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MathGroup Archive 1992

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Re: Determinant Function (Final Version?)

  • To: fateman at peoplesparc.berkeley.edu, mathgroup at yoda.physics.unc.edu
  • Subject: Re: Determinant Function (Final Version?)
  • From: David Withoff <withoff>
  • Date: Wed, 22 Apr 1992 11:29:47 -0500

Richard Fateman writes:

> There is a rather extensive literature on computing the determinant of
> a symbolic matrix.  The generation of the determinant of a matrix
> where the i,j element is a unique symbol is a rather special case.
> What you'd like is a method that takes advantage of sparseness in
> a sparse matrix; that minimizes intermediate expression growth
> (esp. in the case that Det=0) ; that maintains factored forms if
> plausible; minimizes GCD computation if the elements are rational,
> etc.
> 
> Rearranging the Mathematica computation does not affect the asymptotic
> running speed (O(n!) for an n x n).  Taking advantage of special
> structure can sometimes do much better.  Since the Mathematica book
> refuses to provide references to algorithms, people who want to know
> more will have to look elsewhere.  
>  Richard Fateman, UC Berkeley

Almost any elementary linear algebra book would be an appropriate reference
for the algorithm used by Det for general symbolic matrices.  Symbolic
determinants are expanded in minors along the row with the most zeros,
unless there is a column with more zeros.  Each non-zero minor is expanded
and cached for later use, and Together is called on the final result.

Other algorithms are used for matrices of reals, integers, rationals, etc.

Dave Withoff
withoff at wri.com





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