Root of Non-Linear Eqn (Numerical Method)

[Apteronotus]

Hi,

I have a non-linear function $$F: \Re^{3}\rightarrow\Re$$. I would like to find the roots of this equation numerically, since an explicit formula cannot be derived.

As far as I am aware Newton's method can only be utilized when the domain and the range of the function are of the same degree. (i.e. $$F: \Re^{n}\rightarrow\Re^{n}$$)

Is there a method that can used for the case$$F: \Re^{n}\rightarrow\Re^{m}$$ with $$m \neq n$$?

Thanks in advance,

 

[HallsofIvy]

So you have something like f(x,y,z)= 0 to solve? I see no reason why Newton's method could not be generalized to this. Choose some starting point [itex](x_0, y_0, z_0). The tangent (hyper-)plane to u= f(x,y,z) at that point is $u= \nabla f(x_0,y_0,z_0)\cdot(x- x_0,y-y_0,z- z_0)+ f(x_0, y_0, z_0)$. Set that equal to 0 and solve for (x, y, z) as the next approximation.