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Re: Determinant problem
*To*: mathgroup at smc.vnet.net
*Subject*: [mg55082] Re: Determinant problem
*From*: Maxim <ab_def at prontomail.com>
*Date*: Fri, 11 Mar 2005 04:21:30 -0500 (EST)
*References*: <d0p7d6$j22$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On Thu, 10 Mar 2005 10:26:46 +0000 (UTC), Nodar Shubitidze
<shubi at nusun.jinr.ru> wrote:
> Hi All !
>
> I have a problem with calculation of determinant of 24*24
> matrix:
> s1={{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1},
> {1,1,1,1,1,1,1,1,1,1,1,1,-t1,-t1,-t1,-t1,-t1,-t1,-t1,-t1,-t1,-t1,-t1,-t1},
> {1,1,1,1,1,1,-t2,-t2,-t2,-t2,-t2,-t2,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,-t3,-t3,-t3,-t3,-t3,-t3},
> {1,1,1,-t4,-t4,-t4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,1,1,1,-t5,-t5,-t5,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,-t6,-t6,-t6,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,1,1,1,-t7,-t7,-t7},
> {1,1,-t8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,1,1,-t9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,1,1,-t10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,1,1,-t11,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,1,1,-t12,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,-t13,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,-t14,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,-t15},
> {1,-t16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,1,-t17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,1,-t18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,1,-t19,0,0,0,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,1,-t20,0,0,0,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-t21,0,0,0,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-t22,0,0,0,0},
> {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-t23,0}};
> s2=Det[s1];
> Mathematica 5.0 on my computer (AMD Atlon XP, 1.4GHz)
> cannot calculate it after 12 hours.
> It is strange, therefore I calculate more complicated (with
> less number of zeros) 68*68 matrix during the seconds.
> Please help establish the reason.
> Best regards
> Nodar Shubitidze
>
I believe that in this case the elimination process may encounter a hidden
zero in the denominator (something like 1/(2(t1-1)-2t1+2)), which causes
Det to lock up, perhaps because numerical approximations are used. But it
is hardly possible to be certain about it; for example,
Product[1+t[i],{i,25}]+O[x] appears to lock up too (in version 5.1.0),
after taking up more than 1Gb of memory -- in that case most likely some
unnecessary simplification is taking place, and probably the same thing
happens with Det. Here's a straightforward implementation of Det that
works for your matrix:
In[2]:=
Clear[myDet]
myDet[$A_ /; MatrixQ@ $A && Equal @@ Dimensions@ $A] :=
Module[{A = $A, n = Length@ $A, s = 1, i},
Do[
i = Select[Range[j, n], !TrueQ[Together[A[[#, j]]] == 0]&, 1];
If[i === {}, Return[0, Module]];
If[(i = i[[1]]) != j,
A[[{i, j}]] = A[[{j, i}]]; s = -s];
A[[#]] -= A[[j]]*A[[#, j]]/A[[j, j]]& /@ Range[j + 1, n],
{j, n - 1}
];
s*Tr[A, Times]
]
In[4]:=
myDet[s1] // Simplify
Out[4]=
-((1 + t1)*(-1 - t10)*(-1 - t11)*(1 + t12)*(-1 - t13)*(-1 - t14)*(-1 -
t15)*(-1 - t16)*(-1 - t17)*(-1 - t18)*(-1 - t19)*(-1 - t2)*(-1 - t20)*(-1
- t21)*(-1 - t22)*(-1 - t23)*(1 + t3)*(-1 - t4)*(-1 - t5)*(-1 - t6)*(1
+ t7)*(-1 - t8)*(-1 - t9))
This uses Together to avoid the hidden zero problem, but there's no
guarantee that it'll work for more complicated expressions (alternatively
we could use a heuristic numerical check instead of Together). The result
can be simplified a little further:
In[5]:=
% /. (-1 - x_) -> s*(1 + x) /. s -> -1
Out[5]=
(1 + t1)*(1 + t10)*(1 + t11)*(1 + t12)*(1 + t13)*(1 + t14)*(1 + t15)*(1
+ t16)*(1 + t17)*(1 + t18)*(1 + t19)*(1 + t2)*(1 + t20)*(1 + t21)*(1
+ t22)*(1 + t23)*(1 + t3)*(1 + t4)*(1 + t5)*(1 + t6)*(1 + t7)*(1 + t8)*(1
+ t9)
Maxim Rytin
m.r at inbox.ru
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