Applications of FMM to BIEM are investigated by many authors: by Fukui and Hattori [23] and Miyakoshi et al.[60] to two-dimensional Laplace's equation, by Fukui and Mochida[28] and Helsing[42] to two-dimensional elastostatics, by Nishimura et al.[65] and Nishida and Hayami[62] to three-dimensional Laplace's equation, by Yoshida et al.[84,87], Takahashi et al.[78], Fukui and Kutsumi[27] and Fu et al.[18,19] to three-dimensional elastostatics, by Chew et al.[10], Fujiwara[21] and Iritani et al.[47] to two-dimensional elastodynamics, by Fujiwara[22] and Yoshida et al.[86] to three-dimensional elastodynamics, and by Gomez and Power[30,31], Mammoli and Ingber[57,58], Fu and Rodin[20] and Takahashi et al.[79] to fluid mechanics. BIEM are particularly suitable for wave analyses in the infinite domain. Because of this advantage, applications of FM-BIEM to large-scale wave problems, particularly to acoustical and electromagnetic scattering problems, are vigorously investigated by many researchers: by Rokhlin[70], Lu and Chew[54,55], Song and Chew[77], Fukui and Katsumoto[25] and Yoshida et al.[85].

Recently several techniques to enhance the efficiencies of FMM are proposed by several authors. Rokhlin[70,71,72] proposed such a technique called the ``diagonal form''. We shall call this technique ``Rokhlin's diagonal form''. Epton and Dembart[17] and Elliot and Board[16] also proposed similar techniques. However, Rokhlin's diagonal form is known to have numerical instabilities in dealing with static problems and low frequency problems. Hence, Greengard et al.[36,11,34] and Hrycak and Rokhlin[44] proposed techniques based on the integral representation of a fundamental solution. In this thesis we call their techniques the ``new FMM''. The new FMM resolves problems of Rokhlin's diagonal form. Besides above techniques, several other techniques to improve the efficiency are proposed by White and Head-Gordon[82], Hu et al.[45], etc.

Applications of these techniques to FM-BIEM
are found in several papers mentioned above, Koc and
Chew[50], Hu et al.[46], Gyure and
Stalzer[37], Dembart and Yip[13], Fukui and
Katsumoto[26], Nishimura et al.[64] and Yoshida et
al.[88,89]