A comparison between Fast Multipole Algorithm and Tree-Code to evaluate gravitational forces in 3-D

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Abstract
We present tests of comparison between our versions of the Fast Multipole Algorithm (FMA) and ``classic'' tree-code to evaluate gravitational forces in particle systems. We have optimized the Greengard's original version of FMA allowing for a more efficient criterion of well-separation between boxes, to improve the adaptivity of the method (which is very important in highly inhomogeneous situations) and to permit the smoothing of gravitational interactions. The results of our tests indicate that the tree-code is almost three times faster than FMA for both a homogeneous and a clumped distribution, at least in the interval of N (N< 10^5) here investigated and at the same level of accuracy (error ~ 10^{-3)). This order of accuracy is generally considered as the best compromise between CPU-time consumption and precision for astrophysical simulation. Moreover, the claimed linear dependence on N of the CPU-time of FMA is not confirmed and we give a ``theoretical'' explanation for that.