Gaussian quadrature rule for arbitrary weight function and interval

Fukuda, Hiroshi; Katuya, Mituaki; Alt, E. O.; Матвеенко, А. В.

doi:10.1016/j.cpc.2004.12.010

Cited by 6 publications

(3 citation statements)

References 2 publications

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“…System of the equations is then solved numerically using arbitrary precision arithmetic. Results were verified using method of [6].…”

Section: Case With Fixed Parametermentioning

confidence: 99%

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Gaussian integration with rescaling of abscissas and weights

Odrzywołek

2011

Computer Physics Communications

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“…System of the equations is then solved numerically using arbitrary precision arithmetic. Results were verified using method of [6].…”

Section: Case With Fixed Parametermentioning

confidence: 99%

“…However, this procedure is inherently sequential, while using parameterized guess one can do all calculations in parallel. Mathematica script of[6] also do not need any guess points, but require 500 seconds to find 10-point quadrature for(2). Our method do the same job in 2 to 3 seconds on the same machine.…”

mentioning

confidence: 99%

Gaussian integration with rescaling of abscissas and weights

Odrzywołek

2011

Computer Physics Communications

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“…One of these methods (with some small modification) has also been implemented in C++ code by computational biologists, see Fernandes & Atchley (2006). Fukuda et al (2005) used both the symbolic algebra and high-precision arithmetic capabilities of Mathematica to compute custom-made Gauss quadrature rules.…”

Section: Introductionmentioning

confidence: 99%

Custom-made Gauss quadrature for statisticians

Kabaila¹

2022

Preprint

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The theory and computational methods for custom-made Gauss quadrature have been described in Gautschi's 2004 monograph. Gautschi has also provided Fortran and MATLAB code for the implementation and illustration of these methods. We have written an R package, implemented in the high-precision arithmetic provided by the R package Rmpfr, that uses a moment-based method via moment determinants to compute a Gauss quadrature rule, with up to 33 nodes, provided that the moments can be computed to arbitrary precision using the standard mathematical functions provided by the Rmpfr package. Our hope is that the provision of our free R package and the numerical results that we present will encourage other statisticians to also consider the custom-made construction of Gauss quadrature rules.

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High‐order dual‐parametric finite element methods for cavitation computation in nonlinear elasticity

Huang

Liang

et al. 2020

Numerical Methods Partial

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In this paper, we present the numerical analysis on high order dual parametric finite element methods for the cavitation computation problems in nonlinear elasticity, which leads to a meshing strategy assuring high efficiency on numerical approximations to cavity deformations. Furthermore, to cope with the high order approximation of the finite element methods, properly chosen weighted Gaussian type numerical quadrature is applied to the singular part of the elastic energy. Our numerical experiments show that the high order dual parametric finite element methods work well when coupled with properly designed weighted Gaussian type numerical quadratures for the singular part of the elastic energy, and the convergence rates of the numerical cavity solutions are shown to be significantly improved as expected.

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Gaussian quadrature rule for arbitrary weight function and interval

Cited by 6 publications

References 2 publications

Gaussian integration with rescaling of abscissas and weights

Gaussian integration with rescaling of abscissas and weights

Custom-made Gauss quadrature for statisticians

High‐order dual‐parametric finite element methods for cavitation computation in nonlinear elasticity

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