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MathGroup Archive 2005

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Numerical solution of quadratic equations set.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58229] Numerical solution of quadratic equations set.
  • From: "Stepan Yakovenko" <yakovenko-mg at ngs.ru>
  • Date: Thu, 23 Jun 2005 05:34:22 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Dear MathGroup experts!

I've got a system of quadratic equations with many (57) 
variables. Number of equations is less (38), so there may 
be an infinite set of solutions. Also I've got an 
aproximate solution that gives a good discrepancy. I want 
Mathematica to find some solution or/and improve the 
existing one. I'm interested in real (not complex) 
solutions.

Here's what I've tried with no result:

NSolve[eq == 0, var] - gives no solutions.
FindInstance[eq==0,var,Reals] - gives no solutions.
FindRoot[] says that there's not enough equations (yes, 
there isn't, but I'm interested only in one solution).

I guess there are some options, I've no idea of, that make 
these functions work fine. Or may be I'm doing something 
wrong?

I'd be very thankful if you spend some minutes on my 
problem if you've got experience in using Mathematica 
built-in solvers.

And, of course the equations and the approximate solution 
(just CopyPaste them).

eq/.sol says that the solution is good.

eq={x1^2+x2^2+x3^2-1,x4^2+x5^2+x6^2-1,x7^2+x8^2+x9^2-1,x1*x4+x2*x5+x3*x6,
       x1*x7+x2*x8+x3*x9,
       x4*x7+x5*x8+
         x6*x9,(1*x1^1*x5^1*x9^1)+(1*x2^1*x6^1*x7^1)+(1*x3^1*x4^1*x8^1)-(1*
             x3^1*x5^1*x7^1)-(1*x2^1*x4^1*x9^1)-(1*x1^1*x6^1*
             x8^1)-(1),-174.0768153453*x1+77.1294448808052*x2-197.092581590381*
           x3+x10-x37,-174.0768153453*x4+77.1294448808052*x5-197.092581590381*
           x6+x11-x38,-174.0768153453*x7+77.1294448808052*x8-197.092581590381*
           x9+x12-x39,-0.777572718750928*x2+0.628793024018468*x3-
         x40,-0.777572718750928*x5+0.628793024018468*x6-
         x41,-0.777572718750928*x8+0.628793024018468*x9-x42,
       185.9231846547*x1+77.1294448808052*x2-197.092581590381*x3+x10-x43,
       185.9231846547*x4+77.1294448808052*x5-197.092581590381*x6+x11-x44,
       185.9231846547*x7+77.1294448808052*x8-197.092581590381*x9+x12-x45,-x1-
         x46,-x4-x47,-x7-x48,-x49+x37+x55*x40,-x50+x38+x55*x41,-x51+x39+
         x55*x42,-x52-28.6516272343591+0.0316394681497087*
           x56,-x53-270.675972456571+0.99269490646048*
           x56,-x54+47.0508868216556+0.116429234913844*x56,
       x49*x40-x52*x40+x50*x41-x53*x41+x51*x42-x54*x42,
       0.0316394681497087*x49-0.0316394681497087*x52+0.99269490646048*
           x50-0.99269490646048*x53+0.116429234913844*x51-0.116429234913844*
           x54,100*x49^2-200*x49*x52+100*x52^2+100*x50^2-200*x50*x53+100*
           x53^2+100*x51^2-200*x51*x54+100*
           x54^2,-x43-18.5269974264523+0.927403345664447*
           x57,-x44+46.9863976107822-0.0725966543355525*
           x57,-x45+84.3082419940857+0.366950623694352*x57,
       x46+0.927403345664447,x47-0.0725966543355525,x48+0.366950623694352,
       x25+0.0318722982698898*x26+20.0245619057308,
       x27+0.117286020262742*x26-78.7973944118342,
       x32-0.0782794828969912*x31-45.5361138326056,
       x33+0.395675328765874*x31-91.6389177918417};

sol={x1 -> 0.927403345664447`, x2 -> 0.0725966543355521`,
   x3 -> -0.36695062369435`, x4 -> -0.0725966543355525`,
   x5 -> 0.997265609073176`, x6 -> 0.0138213870212883`,
    x7 -> 0.366950623694352`, x8 -> 0.0138213870212883`, 
x9 -> \
0.930137736591272`, x10 -> 64.98983200206`, x11 -> 
-39.8454462776897`, x12 -> \
330.443047358837`, x13 -> 1, x14 -> 0, x15 -> 0, x16 -> 0, 
x17 -> 1, x18 ->
   0, x19 -> 0, x20 -> 0, x21 -> 1, x22 -> 0, x23 -> 0,
    x24 -> 0, x25 -> -11.3974965771025`, x26 -> 
-270.675972456571`,
     x27 -> 110.543902002013`, x28 -> 0.0316394681497087`, 
x29 -> \
0.99269490646048`, x30 -> 0.116429234913844`, x31 -> 
-18.5269974264523`, x32 \
-> 44.085830054429`, x33 -> 98.9695935895978`, x34 -> 
0.927403345664447`,
     x35 -> -0.0725966543355525`, x36 -> 
0.366950623694352`, x37 -> \
-18.5270035625587`, x38 ->
    46.9863980911134`, x39 -> 84.3082395661798`, x40 -> 
-0.28718517022215`,
     x41 -> -0.766755739222584`, x42 -> 
0.574116986661867`, x43 ->
    315.338200876642`, x44 -> 20.8516025303145`, x45 -> 
216.410464096147`, x46 \
-> -0.927403345664447`, x47 -> 0.0725966543355525`, x48 -> 
\
-0.366950623694352`, x49 -> -18.5270035625586`, x50 -> 
46.9863980911134`, x51 \
-> 84.3082395661798`, x52 -> -18.5269974264523`, x53 ->
    46.9863976107822`, x54 -> 84.3082419940857`, x55 ->
    3.3546139097961`*^-14, x56 -> 320, x57 -> 
359.999993383562`};

var = Table[ToExpression["x" <> ToString[i]], {i, 1, 57}];
 
 
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