Numerical evaluation for two-dimensional integral using higher order Gaussian quadrature
DOI:
https://doi.org/10.33292/amm.v2i1.17Keywords:
Gaussian quadrature, Integral, Rational FunctionsAbstract
Presented in this paper is a computational approach that uses higher order Gaussian quadrature to improve the accuracy of the evaluation of an integral. The transformation from ξη space (standard Gaussian) to st space (higher order Gaussian) were shown throughout this paper. Not even that, the efficacy of this higher order Gaussian quadrature were tested by implementing and comparing it with standard Gaussian quadrature over the same integral. Results shown that the evaluation of an integral by using higher order Gaussian quadrature provide accurate and converge results compared to an integral using standard Gaussian quadrature.
References
Butler, J. S., & Moffitt, R. (1982). A computationally efficient quadrature procedure for the one- factor multinomial probit model. Econometrica: Journal of the Econometric Society, 761–764.
Graglia, R. D., & Lombardi, G. (2008). Machine precision evaluation of singular and nearly singular potential integrals by use of Gauss quadrature formulas for rational functions. IEEE Transactions on Antennas and Propagation, 56(4), 981–998.
Hussain, F., Karim, M. S., & Ahamad, R. (2012). Appropriate Gaussian quadrature formulae for triangles. International Journal of Applied Mathematics and Computation, 4(1), 24–38.
Johnson, S. G. (2019). Accurate solar-power integration: Solar-weighted Gaussian quadrature.
ArXiv Preprint ArXiv:1912.06870.
Kaneko, H., & Xu, Y. (1994). Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind. Mathematics of Computation, 62(206), 739–753.
Kloeden, P., & Shardlow, T. (2017). Gauss-quadrature method for one-dimensional mean-field SDEs. SIAM Journal on Scientific Computing, 39(6), A2784–A2807.
Ma, J., Rokhlin, V., & Wandzura, S. (1996). Generalized Gaussian quadrature rules for systems of arbitrary functions. SIAM Journal on Numerical Analysis, 33(3), 971–996.
Monegato, G., & Scuderi, L. (2005). Numerical integration of functions with endpoint singularities and/or complex poles in 3D Galerkin boundary element methods. Publications of the Research Institute for Mathematical Sciences, 41(4), 869–895.
Rathod, H. T., & Karim, M. S. (2002). An explicit integration scheme based on recursion for the curved triangular finite elements. Computers & Structures, 80(1), 43–76.
Schwartz, C. (1969). Numerical integration of analytic functions. Journal of Computational Physics, 4(1), 19–29.
Teh, C. R. C. (2009). Numerical Method Algorithm and Matlab Programming.
