Improving the Double Exponential Quadrature Tanh-Sinh, Sinh-Sinh and Exp-Sinh Formulas ====================================================================================== <p align="right"><i>by Dr. Robert van Engelen, June 27, 2021.<br>Genivia Research Labs</i></p> Introduction ------------ *Tanh-Sinh quadrature* is a method for numerical integration introduced by Hidetoshi Takahashi and Masatake Mori . The method uses the tanh and sinh hyperbolic functions in a change of variable to transform the (−1, +1) open interval of the integral to an open interval on the entire real line (−∞,+∞). Singularities at one or both endpoints of the (−1, +1) interval are mapped to the (−∞,+∞) endpoints of the transformed interval, forcing the endpoint singularities to vanish. This makes the method quite insensitive to endpoint behavior, resulting in a significant enhancement of the accuracy of the numerical integration procedure compared to quadrature formulas that are based on the *trapezoidal* or *midpoint* rules with equidistant grids . In most cases, the transformed integrand displays a rapid roll-off (decay) at a *double exponential* rate, enabling the numerical integrator to quickly achieve convergence . This method is therefore also known as the *Double Exponential* (DE) formula [2,4]. Implementations of the DE methods can be found in popular open source *math libraries*, such as Boost for C++ and mpmath for Python, as well as in popular open source calculator software such as for the WP-34S. A modification of the *Tanh-Sinh* formula was introduced by Krzysztof Michalski and Juan Mosig . This modification simplifies the formulas for the abscissas and weights. This modification requires fewer arithmetic operations to speed up numerical integration. This article presents effective improvements to the Michalski & Mosig *Tanh-Sinh quadrature* method. The improvements are compared the Boost, mpmath, and WP-34S implementations of the Tanh-Sinh method. In addition, a new *Exp-Sinh pre-conditioning* step is proposed to compute an optimal splitting point on the integration interval for this quadrature method. <br> ![Download](images/pdf.png) [View PDF article](files/qthsh.pdf) <br> References ----------  Takahasi, Hidetosi; Mori, Masatake (1974), "Double Exponential Formulas for Numerical Integration", Publications of the Research Institute for Mathematical Sciences, 9 (3): 721–741 <http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=9&iss=3&rank=12>  Mori, Masatake (2005), "Discovery of the Double Exponential Transformation and Its Developments", Publications of the Research Institute for Mathematical Sciences, 41 (4): 897–935, doi:10.2977/prims/1145474600, ISSN 0034-5318  Krzysztof Michalski and Juan Mosig “Efficient computation of Sommerfeld integral tails – methods and algorithms” Journal of Electromagnetic Waves and Applications 2016 https://doi.org/10.1080/09205071.2015.1129915  Evans G.A., Forbes R.C., Hyslop J. "The tanh transformation for singular integrals" International Journal of Computational Mathematics. 1984;15:339–358  Tanh-Sinh quadrature, Wikipedia, <https://en.wikipedia.org/wiki/Tanh-sinh_quadrature> <p align="right"><i>Copyright (c) 2021, Robert van Engelen, Genivia Inc. All rights reserved.</i></p>