       Re: Strange Det function behavior.

• To: mathgroup at smc.vnet.net
• Subject: [mg65459] Re: [mg65445] Strange Det function behavior.
• From: "Carl K. Woll" <carlw at wolfram.com>
• Date: Mon, 3 Apr 2006 06:59:23 -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.
>
>
> Alexander.

I won't try to give you an explanation, but for your particular problem
there is an undocumented workaround. The function
CharacteristicPolynomial can accept a list of two matrices as its first
argument:

CharacteristicPolynomial[{m,a},t]

returns

Det[m - a t]

So, we can use CharacteristicPolynomial for your problem. For example,
here is a 12x12 matrix:

matrix = Table[ Random[]*t + i/j, {i, 12}, {j, 12} ];

Extract the {m,a} matrices:

{m,a} = {matrix /. t->0, -D[matrix,t]};

Use CharacteristicPolynomial:

In:=
CharacteristicPolynomial[{m,a},t]//Chop//Timing

Out=
11              12
{0. Second, 1.24466 t   + 0.0082288 t  }

An instantaneous result for 12x12 matrices.

Carl Woll
Wolfram Research

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