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Double-Exponential transformation: A quick review of a Japanese tradition

Kazuo Murota, Takayasu Matsuo

TL;DR

The paper surveys the double exponential transformation and its role in fast, robust numerical quadrature and Sinc-based approximation, emphasizing the Japanese development of tanh-sinh quadrature and DE-Sinc methods. It details the DE integration formula $I_h^{(N)} = h \sum_{k=-N}^{N} f(\phi(kh)) \phi'(kh)$ with $\phi(t) = \tanh\left(\frac{\pi}{2}\sinh t\right)$, discusses error balance and optimality results, and extends the framework to Fourier-type integrals via Ooura–Mori transforms and the IMT rule. The survey covers DE-Sinc theory and convergence rates, including $\exp(-C N/\log N)$ for DE-Sinc and $\exp(-C\sqrt{N})$ for SE-Sinc, and demonstrates broad applicability to indefinite integration, initial/boundary value problems, Volterra and Fredholm integral equations, and matrix-function computations. Overall, the work highlights the practical impact of DE methods on accurate, rapidly convergent quadrature and function approximation, with deep connections to optimality theory and historical development in Japan.

Abstract

This paper is a short introduction to numerical methods using the double exponential (DE) transformation, such as tanh-sinh quadrature and DE-Sinc approximation. The DE-based methods for numerical computation have been developed intensively in Japan and the objective of this paper is to describe their history in addition to the underlying mathematical ideas.

Double-Exponential transformation: A quick review of a Japanese tradition

TL;DR

The paper surveys the double exponential transformation and its role in fast, robust numerical quadrature and Sinc-based approximation, emphasizing the Japanese development of tanh-sinh quadrature and DE-Sinc methods. It details the DE integration formula with , discusses error balance and optimality results, and extends the framework to Fourier-type integrals via Ooura–Mori transforms and the IMT rule. The survey covers DE-Sinc theory and convergence rates, including for DE-Sinc and for SE-Sinc, and demonstrates broad applicability to indefinite integration, initial/boundary value problems, Volterra and Fredholm integral equations, and matrix-function computations. Overall, the work highlights the practical impact of DE methods on accurate, rapidly convergent quadrature and function approximation, with deep connections to optimality theory and historical development in Japan.

Abstract

This paper is a short introduction to numerical methods using the double exponential (DE) transformation, such as tanh-sinh quadrature and DE-Sinc approximation. The DE-based methods for numerical computation have been developed intensively in Japan and the objective of this paper is to describe their history in addition to the underlying mathematical ideas.
Paper Structure (9 sections, 24 equations, 2 figures)

This paper contains 9 sections, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. DE formula for numerical integration
  3. Quadrature formula
  4. Optimality
  5. Fourier-type integrals
  6. IMT rule
  7. DE-Sinc methods
  8. Sinc approximation
  9. Application to other problems

Figures (2)

  • Figure 1: Comparison of the efficiency of several variable transformations for the integral $I = \int_{-1}\sp{1} {\rm d}x/\{ (x-2)(1-x)\sp{1/4}(1+x)\sp{3/4} \}$. The figure is taken from Mori Mor05 with permission from Publications RIMS, Research Institute for Mathematical Sciences, Kyoto University; $u$ and $N$ in the figure correspond, respectively, to $t$ and $2N+1$ in the present notation.
  • Figure 2: Errors in the Sinc approximations for $x\sp{1/2} (1-x)\sp{3/4}$ using \ref{['transSEsinc']} and \ref{['transDEsinc']} and the Chebyshev interpolation. "Ordinary-Sinc" means the SE-Sinc approximation using \ref{['transSEsinc']}. The figure is taken from Sugihara and Matsuo SM04sinc with permission from Elsevier; $n$ in the figure correspond to $N$ in \ref{['Sincappr']}.