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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: [mg58238] Re: [mg58229] Numerical solution of quadratic equations set.
  • From: Pratik Desai <pdesai1 at umbc.edu>
  • Date: Fri, 24 Jun 2005 02:49:57 -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
>
>  
>
The best I could do is

Reduce[Thread[eq == 0], var]

This gives you a couple of solutions. Incidently the value for x1  in 
the first solution matches what you have, now I don't know how to get 
the equations we get from reduce to further solve for the other 
variables by substituting one of the explicit solutions like x1 etc . I 
tried using cut and paste but there must be a better way to further use 
the results from Reduce.

Best regards

-- 
Pratik Desai
Graduate Student
UMBC
Department of Mechanical Engineering
Phone: 410 455 8134



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