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Singular and Maxima gives different solutions
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1937		 Page 1 of 1
Author:  songuke [ Thu Jun 30, 2011 5:38 am ]
Post subject:  Singular and Maxima gives different solutions
Hi,

I've a system of linear-quadratic equations to solve. I tried Maxima and it gives me the correct solution. Singular gives me a complex solution, which is not accurate. Below is my code and outputs in Maxima and Singular.

Singular:
Code:
number rax, ray, raz = 8.076651, -5.766802, 3.363392;
number rbx, rby, rbz = 6.532685, -6.184719, 4.475940;
number rcx, rcy, rcz = 7.850585, -4.163334, 4.243473;
number dax, day, daz = 10.079651, -2.603802, 0.338392;
number dbx, dby, dbz = 8.535685, -3.021719, 1.450940;
number dcx, dcy, dcz = 9.853585, -1.000334, 1.218473;
number la = 5.189281;
number lb = 2.145810;
number lc = 2.236297;
number ta = 4.788316;
number tb = 0.935814;
number tc = 4.252196;
number fab = 0.093240;
number fac = 0.512108;
number fbc = 1.061882;

poly f1 = raz*maz+ray*may+rax*max - la;
poly f2 = daz*maz+day*may+dax*max - ta;
poly f3 = rbz*mbz+rby*mby+rbx*mbx - lb;
poly f4 = dbz*mbz+dby*mby+dbx*mbx - tb;
poly f5 = rcz*mcz+rcy*mcy+rcx*mcx - lc;
poly f6 = dcz*mcz+dcy*mcy+dcx*mcx - tc;
poly f7 = -raz*rbz*maz*mbz-raz*rby*may*mbz-raz*rbx*max*mbz+la*raz*mbz
                         -ray*rbz*maz*mby-ray*rby*may*mby-ray*rbx*max*mby
                         +la*ray*mby-rax*rbz*maz*mbx-rax*rby*may*mbx
                         -rax*rbx*max*mbx+la*rax*mbx+lb*rbz*maz+lb*rby*may
                         +lb*rbx*max-la*lb+fab
           ;
poly f8 = -raz*rcz*maz*mcz-raz*rcy*may*mcz-raz*rcx*max*mcz+la*raz*mcz
                         -ray*rcz*maz*mcy-ray*rcy*may*mcy-ray*rcx*max*mcy
                         +la*ray*mcy-rax*rcz*maz*mcx-rax*rcy*may*mcx
                         -rax*rcx*max*mcx+la*rax*mcx+lc*rcz*maz+lc*rcy*may
                         +lc*rcx*max-la*lc+fac
           ;
poly f9 = -rbz*rcz*mbz*mcz-rbz*rcy*mby*mcz-rbz*rcx*mbx*mcz+lb*rbz*mcz
                         -rby*rcz*mbz*mcy-rby*rcy*mby*mcy-rby*rcx*mbx*mcy
                         +lb*rby*mcy-rbx*rcz*mbz*mcx-rbx*rcy*mby*mcx
                         -rbx*rcx*mbx*mcx+lb*rbx*mcx+lc*rcz*mbz+lc*rcy*mby
                         +lc*rcx*mbx-lb*lc+fbc
           ;

ideal I = f1, f2, f3, f4, f5, f6, f7, f8, f9;
ideal G = groebner(I);
dim(G);
vdim(G);
G;

def s = solve(I, 12);


Maxima:
Code:
ra : [8.076651, -5.766802, 3.363392];
rb : [6.532685, -6.184719, 4.475940];
rc : [7.850585, -4.163334, 4.243473];
da : [10.079651, -2.603802, 0.338392];
db : [8.535685, -3.021719, 1.450940];
dc : [9.853585, -1.000334, 1.218473];
la : 5.189281;
lb : 2.145810;
lc : 2.236297;
ta : 4.788316;
tb : 0.935814;
tc : 4.252196;
fab : 0.093240;
fac : 0.512108;
fbc : 1.061882;


ma : [max, may, maz];
mb : [mbx, mby, mbz];
mc : [mcx, mcy, mcz];

Hab : expand((ma . rb - la) * (lb - mb . ra));
Hac : expand((ma . rc - la) * (lc - mc . ra));
Hbc : expand((mb . rc - lb) * (lc - mc . rb));

eqns : [ma . ra = la, ma . da = ta, mb . rb = lb, mb . db = tb, mc . rc = lc, mc . dc = tc, Hab + fab = 0, Hac + fac = 0, Hbc + fbc = 0];
eqns; 
sols : solve(eqns, [max, may, maz, mbx, mby, mbz, mcx, mcy, mcz]);

solsf : bfloat(sols);


The ground truth solution is
Code:
ma: 0.397328 -0.288674, 0.093798
mb: -0.007418 -0.283056, 0.099119
mc: 0.459667 0.599035, 0.264318


