How i can solve the "Non-Linear Equation" with "TWO Unknown variables" ?

Hi everyone, I want to solve the"Non-Linear Equation" with "TWO Unknown variables". In my equation i have two eq. Eq7 & Eq.10 non- linear equation and 2 unknown variables (x,y) are there.

>> Eq7=(((1+2*x^2)./(1-x^2))+(4*(1-2*y^2)*sqrt(1-x^2)*(2+x^2))./((1-x^2)*(1-y^2)^0.5*(2+y^2))); >> Eq10=(((4*(y*(2+x^2)*(1-x^2)*sqrt(pi^2*(1-y^2)+4*y^2)))./(x*(2+y^2)*(1-y^2)*sqrt(pi^2*(1- x^2)+4*x^2))-1)); I would like to solve these equations with Newton-Raphson Method. DETAILED INFORMATION IS AVAILABLE IN ATTACHED "PDF". Kindly help me in this regard. Thanks,

SANDEEP SONI,Surat,India E-mail ID - [e-mail]

[John D'Errico]

Why do you think that Newton-Raphson is what is needed to solve them? What if some other method is better? Or is this your homework assignment that you wish us to do for you?

For example, is there a good reason why you would not just use fsolve to solve the problem?

Answer 1

[John D'Errico]

By the way, these functions are rather nasty looking, full of discontinuities and singularities. Newton-Raphson will fail rather miserably.

[xx,yy] = meshgrid(-3:.01:3);

f10 = matlabFunction(Eq10)
f10 = 
    @(x,y)(y.*(x.^2-1.0).*(x.^2+2.0).*1.0./sqrt(x.^2.*(-5.869604401089358)+9.869604401089358).*sqrt(y.^2.*(-5.869604401089358)+9.869604401089358).*4.0)./(x.*(y.^2-1.0).*(y.^2+2.0))-1.0

f7 = matlabFunction(Eq7)
f7 = 
    @(x,y)-(x.^2.*2.0+1.0)./(x.^2-1.0)-(1.0./sqrt(-x.^2+1.0).*(x.^2+2.0).*1.0./sqrt(-y.^2+1.0).*(y.^2.*8.0-4.0))./(y.^2+2.0)

[xx,yy] = meshgrid(-3:.01:3);

z7 = f7(xx,yy);
z7(imag(z7) ~= 0) = NaN;

z10 = f10(xx,yy);

surf(xx,yy,z7)

surf(xx,yy,z10)

I'm sorry, but if you honestly think that Newton-Raphson has a snowball's chance in hell on this problem, then try thinking again.

xy = vpasolve(Eq7,Eq10,[x,y]);

xy.x
ans =
- 0.12987020281246053513502230527952 - 1.2712758599621447488655663518207i

xy.y
ans =
- 0.56101002772745884082718756405031 - 0.54153276698206277641963799470431i