Tanh-sinh quadrature

Tanh-sinh quadrature is a method for numerical integration introduced by Hidetosi Takahasi and Masatake Mori in 1974.^[1] It uses hyperbolic functions in the change of variables

$x = \tanh(\tfrac12 \pi \sinh t)\,$

to transform an integral on the interval x ∈ (−1, +1) to an integral on the entire real line t ∈ (−∞,+∞). After this transformation, the integrand decays with a double exponential rate, and thus, this method is also known as the double exponential (DE) formula.^[2]

For a given step size h, the integral is approximated by the sum

$\int_{-1}^1 f(x) \, dx \approx \sum_{k=-\infty}^\infty w_k f(x_k),$

with the abscissas

$x_k = \tanh(\tfrac12 \pi \sinh kh)$

and the weights

$w_k = \frac{\tfrac12 h \pi \cosh kh}{\cosh^2(\tfrac12 \pi \sinh kh)}.$

Like Gaussian quadrature, tanh-sinh quadrature is well suited for arbitrary-precision integration, where an accuracy of hundreds or even thousands of digits is desired. The convergence is exponential (in the discretization sense) for sufficiently well-behaved integrands: doubling the number of evaluation points roughly doubles the number of correct digits.

Tanh-sinh quadrature is less efficient than Gaussian quadrature for smooth integrands, but unlike Gaussian quadrature tends to work equally well with integrands having singularities or infinite derivatives at one or both endpoints of the integration interval. A further advantage is that the abscissas and weights are relatively easy to compute. The cost of calculating abscissa–weight pairs for n-digit accuracy is roughly n² log² n compared to n³ log n for Gaussian quadrature.

Upon comparing the scheme to Gaussian quadrature and error function quadrature, Bailey et al. (2005) found that the tanh-sinh scheme "appears to be the best for integrands of the type most often encountered in experimental math research".

Bailey and others have done extensive research on tanh-sinh quadrature, Gaussian quadrature and error function quadrature, as well as several of the classical quadrature methods, and found that the classical methods are not competitive with the first three methods, particularly when high-precision results are required. In a conference paper (July 2004) comparing tanh-sinh quadrature with Gaussian quadrature and error function quadrature, Bailey and Li found: "Overall, the tanh-sinh scheme appears to be the best. It combines uniformly excellent accuracy with fast run times. It is the nearest we have to a truly all-purpose quadrature scheme at the present time."

Bailey (2006) found that: "The tanh-sinh quadrature scheme is the fastest known high-precision quadrature scheme, especially when the time for computing abscissas and weights is considered. It has been successfully employed for quadrature calculations of up to 20,000-digit precision. It works well for functions with blow-up singularities or infinite derivatives at endpoints."

Notes

1. Takahasi & Mori (1974)
2. Mori (2005)

References

• David H. Bailey, "Tanh-Sinh High-Precision Quadrature". (2006).
• Pascal Molin, Intégration numérique et calculs de fonctions L (French), doctoral thesis (2010).
• David H. Bailey, Karthik Jeyabalan, and Xiaoye S. Li, "A comparison of three high-precision quadrature schemes". Experimental Mathematics, 14.3 (2005).
• David H. Bailey, Jonathan M. Borwein, David Broadhurst, and Wadim Zudlin, Experimental mathematics and mathematical physics, in Gems in Experimental Mathematics (2010), American Mathematical Society, pp. 41–58.
• Jonathan Borwein, David H. Bailey, and Roland Girgensohn, Experimentation in Mathematics—Computational Paths to Discovery. A K Peters, 2003. ISBN 1-56881-136-5.
• 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. This paper is also available from here.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 4.5. Quadrature by Variable Transformation", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
• Takahasi, Hidetosi; Mori, Masatake (1974), "Double exponential formulas for numerical integration", Publications of the Research Institute for Mathematical Sciences, 9 (3): 721–741, doi:10.2977/prims/1195192451, ISSN 0034-5318. This paper is also available from here.

External links

• John D. Cook, "Double Exponential Integration" with source code.
• Graeme Dennes, "Numerical Integration: Tanh-Sinh Quadrature v4.3" A Microsoft Excel workbook containing thirteen quadrature programs for performing Tanh-Sinh, Double-Exponential, Gauss–Kronrod, Recursive Monotone Stable (RMS) and Romberg numerical integration of functions over the finite interval (a,b), semi-infinite intervals (−∞,b) and (a,∞), and infinite interval (-∞,∞), for periodic and non-periodic functions. Demonstrates the astounding speed and accuracy of the Tanh-Sinh method in particular and the Double-Exponential method in general, all of which are members of the family of Double-Exponential quadrature techniques developed by Takahasi and Mori in 1974. The quadrature programs are demonstrated using an extensive, diverse range of test functions. Full open source code is provided, including extensive documentation.