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MathGroup Archive 2005

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Re: Numerical solution of quadratic equations set.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58242] Re: [mg58229] Numerical solution of quadratic equations set.
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Fri, 24 Jun 2005 02:50:02 -0400 (EDT)
  • References: <200506230934.FAA16416@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Stepan Yakovenko wrote:
> 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}];
>  
>  
> -----------------------------------------------------------
> http://auto.ngs.ru - × ÐÒÏÄÁÖÅ ÂÏÌÅÅ 1200 Á×ÔÏ


Generally speaking one might do this with FindMinimum. Use the sum of 
squares to minimize. Take as start value the solution you already have.

eqsq = eq.eq

{min,vals} = FindMinimum[eqsq,
   Evaluate[Apply[Sequence, Transpose[{var,var/.sol}]]],
   WorkingPrecision->100, PrecisionGoal->100, AccuracyGoal->100]

But this will not do much good in your case, because the original system 
has coefficients that are only specified to machine precision. Hence you 
cannot hope to get a solution with very small residual. Had you, say, 
exact coefficients, then the method indicated above is a viable way to 
do what is called "root polishing".


Daniel Lichtblau
Wolfram Research









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