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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.

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1  
I suspect the "planar algebras" tag is not appropriate, although subfactor/quantum topology people would know better than I do. –  Yemon Choi Aug 18 '10 at 17:33
1  
Almost always for simultaneous nonlinear equations, the best method to use depends on the nature of $f(x,y)$ and $g(x,y)$; without insight into the geometry of your functions near the roots, or even a way to come up with good starting points, you might end up chaotically exploring the plane (which is even more of a risk if the contours of your two bivariate functions have tangencies or near-tangencies to each other). That being said, I wish to direct your attention to Acton's "Numerical Methods that Work", most especially chapter 14. You might be able to pick up something useful there. –  J. M. Aug 18 '10 at 17:38
1  
After some cursory searching: portal.acm.org/citation.cfm?id=2705 & springerlink.com/content/w72615872r512112 ; as to whether you might be able to use them, you'll have to experiment. –  J. M. Aug 18 '10 at 17:48
    
The functions are continuous, one-to-one and monotonic as far as each independent variable, but non-linear. Sometimes they are near linear with a $x^{10/9}$ behavior. –  ja72 Aug 19 '10 at 17:32
1  
Have you looked at the papers I pointed out to you? –  J. M. Aug 19 '10 at 22:12

3 Answers 3

A remarkable generalization of bisection to multiple dimensions is the subgradient method from convex optimization theory. If $f$ and $g$ are convex then $h = f^2 + g^2$ is also convex, and a simultaneous zero of $f$ and $g$ is a minimum of $h$.

Unfortunately, the subgradient method has more theoretical than practical value. But in a two-dimensional problem, it might do okay.

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You might want to consider the vector field

$ \vec{F}(x,y) = (f(x,y), g(x,y)) $

and look for sources and sinks of $\vec{F}$. I think this could be done by recursively dividing up the plane into squares and calculating the winding number of each square. If it is nonzero then you have a critical point within that square (cf Thm 2 this paper) and should divide further.

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Sorry, Dead link. –  ja72 Aug 8 '13 at 13:15
    
drats, maybe this will help: arxiv.org/abs/1004.4485 –  dranxo Aug 8 '13 at 20:37
  1. Check the pair of opposite corners to determine if zeroes lie within each of the four subdivided rectangles (zeroes can be there in more than one of them). Eg. if f(M)>0 and f(A)<0, then AEMG contains zeroes of f. Same is true also if f(G)>0 and f(E)<0.

  2. Do this for all the four sub rectangles, and for both f and g.

  3. There will be atleast one which contains zeroes for both f and g. Zoom into that and repeat.

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