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O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing

Ignace Bogaert (UGent) , Bart Michiels (UGent) and Jan Fostier (UGent)
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Abstract
A self-contained set of algorithms is proposed for the fast evaluation of Legendre polynomials of arbitrary degree and argument is an element of [-1, 1]. More specifically the time required to evaluate any Legendre polynomial, regardless of argument and degree, is bounded by a constant; i.e., the complexity is O(1). The proposed algorithm also immediately yields an O(1) algorithm for computing an arbitrary Gauss-Legendre quadrature node. Such a capability is crucial for efficiently performing certain parallel computations with high order Legendre polynomials, such as computing an integral in parallel by means of Gauss-Legendre quadrature and the parallel evaluation of Legendre series. In order to achieve the O(1) complexity, novel efficient asymptotic expansions are derived and used alongside known results. A C++ implementation is available from the authors that includes the evaluation routines of the Legendre polynomials and Gauss-Legendre quadrature rules.
Keywords
Gauss-Legendre quadrature, Legendre polynomial, fixed complexity, parallel computing, QUADRATURE

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Citation

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

MLA
Bogaert, Ignace, Bart Michiels, and Jan Fostier. “O(1) Computation of Legendre Polynomials and Gauss-Legendre Nodes and Weights for Parallel Computing.” SIAM JOURNAL ON SCIENTIFIC COMPUTING 34.3 (2012): C83–C101. Print.
APA
Bogaert, Ignace, Michiels, B., & Fostier, J. (2012). O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing. SIAM JOURNAL ON SCIENTIFIC COMPUTING, 34(3), C83–C101.
Chicago author-date
Bogaert, Ignace, Bart Michiels, and Jan Fostier. 2012. “O(1) Computation of Legendre Polynomials and Gauss-Legendre Nodes and Weights for Parallel Computing.” Siam Journal on Scientific Computing 34 (3): C83–C101.
Chicago author-date (all authors)
Bogaert, Ignace, Bart Michiels, and Jan Fostier. 2012. “O(1) Computation of Legendre Polynomials and Gauss-Legendre Nodes and Weights for Parallel Computing.” Siam Journal on Scientific Computing 34 (3): C83–C101.
Vancouver
1.
Bogaert I, Michiels B, Fostier J. O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing. SIAM JOURNAL ON SCIENTIFIC COMPUTING. 2012;34(3):C83–C101.
IEEE
[1]
I. Bogaert, B. Michiels, and J. Fostier, “O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing,” SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 34, no. 3, pp. C83–C101, 2012.
@article{3108575,
  abstract     = {{A self-contained set of algorithms is proposed for the fast evaluation of Legendre polynomials of arbitrary degree and argument is an element of [-1, 1]. More specifically the time required to evaluate any Legendre polynomial, regardless of argument and degree, is bounded by a constant; i.e., the complexity is O(1). The proposed algorithm also immediately yields an O(1) algorithm for computing an arbitrary Gauss-Legendre quadrature node. Such a capability is crucial for efficiently performing certain parallel computations with high order Legendre polynomials, such as computing an integral in parallel by means of Gauss-Legendre quadrature and the parallel evaluation of Legendre series. In order to achieve the O(1) complexity, novel efficient asymptotic expansions are derived and used alongside known results. A C++ implementation is available from the authors that includes the evaluation routines of the Legendre polynomials and Gauss-Legendre quadrature rules.}},
  author       = {{Bogaert, Ignace and Michiels, Bart and Fostier, Jan}},
  issn         = {{1064-8275}},
  journal      = {{SIAM JOURNAL ON SCIENTIFIC COMPUTING}},
  keywords     = {{Gauss-Legendre quadrature,Legendre polynomial,fixed complexity,parallel computing,QUADRATURE}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{C83--C101}},
  title        = {{O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing}},
  url          = {{http://dx.doi.org/10.1137/110855442}},
  volume       = {{34}},
  year         = {{2012}},
}

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