Applying matrix rotation to the three-dimensional low-frequency multilevel fast multipole algorithm

The matrix rotation technique is applied to the three-dimensional (3-D) multilevel fast multipole algorithm (MLFMA) at very low frequencies (LF) to save the storage without increasing the order of the floating-point operations. Some symmetrical properties of the translation matrices along the z-direction for the 3-D LF-MLFMA and the static MLFMA are also derived to further reduce the memory requirements. Numerical results shown that the order of errors introduced by the matrix rotation is much smaller than the order of errors from the plain LF-MLFMA or plain static MLFMA. The implementation of the matrix rotation technique does not change the precision order of the LF-MLFMA and the static MLFMA for fixed truncation of multipole expansions.