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November 19th, 2008, 05:28 PM   #1
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check if set of nonlinear equ. has only the trivial solution

Hi guys,
When I solve equations 3,4,5,6 I'm getting small values for x(1),x(2),x(3),x(4). Using fsolve in matlab I tried different starting values x0, and solutions are in the order of 1e-3 and depend on the starting value. How can I tell if the small values are approximating the trivial solution (because of limited accuracy during the calculation), or if there are several solutions close to the trivial solution?
Since I know that there is only the trivial solution when using equation 1,2,3,4,5,6
I calculated x solving all equations and found x in the order of 1e-9. In summary,
I want to find out if there is only the trivial solution when solving equations 3,4,5,6.

c2=-0.27215029;
c3=0.09718148;
c5=2.62373636;
c6=0.80306059;
a1 = -0.2525
a2 = 0.0543
a3 = -0.0264

(1) c2*x(1)+c3*x(3)=0
(2) c2*x(2)+c3*x(4)=0

(3) c5*x(1)+c6*x(3)=0
(4) c5*x(2)+c6*x(4)=0

(5) a1*x(1)*x(2)+a2*x(3)*x(4)-a3*(x(1)*x(4)+x(2)*x(3))=0
(6) a1*(x(2)^2-x(1)^2)+a2*(x(4)^2-x(3)^2)-2*a3*(x(2)*x(4)-x(1)*x(3))=0

Best Regards,
Peter


matlab code:

x0 = [rand(1);rand(1);rand(1);rand(1)];
options=optimset('Display','iter','TolFun',1e-20 ); % Option to display output
[x,fval] = fsolve(@my_fun,x0,options);

function F=my_fun(x)
c2=-0.27215029;
c3=0.09718148;
c5=2.62373636;
c6=0.80306059;
a1 = -0.2525
a2 = 0.0543
a3 = -0.0264

F=[ c2*x(1)+c3*x(3);
c2*x(2)+c3*x(4);
c5*x(1)+c6*x(3);
c5*x(2)+c6*x(4);
a1*x(1)*x(2)+a2*x(3)*x(4)-a3*(x(1)*x(4)+x(2)*x(3));
a1*(x(2)^2-x(1)^2)+a2*(x(4)^2-x(3)^2)-2*a3*(x(2)*x(4)-x(1)*x(3))];
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