Output of Singular:
Code:
[1]:
   [1]:
      (0.396501063225-i*0.0585142482085)
   [2]:
      (-0.292459889853-i*0.267965150227)
   [3]:
      (0.0892914674573-i*0.318934797814)
   [4]:
      (-0.0228603746306-i*0.0180139060626)
   [5]:
      (-0.380524257884-i*0.113697415565)
   [6]:
      (-0.0130221380534-i*0.130812174086)
   [7]:
      (0.467285747848+i*0.00508868957766)
   [8]:
      (0.302313482452-i*0.198182075022)
   [9]:
      (-0.0408951640129-i*0.20385360867)
[2]:
   [1]:
      (0.396501063225+i*0.0585142482085)
   [2]:
      (-0.292459889853+i*0.267965150227)
   [3]:
      (0.0892914674573+i*0.318934797814)
   [4]:
      (-0.0228603746306+i*0.0180139060626)
   [5]:
      (-0.380524257884+i*0.113697415565)
   [6]:
      (-0.0130221380534+i*0.130812174086)
   [7]:
      (0.467285747848-i*0.00508868957766)
   [8]:
      (0.302313482452+i*0.198182075022)
   [9]:
      (-0.0408951640129+i*0.20385360867)


Output of Maxima:
Code:
[[max = 3.463461b-1,may = -5.2214398b-1,maz = -1.8408099b-1,
          mbx = -2.3113602b-2,mby = -3.8212235b-1,mbz = -1.4860691b-2,
          mcx = 4.6593653b-1,mcy = 3.5486107b-1,mcz = 1.3156148b-2],

         [max = 3.973295b-1,may = -2.8866626b-1,maz = 9.3806532b-2,
          mbx = -7.4168349b-3,mby = -2.8304991b-1,mbz = 9.9125014b-2,
          mcx = 4.5966645b-1,mcy = 5.9905295b-1,mcz = 2.6433629b-1]]


Did I do something wrong in Singular? Thanks for your help.
Author:  songuke [ Fri Jul 01, 2011 9:23 am ]
Post subject:  Re: Singular and Maxima gives different solutions
Previously, the declared ring is
Code:
ring r1 = real, (max, may, maz, mbx, mby, mbz, mcx, mcy, mcz), lp;


It seems like there are something wrong when the ring is declared with real type. I switch to the following ring
Code:
ring r1 = 0, (max, may, maz, mbx, mby, mbz, mcx, mcy, mcz), lp;

and convert all floating point numbers to rational form. The solver gives solution similar to Maxima now.

The full working code is as follows:
Code:
ring r1 = 0, (max, may, maz, mbx, mby, mbz, mcx, mcy, mcz), lp;

number rax, ray, raz = 2318/287, -1459/253, 2499/743;
number rbx, rby, rbz = 2698/413, -971/157, 837/187;
number rcx, rcy, rcz = 8722/1111, -1249/300, 6501/1532;
number dax, day, daz = 1139/113, -1781/684, 223/659;
number dbx, dby, dbz = 10166/1191, -3200/1059, 695/479;
number dcx, dcy, dcz = 3163/321, -2994/2993, 686/563;
number la = 1453/280;
number lb = 1972/919;
number lc = 1836/821;
number ta = 6311/1318;
number tb = 1283/1371;
number tc = 1450/341;
number fab = 491/5266;
number fac = 571/1115;
number fbc = 2042/1923;

poly f1 = raz*maz+ray*may+rax*max - la;
poly f2 = daz*maz+day*may+dax*max - ta;
poly f3 = rbz*mbz+rby*mby+rbx*mbx - lb;
poly f4 = dbz*mbz+dby*mby+dbx*mbx - tb;
poly f5 = rcz*mcz+rcy*mcy+rcx*mcx - lc;
poly f6 = dcz*mcz+dcy*mcy+dcx*mcx - tc;
poly f7 = -raz*rbz*maz*mbz-raz*rby*may*mbz-raz*rbx*max*mbz+la*raz*mbz
                         -ray*rbz*maz*mby-ray*rby*may*mby-ray*rbx*max*mby
                         +la*ray*mby-rax*rbz*maz*mbx-rax*rby*may*mbx
                         -rax*rbx*max*mbx+la*rax*mbx+lb*rbz*maz+lb*rby*may
                         +lb*rbx*max-la*lb+fab
           ;
poly f8 = -raz*rcz*maz*mcz-raz*rcy*may*mcz-raz*rcx*max*mcz+la*raz*mcz
                         -ray*rcz*maz*mcy-ray*rcy*may*mcy-ray*rcx*max*mcy
                         +la*ray*mcy-rax*rcz*maz*mcx-rax*rcy*may*mcx
                         -rax*rcx*max*mcx+la*rax*mcx+lc*rcz*maz+lc*rcy*may
                         +lc*rcx*max-la*lc+fac
           ;
poly f9 = -rbz*rcz*mbz*mcz-rbz*rcy*mby*mcz-rbz*rcx*mbx*mcz+lb*rbz*mcz
                         -rby*rcz*mbz*mcy-rby*rcy*mby*mcy-rby*rcx*mbx*mcy
                         +lb*rby*mcy-rbx*rcz*mbz*mcx-rbx*rcy*mby*mcx
                         -rbx*rcx*mbx*mcx+lb*rbx*mcx+lc*rcz*mbz+lc*rcy*mby
                         +lc*rcx*mbx-lb*lc+fbc
           ;

ideal I = f1, f2, f3, f4, f5, f6, f7, f8, f9;
def s = solve(I, 12);


Could anybody help explain why I cannot define a ring with real data type to use with the solve function? Thanks.
Author:  hannes [ Mon Jul 04, 2011 11:14 am ]
Post subject:  Re: Singular and Maxima gives different solutions
As soon as one calls groebner or std in a ring with inexact coefficients
(like real or complex),
most probably the result will be unusable.
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