Calculation of the Impedance Matrix Inner Integral of the Magnetic-Field Integral Equation to Prescribed Precision

We present a new method to evaluate the inner integral of the impedance matrix elements in the traditional Rao–Wilton–Glisson formulation of the method of moments for the magnetic-field integral equation. The evaluation is to a prescribed precision. We show that we can write the inner integral in terms of two scalar integrals: 1) a surface integral over the integration triangle (IT) and 2) a line integral over the boundary of the IT. Just as with the original integral, neither of these integrals can be evaluated analytically. In our method, we bypass this obstacle by replacing the original integrand (modified by a constant phase factor) by its Taylor series and by keeping enough terms to guarantee a number of significant digits in the integration outcome. We have accomplished this for the surface and the line integral. We present a systematic derivation of the formulas for the two integrals and we conduct extensive testing of our results.