A fast algorithm for particle simulations
Journal of Computational Physics
A fast algorithm for the evaluation of Legendre expansions
SIAM Journal on Scientific and Statistical Computing
Fast algorithms for polynomial interpolation, integration, and differentiation
SIAM Journal on Numerical Analysis
Matrix computations (3rd ed.)
A fast spherical filter with uniform resolution
Journal of Computational Physics
Journal of Computational Physics
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
A fast spherical harmonics transform algorithm
Mathematics of Computation
Using a Markov network model in a univariate EDA: an empirical cost-benefit analysis
GECCO '05 Proceedings of the 7th annual conference on Genetic and evolutionary computation
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The fast multipole method (FMM), which is originally an algorithm for fast evaluation of particle interactions, is also effective for accelerating several numerical computations. Yarvin and Rokhlin proposed "generalized" FMM using the singular value decomposition (SVD), which gives the optimum low-rank approximation. Their algorithm reduces the computational costs of the FMM evaluation and frees the FMM from analytical approximation formulae. However, the computational complexity of the preprocessing for an N × N matrix is O(N3) because of the SVD, and it requires orthogonal matrices of the low-rank approximations. In this paper we propose another preprocessing algorithm for the generalized FMM. Our algorithm runs in time O(N2) even with the SVD and releases the low-rank approximations from orthogonal matrices. The triangularization by the QR decomposition with sparsification, which reduces the costs of the FMM more than the diagonalization, is enabled. Although the algorithm by Yarvin and Rokhlin can be accelerated to O(N2) using the QR decomposition, our preprocessing algorithm outperforms it in fast spherical filter, fast polynomial interpolation and fast Legendre transform.