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MathGroup Archive 2006

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Re: Strange Det function behavior.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65482] Re: [mg65445] Strange Det function behavior.
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Wed, 5 Apr 2006 06:55:21 -0400 (EDT)
  • References: <200604020900.FAA01592@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Alexander wrote:
> Dear MathGroup!
> 
> Suppose we have defined the following function:
> 
> f[x_Integer] := Det[ Table[ Random[]*t + i/j, {i, x}, {j, x} ] ];
> 
> Now try to make a table for different values of argument (matrix
> dimension):
> 
> Table[f[i], {i, 12}] // TableForm
> 
> It takes a considerable time on my Celeron 1700 with Mathematica 5.2
> under WinXP to make such a table.
> 
> Once we made this table we see very strange bihavior, all results
> before x=12 dimension are polynoms,
> but starting from x=12 result become very big and very
> strange, and in fact it's not even a polynom.
> 
> I spend several hours try to understand why system acts so strange, and
> finally came to idea that the answer is in numbers representation and
> inner
> system algorithms used to evaluate Det function.
> 
> It would be very interesting to see explanations of this result.
> 
> Thanks for your answers!
> 
> Alexander.

I realize several helpful responses have already appeared. I just wanted 
to confirm that there is indeed a cause for the behavior you observed.

Det will use explicit cofactor expansion through dimension 11 x 11. 
After that it will resort to a variant that uses one-step row reduction. 
In some cases this might leave a denominator. I'm not sure if this will 
ever happen for exact inputs but it certainly can happen for matrices 
containing approximate numbers (because cancellations will not always be 
exact).

I'll point out that the heuristics determining what method to use are 
far from perfect (or, to paraphrase an old quote on the topic, if they 
were flawless, they wouldn't be heuristics). Something on my neverending 
to-do list is to have Det take a Method option. Even then it might be a 
challenge to do it "perfectly" insofar as small variations in internals 
of various methods can lead to noticeable performance differences.


Daniel Lichtblau
Wolfram Research


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