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Iteration-free computation of Gauss-Legendre quadrature nodes and weights

Ignace Bogaert (UGent)
(2014) SIAM JOURNAL ON SCIENTIFIC COMPUTING. 36(3). p.A1008-A1026
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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 online.
Keywords
BARYCENTRIC LAGRANGE INTERPOLATION, ALGORITHM, Legendre polynomial, parallel computing, asymptotic series, Gauss-Legendre quadrature

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Citation

Please use this url to cite or link to this publication:

MLA
Bogaert, Ignace. “Iteration-free Computation of Gauss-Legendre Quadrature Nodes and Weights.” SIAM JOURNAL ON SCIENTIFIC COMPUTING 36.3 (2014): A1008–A1026. Print.
APA
Bogaert, Ignace. (2014). Iteration-free computation of Gauss-Legendre quadrature nodes and weights. SIAM JOURNAL ON SCIENTIFIC COMPUTING, 36(3), A1008–A1026.
Chicago author-date
Bogaert, Ignace. 2014. “Iteration-free Computation of Gauss-Legendre Quadrature Nodes and Weights.” Siam Journal on Scientific Computing 36 (3): A1008–A1026.
Chicago author-date (all authors)
Bogaert, Ignace. 2014. “Iteration-free Computation of Gauss-Legendre Quadrature Nodes and Weights.” Siam Journal on Scientific Computing 36 (3): A1008–A1026.
Vancouver
1.
Bogaert I. Iteration-free computation of Gauss-Legendre quadrature nodes and weights. SIAM JOURNAL ON SCIENTIFIC COMPUTING. 2014;36(3):A1008–A1026.
IEEE
[1]
I. Bogaert, “Iteration-free computation of Gauss-Legendre quadrature nodes and weights,” SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 36, no. 3, pp. A1008–A1026, 2014.
@article{5683230,
  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 online.}},
  author       = {{Bogaert, Ignace}},
  issn         = {{1064-8275}},
  journal      = {{SIAM JOURNAL ON SCIENTIFIC COMPUTING}},
  keywords     = {{BARYCENTRIC LAGRANGE INTERPOLATION,ALGORITHM,Legendre polynomial,parallel computing,asymptotic series,Gauss-Legendre quadrature}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{A1008--A1026}},
  title        = {{Iteration-free computation of Gauss-Legendre quadrature nodes and weights}},
  url          = {{http://dx.doi.org/10.1137/140954969}},
  volume       = {{36}},
  year         = {{2014}},
}

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