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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```

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