This is a simple introduction to fast multipole methods for the N-body summation problems. Low rank approximation plus hierarchical decomposition leads to fast O(N) or O(N logN) algorithms for the summation problem or equivalently the computation of a matrix-vector product. Code for 1-D problem is presented for a better illustration. We consider a summation problem in the form:

This paper presents a flexible mixture model (FMM) for collaborative filtering. FMM extends existing partitioning/clustering algorithms for collaborative filtering by clustering both users and items together simultaneously without assuming that each user and item should only belong to a single cluster. Furthermore, with the introduction of ‘preference’ nodes, the proposed framework is able to e...

Preface The authors would like to thank V. Rokhlin both for discovering much of the content of later parts of these notes and for elaborating the material for us while we prepared the notes. We would also like to thank R. R. Coifman and L. Greengard for many discussions related to the content of these notes, and S. Zucker for proposing that we develop the course which stimulated us to produce t...

In this paper, we focus on the truncation error of the multipole expansion for the fast multipole method and the multilevel fast multipole algorithm. When the buffer size is large enough, the error can be controlled and minimized by using the conventional selection rules. On the other hand, if the buffer size is small, the conventional selection rules no longer hold, and the new approach which ...

A theoretical method for the systematic definition and determination of Cartesian and spherical electromagnetic (onshell) formfactors and multipole moments for particles or composite systems from electromagnetic Breit-frame current distributions is presented. The method presented is free of sign ambiguities and is not based on the underlying analytical substructure of the electromagnetic curren...