The computational error of the multilevel fast multipole algorithm is studied. The error convergence rate, achievable minimum error, and error bound are investigated for various element distributions. We will discuss the boundary between the large and small buffer cases in terms of machine precision. The needed buffer size to reach double precision accuracy will be clarified.

This paper describes a method for the fast evaluation of the Sommerfeld integrals for modeling a vertical dipole antenna array in a borehole. When we analyze the antenna inside a medium modeled by multiple cylindrical layers with the Method of Moment (MoM), we need a Green's function including the scattered field from the cylindrical boundaries. We focus on the calculation of Green's functions under the condition that both the detector and the source are situated in the innermost layer, since the Green's functions are used to form the impedance matrix of the antenna. Considering bounds on the location of singularities on a complex wave number plane, a fast convergent integration path where pole tracking is unnecessary is considered for numerical integration. Furthermore, as an approximation of the Sommerfeld integral, we describe an asymptotic expansion of the integrals along the branch cuts. The pole contribution of TM01 and HE11 modes are considered in the asymptotic expansion. To obtain numerical results, we use a fast convergent integration path that always proves to be accurate and efficient. The asymptotic expansion works well under specific conditions. The Sommerfeld integral values calculated with the fast evaluation method is used to model the array antenna in a borehole with the MoM. We compare the MoM data with experimental data, and we show the validity of the fast evaluation method.