cross post in StackOverflow
I need an algorithm to perform a 2D bisection method for solving a $2$x$2$ non-linear problem. Example: two equations $f(x,y)=0$ and $g(x,y)=0$ which I want to solve simultaneously. I have very familiar with the 1D bisection ( as well as other numerical methods ). Assume I already know the solution lies between the bounds $x_1 < x < x_2$ and $y_1 < y < y_2$.
In a grid the starting bounds are:
^ | C D y2 -+ o-------o | | | | | | | | | y1 -+ o-------o | A B o--+------+----> x1 x2
and I know the values at $f(A)$, $f(B)$, $f(C)$ and $f(D)$ as well as $g(A)$, $g(B)$, $g(C)$ and $g(D)$. I might even know for which edges $f=0$ and for which $g=0$.
To start the bisection I guess we need to divide the points out along the edges as well as the middle.
^ | C F D y2 -+ o---o---o | | | |G o o M o H | | | y1 -+ o---o---o | A E B o--+------+----> x1 x2
Now considering the possibilities of combinations such as checking if $f(G)*f(M)<0$
AND $g(G)*g(M)<0$ seems overwhelming. Maybe I am making this a little too complicated, but I think there should be a multidimensional version of the Bisection, just as Newton-Raphson can be easily be multidimed using gradient operators.
Any clues, comments, or links are welcomed.