$\mathcal{O}(1)$ Computation of Legendre Polynomials and Gauss--Legendre Nodes and Weights for Parallel Computing

Bogaert, Ignace; Michiels, Bart; Fostier, Jan

doi:10.1137/110855442

Cited by 49 publications

(76 citation statements)

References 10 publications

(19 reference statements)

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“…As said before, expansions (2.24) and (3.8) are only useful for n sufficiently large, such that tabulated values should be used for small n. Because the tables from [10] can be reused here, tabulation has been chosen for all n ≤ 100. With this choice, M = 3 in (2.24) and (3.8) is sufficient for obtaining machine precision.…”

Section: Numerical Results For Double Precision the Results Depicted Inmentioning

confidence: 99%

“…As explained in [10], the nodes exhibit quadratic clustering near the points ±1 for large n. This leads to the conclusion that, in a fixed-precision floating point format, it is numerically advantageous to compute not x n,k but rather θ n,k such that…”

Section: Expansion For the Gl Nodesmentioning

confidence: 98%

“…This approach can be used to get arbitrarily high-order corrections to the initial approximation, such that the error on the GL nodes can be controlled for n sufficiently large. For small n, the expansions cannot be used and a tabulation strategy similar to [10] is adopted. expansion used in [8] (equation 3.12):…”

Section: Expansion For the Gl Nodesmentioning

confidence: 99%

“…the same code runs for all relevant nodes and weights in a GL quadrature rule. This is in contrast with [10] and [8], where two separate expansions for the Legendre polynomials were used to cover the full range [−1, 1]. In a parallel computing environment, this can be very advantageous for achieving a good load balancing.…”

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confidence: 99%

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Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

Bogaert¹

2014

SIAM J. Sci. Comput.

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Abstract. Gauss-Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss-Legendre weights is derived. Together, these two expansions provide a practical and fast iteration-free method to compute individual Gauss-Legendre node-weight pairs in O (1) complexity and with double precision accuracy. An expansion for the barycentric interpolation weights for the Gauss-Legendre nodes is also derived. A C++ implementation is available on line.

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Section: Numerical Results For Double Precision the Results Depicted Inmentioning

confidence: 99%

Section: Expansion For the Gl Nodesmentioning

confidence: 98%

Section: Expansion For the Gl Nodesmentioning

confidence: 99%

mentioning

confidence: 99%

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Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

Bogaert¹

2014

SIAM J. Sci. Comput.

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“…L. For the spherical Hankel functions, such libraries already exist (e.g., Amos). For the Legendre polynomials, such a method has recently been developed in [5].…”

Section: Parallel Algorithmmentioning

confidence: 99%

Scalable parallel computation of the translation operator in three dimensions

Michiels

Bogaert

Fostier

et al. 2013

CEM'13 Computational Electromagnetics International Workshop

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Abstract-We propose a novel algorithm for the parallel, distributed-memory computation of the translation operator in the three-dimensional Multilevel Fast Multipole Algorithm (MLFMA). Sequential algorithms can compute the translation operator with L multipoles and O(L 2 ) sampling points in O(L 2 ) time. State-of-the-art hierarchical parallelization schemes of the MLFMA rely on the distribution of radiation patterns and associated translation operators among P = O(L 2 ) parallel processes, necessitating the development of distributed-memory algorithms for the computation of the translation operator. Whereas a baseline parallel algorithm computes this translation operator in O(L) time, we propose an algorithm that achieves this in only O(log L) time. For large translation operators and a high number of parallel processes, our algorithm proves to be roughly ten times faster than the baseline algorithm.

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Fast, reliable and unrestricted iterative computation of Gauss–Hermite and Gauss–Laguerre quadratures

2019

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Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and they are fast due to their fourth-order convergence and its asymptotic exactness for an appropriate selection of the variables. For Gauss-Hermite and Gauss-Laguerre quadratures, local Taylor series can be used for computing efficiently the orthogonal polynomials involved, with exact initial values for the Hermite case and first values computed with a continued fraction for the Laguerre case. The resulting algorithms have almost unrestricted validity with respect to the parameters. Full relative precision is reached for the Hermite nodes, without any accuracy loss and for any degree, and a mild accuracy loss occurs for the Hermite and Laguerre weights as well as for the Laguerre nodes. These fast methods are exclusively based on convergent processes, which, together with the high order of convergence of the underlying iterative method, makes them particularly useful for high accuracy computations. We show examples of very high accuracy computations (of up to 1000 digits of accuracy).

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$\mathcal{O}(1)$ Computation of Legendre Polynomials and Gauss--Legendre Nodes and Weights for Parallel Computing

Cited by 49 publications

References 10 publications

Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

Scalable parallel computation of the translation operator in three dimensions

Fast, reliable and unrestricted iterative computation of Gauss–Hermite and Gauss–Laguerre quadratures

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