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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

The Double Exponential Formula as a Gauss Quadratures Replacement for Numerical Integration

Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 499--509 | DOI:10.5890/DNC.2015.11.011

Dariusz W. Brzeziński; Piotr Ostalczyk

Institute of Applied Computer Science, Lodz University of Technology, 18/22 Stefanowskiego St., 90-924 Łodź, Poland

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Abstract

We propose to replace the Gauss Quadratures with a numerical integration method known as the Double Exponential (DE) Formula. The numerical quadrature built upon it is at least equivalently accurate and much simpler to customize and apply in situations when tabulated values of the Gauss Quadratures’ nodes and weights can not be applied. The DE Formula was developed for integrals with endpoint singularities. However, we confirm that it can be successfully applied to any integral and interval, for which the Gauss Quadratures have been usually selected. To remain compact, the following presentation focuses only on the most difficult integrals, e.g. the improper integrals and the integrals with endpoint singularities. The main part of the paper consists of the calculations accuracy comparison between numerical quadrature based upon the DE Formula and the Gauss-Laguerre, the Gauss-Hermite or Gauss-Chebyshev Quadratures.

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