Showing papers by "Leslie Greengard published in 1991"
TL;DR: In this article, an algorithm is presented which evaluates the sum of N Gaussians at M arbitrarily distributed points in $C \cdot (N + M)$ work, where C depends only on the precision required.
Abstract: Many problems in applied mathematics require the evaluation of the sum of N Gaussians at M points in space. The work required for direct evaluation grows like $NM$ as N and M increase; this makes it very expensive to carry out such calculations on a large scale. In this paper, an algorithm is presented which evaluates the sum of N Gaussians at M arbitrarily distributed points in $C \cdot (N + M)$ work, where C depends only on the precision required. When $N = M = 100,000$, the algorithm presented here is several thousand times faster than direct evaluation. It is based on a divide-and-conquer strategy, combined with the manipulation of Hermite expansions and Taylor series.
481 citations
Journal Article•
TL;DR: An algorithm is presented which evaluates the sum of N Gaussians at M arbitrarily distributed points in $C \cdot (N + M)$ work, where C depends only on the precision required.
Abstract: Many problems in applied mathematics require the evaluation of the sum of N Gaussians at M points in space. The work required for direct evaluation grows like $NM$ as N and M increase; this makes it very expensive to carry out such calculations on a large scale. In this paper, an algorithm is presented which evaluates the sum of N Gaussians at M arbitrarily distributed points in $C \cdot (N + M)$ work, where C depends only on the precision required. When $N = M = 100,000$, the algorithm presented here is several thousand times faster than direct evaluation. It is based on a divide-and-conquer strategy, combined with the manipulation of Hermite expansions and Taylor series.
475 citations
TL;DR: A numerical method for two-point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for Chebyshev nodes, and achieves superalgebraic convergence.
Abstract: A numerical method for two-point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for Chebyshev nodes. The method is stable, achieves superalgebraic convergence, and requires $O(N\log N)$ operations, where N is the number of nodes in the discretization. Although stable spectral methods have been constructed in the past, they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill-conditioning introduced by that process.
178 citations