[Date Index]
[Thread Index]
[Author Index]
Re: Strange Det function behavior.
*To*: mathgroup at smc.vnet.net
*Subject*: [mg65477] Re: Strange Det function behavior.
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Mon, 3 Apr 2006 06:59:55 -0400 (EDT)
*Organization*: The University of Western Australia
*References*: <e0o513$1to$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <e0o513$1to$1 at smc.vnet.net>,
"Alexander" <beginning.physst at mail.ru> wrote:
> 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.
It is not suprising that this is is time-consuming. Effectively you are
computing the _symbolic_ determinant of n x n matrices involving t.
Defining
mat[n_] := Table[a[i,j], {i, n}, {j, n}]
rat[n_] := Table[i/j, {i, n}, {j, n}]
sign[n_] := Table[(-1)^(i + j), {i, n}, {j, n}]
then, effectively, the determinant you are computing can be expressed as
Det[ mat[n] t + rat[n] ] == Det[ mat[n] ] t^n + Total[
Map[Reverse, Minors[ mat[n] ], {0, 1}] * rat[n] * sign[n], 2] t^(n-1)
as you can easily confirm for small n. Note that
Map[Reverse, Minors[mat], {0,1}]
makes the (i,j)-th element of Minors[mat] correspond to deleting the
i-th row and j-th column of mat (from the online help for Minors).
The important point about the right-hand side of this expression is that
no symbolic matrix computations are required. Hence your original
function can be written as
f[n_Integer] :=
Module[ { m = Table[Random[], {i, n}, {j, n}]}, Det[m] t^n +
Total[Map[Reverse, Minors[m], {0, 1}] * rat[n] * sign[n], 2] t^(n-1)]
Now the time to compute f[n] is quite reasonable.
> 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.
Are you sure? I think that the determinant has not been expanded out.
> 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 might have been better spending this time on obtaining a better
representation for the original function!
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
Prev by Date:
**Re: Typing special symbols in text mode?**
Next by Date:
**Re: simplify a trig expression**
Previous by thread:
**Re: Strange Det function behavior.**
Next by thread:
**Re: Convex Hull 3D**
| |