Multi-Dimensional Bisection Method for finding the roots of non-linear implicit equation systems Introduction The bisection method - or the so-called interval halving method - is one of the simplest root-finding algorithms which is used to find zeros of continuous non-linear functions. This method is very robust and it always tends to the solution if the signs of the function values are different at the borders of the chosen initial interval. Geometrically, root-finding algorithms of f(x)=0f(x)=0 find one intersection point of the graph of the function and the axis of the independent variable. In many applications, this 1-dimensional intersection problem must be extended to higher dimensions, e.g.: intersections of surfaces in a 3D space (volume), which can be described as a system on non-linear equations. In higher dimensions, the existence of multiple solutions becomes very important, since the intersections of two surfaces can create multiple intersection lines. Multiple solutions The original form of the bisection method can easily be extended to find numerous roots of a non-linear equation in a given interval. If the function values are computed in an initial mesh on the examined interval, then the original method can be used for each neighboring points where the sign of the function values are different. This way, some roots may be omitted if the initial mesh is not fine enough, and even number of roots are placed inside one interval. Generalization for higher dimensions In many applications, the roots of a system of non-linear implicit equations
