# Testing and Solving the Integral Equation

The testing of the integral equation applies the integral equation over each triangle edge to obtain N equations with N unknowns which can readily be solved on a computer.

For arbitrarily shaped bodies the integral operation is much more complicated than that for a straight wire. It involves several mathematically complex derivations and pitfalls to navigate around of which some are as follows:
• When the integral equations are tested, the so-called self-terms are problematic. The testing of the integral equation at or very near the same position as the unknown leads to a (near) singularity in the matrix equation. In Feko, a computationally efficient methodology is adopted to deal with this problem.
• The testing of the integral equation applies the boundary condition (zero tangential electric field all over the surface of the conductor) at discrete points. Between these points the boundary conditions are not satisfied and this deviation is denoted the “residual”. Naturally, this residual introduces deviations from the exact physical solution. One way to minimize the residual is to minimize the average residual all over the structure. For this purpose, a set of vector weighting functions are defined. Different weighting functions were proposed and the implementation in Feko is beyond the scope of this document.
Note: Minimizing the deviation from the boundary conditions is denoted the “method of weighted residuals” or more commonly, the method of moments.

The testing of Equation 2 results in a square matrix very similar to Equation 7.

This equation can be solved for the current coefficients by using LU decomposition routines.