### An Accurate Quadrature for the Numerical Integration of Polynomial Functions

Received: 15 December 2017     Accepted: 8 January 2018     Published: 19 January 2018
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Abstract

The numerical integration of polynomial functions is one of the most interesting processes for numerical calculus and analyses, and represents thus, a compulsory step especially in finite elements analyses. Via the Gauss quadrature, the users concluded a great inconvenience that is processing at certain points which not required the based in finite element method points for deducting the form polynomials constants. In this paper, the same accuracy and efficiency as the Gauss quadrature extends for the numerical integration of the polynomial functions, but as such at the same points and nods have chosen for the determination of the form polynomials. Not just to profit from the values of the polynomials at those points and nods, but also from their first derivatives, the chosen points positions are arbitrary and the resulted deducted formulas are therefore different, as will be presented bellow and implemented.

 Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 4, Issue 1) DOI 10.11648/j.ijamtp.20180401.11 Page(s) 1-7 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2018. Published by Science Publishing Group
Keywords

Integration Quadrature, Numerical Methods, Numerical Integration, Polynomial Functions, Accurate Methods

References
 [1] Baldoni, V. N. Berline, De Loera, J. A. K¨oppe, M. and Vergne M. (2010). How to integrate a polynomial over a simplex. Mathematics of Computation, DOI: 10.1090/S0025-5718-2010-02378-6. [2] Bernardini F. (1991) Integration of polynomials over n-dimensional polyhedra. Comput Aided Des 23 (1): 51–58. [3] Davis, P. J. and Rabinowitz, P. (1984a). Methods of Numerical Integration. Academic Press, San Diego, 2nd edition. [4] Davis, P. J. and Rabinowitz, P. (1984b). Methods of Numerical Integration. Academic Press, San Diego, 2nd edition. [5] Dunavant D. A. (1985) High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int J Numer Methods Eng 21: 1129–1148. [6] Holdych D. J., Noble D. R., Secor R. B. (2008) Quadrature rules for triangular and tetrahedral elements with generalized functions. Int J Numer Methods Eng 73: 1310–1327. [7] Keast P. (1986) Moderate-degree tetrahedral quadrature formulas. Comput Methods Appl Mech Eng 55: 339–348. [8] Liu Y. and Vinokur M. (1998). Exact integrations of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids. Journal of Computational Physics, 140: 122–147. [9] Lubinsky, D. S. and Rabinowitz, P. (1984). Rates of convergence of Gaussian quadrature for singular integrands. Mathematics of Computation, 43 (167): 219–242. Press. [10] Lyness J. N., Jespersen D. (1975) Moderate degree symmetric quadrature rules for the triangle. J Inst Math Appl 15: 19–32. [11] Lyness J. N., Monegato G. (1977) Quadrature rules for regions having regular hexagonal symmetry. SIAM J Numer Anal 14 (2): 283–295. [12] Mousavi S. E., Xiao H., Sukumar N. (2010) Generalized Gaussian quadrature rules on arbitrary polygons. Int J Numer Methods Eng 82 (1): 99–113. [13] Mousavi S. E., Sukumar N. (2010) Generalized Duffy transformation for integrating vertex singularities. Comput Mech 45 (2–3): 127–140. [14] Mousavi S. E., Sukumar N. (2010) Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method. Comput Methods Appl Mech Eng 199 (49–52): 3237–3249. [15] Rathod H. T., Govinda Rao H. S. (1997) Integration of polynomials over n-dimensional linear polyhedra. Comput Struct 65 (6): 829–847 [16] Silvester P. (1970) Symmetric quadrature formulae for simplexes. Math Comput 24 (109): 95–100. [17] Sunder K. S., Cookson R. A. (1985) Integration points for triangles and tetrahedrons obtained from the Gaussian quadrature points for a line. Comput Struct 21 (5): 881–885. [18] Ventura G. (2006) On the elimination of quadrature subcells for discontinuous functions in the eXtended finite-element method. Int J Numer Methods Eng 66: 761–795. [19] Wandzura S., Xiao H. (2003) Symmetric quadrature rules on a triangle. Comput Math Appl 45: 1829–1840. [20] Wong, R. (1989). Asymptotic approximation of integrals. Academic Press, San Diego. [21] Xiao H., Gimbutas Z. (2010) A numerical algorithm for the construction of efficient quadratures in two and higher dimensions. Comput Math Appl 59: 663–676.
Cite This Article
• APA Style

Tahar Latrache. (2018). An Accurate Quadrature for the Numerical Integration of Polynomial Functions. International Journal of Applied Mathematics and Theoretical Physics, 4(1), 1-7. https://doi.org/10.11648/j.ijamtp.20180401.11

ACS Style

Tahar Latrache. An Accurate Quadrature for the Numerical Integration of Polynomial Functions. Int. J. Appl. Math. Theor. Phys. 2018, 4(1), 1-7. doi: 10.11648/j.ijamtp.20180401.11

AMA Style

Tahar Latrache. An Accurate Quadrature for the Numerical Integration of Polynomial Functions. Int J Appl Math Theor Phys. 2018;4(1):1-7. doi: 10.11648/j.ijamtp.20180401.11

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author = {Tahar Latrache},
title = {An Accurate Quadrature for the Numerical Integration of Polynomial Functions},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {4},
number = {1},
pages = {1-7},
doi = {10.11648/j.ijamtp.20180401.11},
url = {https://doi.org/10.11648/j.ijamtp.20180401.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20180401.11},
abstract = {The numerical integration of polynomial functions is one of the most interesting processes for numerical calculus and analyses, and represents thus, a compulsory step especially in finite elements analyses. Via the Gauss quadrature, the users concluded a great inconvenience that is processing at certain points which not required the based in finite element method points for deducting the form polynomials constants. In this paper, the same accuracy and efficiency as the Gauss quadrature extends for the numerical integration of the polynomial functions, but as such at the same points and nods have chosen for the determination of the form polynomials. Not just to profit from the values of the polynomials at those points and nods, but also from their first derivatives, the chosen points positions are arbitrary and the resulted deducted formulas are therefore different, as will be presented bellow and implemented.},
year = {2018}
}
```
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JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
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AB  - The numerical integration of polynomial functions is one of the most interesting processes for numerical calculus and analyses, and represents thus, a compulsory step especially in finite elements analyses. Via the Gauss quadrature, the users concluded a great inconvenience that is processing at certain points which not required the based in finite element method points for deducting the form polynomials constants. In this paper, the same accuracy and efficiency as the Gauss quadrature extends for the numerical integration of the polynomial functions, but as such at the same points and nods have chosen for the determination of the form polynomials. Not just to profit from the values of the polynomials at those points and nods, but also from their first derivatives, the chosen points positions are arbitrary and the resulted deducted formulas are therefore different, as will be presented bellow and implemented.
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IS  - 1
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Author Information
• Department of Civil Engineering, University of Tebessa, Tebessa, Algeria

• Sections