A new version of the fast multipole method (FMM) for the evaluation of potential fields on the line is introduced. It uses a new representation of potential fields, based on generalized Gaussian quadratures [J. Ma, V. Rokhlin, and S. Wandzura, SIAM J. Numer. Anal., 33 (1996), pp. 971--996; N. Yarvin and V. Rokhlin, SIAM J. Sci. Comput., 20 (1999), pp. 699--718]. In this representation, most translation operators are diagonal. To efficiently incorporate this representation into the FMM, an apparatus is introduced for transforming between different types of expansions. This apparatus is highly general and is based on formulae for the least squares approximation of linear operators. The performance of the method is illustrated with several numerical examples; it is roughly twice as fast as previously published algorithms.
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Yarvin et al. (1999) studied this question.
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