Tanh-sinh quadrature

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

x = \tanh(\pi/2 \,\sinh t)\,

to transform an integral on (−1, 1) to an integral on the entire real line. 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(\pi/2 \,\sinh kh)

and the weights

w_k = \frac{\pi/2 \,\cosh kh}{\cosh^2[\pi/2 \,\sinh kh]}.

Like Gaussian quadrature, tanh-sinh quadrature is well suited for arbitrary-precision integration, where an accuracy of hundreds or thousands of digits is desired. The convergence is quadratic 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. found that the tanh-sinh scheme "appears to be the best for integrands of the type most often encountered in experimental math research".

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