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