I am implementing an equation solver into Omicron.

If I have an equation on the form:

f(x) = 0

If quite easy to solve using Newton-Raphson's method. This part of the equation solver works fine.

This equation system is a bit trickier:

f(x,y) = 0

g(x,y) = 0

That should be possible to solve using a similar method. But as of yet, I haven't studied any multi-dimensional analysis, so could someone please help me with an algorithm?

And what about the general case:

f1(x1,x2,...,xn) = 0

f2(x1,x2,...,xn) = 0

...

fn(x1,x2,...,xn) = 0