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>*Reply-to*: drbob at bigfoot.com*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. >> 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 > > > > -- DrBob at bigfoot.com

**References**:**Re: Determinant problem***From:*Maxim <ab_def@prontomail.com>