Re: Re: Determinant problem

• To: mathgroup at smc.vnet.net
• Subject: [mg55111] Re: [mg55082] Re: Determinant problem
• From: DrBob <drbob at bigfoot.com>
• Date: Sun, 13 Mar 2005 04:57:31 -0500 (EST)
• References: <d0p7d6\$j22\$1@smc.vnet.net> <200503110921.EAA02259@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```>> Product[1+t[i],{i,25}]+O[x] appears to lock up too
>> (in version 5.1.0)

And version 5.1.1, as well.

Bobby

On Fri, 11 Mar 2005 04:21:30 -0500 (EST), Maxim <ab_def at prontomail.com> wrote:

> 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.
>> 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
>
> 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
>
>
>
>

--
DrBob at bigfoot.com

```

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