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Re: Numerical solution of quadratic equations set.
*To*: mathgroup at smc.vnet.net
*Subject*: [mg58259] Re: Numerical solution of quadratic equations set.
*From*: Andrzej Kozlowski <andrzej at akikoz.net>
*Date*: Sat, 25 Jun 2005 01:56:17 -0400 (EDT)
*References*: <200506230934.FAA16416@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On 23 Jun 2005, at 18:34, 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
>
>
I am sure that the equations as you have written them down have no
solutions. Certainly, the rationalised equations have no exact solution:
With your definition of eq we have:
In[2]:=
vars=Variables[eq];
In[3]:=
Length[vars]
Out[3]=
39
That by the way shows that there are only 39 and not 57 distinct
variables. We can rationalize the equations:
=
eq=Rationalize[eq,0];
GroebnerBasis[eq, vars,MonomialOrder->DegreeReverseLexicographic]
{1}
This shows that without any doubt the rationalised equations have no
solutions. On the other hand, since you are using MachinePrecision
numbers I would not be surprised if you could fine a "solution" that
looked like one purely due to loss of precision in the verification
process.
Andrzej Kozlowski
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