. Of 
> interest is that they note that higher order polynomial terms (> 10) 
> end up canceling each other out, which made me optimistic that it was 
> possible to keep it all symbolic. But I guess the intermediate steps 
> that Maxima has to do will take an extremely long amount of time 
> before it can do any cancellation of terms.
>
>     ------------------------------------------------------------------------
>     *From:* Barton Willis <willisb at unk.edu>
>     *To:* nghiaho12 at yahoo.com
>     *Cc:* bernard at marcade.biz; cladelpino at gmail.com;
>     maxima at math.utexas.edu
>     *Sent:* Sunday, 7 August 2011 8:18 PM
>     *Subject:* Re: [Maxima] Help with very very slow determinant
>     calcuation
>
>     The cost for the function determinant is comparable to the cost of
>     a LU factorization. So, no
>     computing the determinant by means of the LU factorization is
>     unlikely to help. Nghia Ho already
>     tried setting ratmx to true and newdet. And that's about all that
>     might speed up the
>     calculation.
>
>     For symbolic matrices, the standard big oh time estimates are
>     misleading. The time for addition,
>     for example, isn't constant, but the time depends on the
>     complexity of the expressions.
>
>     It's likely that if Maxima was able to find the determinant of
>     this 10x10 symbolic matrix, the
>     result would be so huge that it wouldn't be useful. I would
>     suggest a different approach.
>
>     --Barton
>
>     -----maxima-bounces at math.utexas.edu
>     <mailto:-----maxima-bounces at math.utexas.edu> wrote: -----
>
>     Can you elaborate more on how factorizing the matrix can help? My
>     background isn't in mathematics, I only did basic engineering
>     level maths.
>
>
>
>
>
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