title: Multivariate Bisection Method
body: <p>I need an algorithm to perform a 2D bisection method for solving a 2x2 non-linear problem. Example: two equations <code>f(x,y)=0</code> and <code>g(x,y)=0</code> which I want to solve simultaneously. I am very familiar with the 1D bisection ( as well as other numerical methods ). Assume I already know the solution lies between the bounds <code>x1 &lt; x &lt; x2</code> and <code>y1 &lt; y &lt; y2</code>.</p>
  <p>In a grid the starting bounds are:</p>
  <pre><code>    ^
      |   C       D
  y2 -+  o-------o
      |  |       |
      |  |       |
      |  |       |
  y1 -+  o-------o
      |   A       B
      o--+------+----&gt;
         x1     x2
  </code></pre>
  <p>and I know the values <code>f(A), f(B), f(C) and f(D)</code> as well as <code>g(A), g(B), g(C) and g(D)</code>. To start the bisection I guess we need to divide the points out along the edges as well as the middle.</p>
  <pre><code>    ^
      |   C   F   D
  y2 -+  o---o---o
      |  |       |
      |G o   o M o H
      |  |       |
  y1 -+  o---o---o
      |   A   E   B
      o--+------+----&gt;
         x1     x2
  </code></pre>
  <p>Now considering the possibilities of combinations such as checking if <code>f(G)*f(M)&lt;0 AND g(G)*g(M)&lt;0</code> 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.</p>
  <p>Any clues, comments, or links are welcomed.</p>