Double-Exponential Transformation

• Double-exponential transformation is a technique that maps intervals to the real line, inducing decay as exp(–c exp(|t|)) for enhanced convergence.
• It underpins methods like tanh-sinh quadrature and DE–Sinc approximation, effectively handling endpoint singularities and oscillatory integrals.
• Recent advances include parametrized maps and robust error analysis, broadening its applications in spectral methods, linear algebra, and fractional diffusion problems.

The double-exponential transformation is a mathematical and computational technique centered on variable transformations that induce double-exponential decay in transformed functions. Originating in numerical integration and approximation theory, it plays a central role in high-precision quadrature, function approximation with endpoint singularities, spectral methods for differential equations, and computational linear algebra. The defining feature is a change of variables such that the transformed integrand or approximand decays as $\exp(-c\exp(|t|))$ for large $|t|$. This enables rapid convergence of discretization schemes, even in the presence of singularities or rapid variations, and underpins advanced methods such as tanh-sinh quadrature, DE-Sinc approximation, and their numerous recent generalizations.

1. Mathematical Foundations of the Double-Exponential Transformation

The double-exponential (DE) transformation involves mapping a finite, semi-infinite, or infinite interval (often with endpoint singularities) to the entire real axis using a carefully constructed conformal transformation, $\phi$. For instance, for integration over $[-1,1]$, a standard DE mapping is

$$x = \phi(t) = \tanh\left( \frac{\pi}{2} \sinh t \right),$$

with derivative

$$\phi'(t) = \frac{\frac{\pi}{2} \cosh t}{\cosh^2\left(\frac{\pi}{2} \sinh t\right)}.$$

For large $|t|$, $\phi'(t)$ behaves as $\exp(-\frac{\pi}{2} \exp |t|)$, inducing double-exponential decay.

This transformation exponentially clusters quadrature points near potential boundary singularities or regions of interest, enabling high-order accuracy for challenging integrals and function approximation tasks. The double-exponential property ensures negligible contributions from the tails, so simple quadrature formulas such as the trapezoidal rule or Sinc approximation attain superalgebraic convergence rates for classes of analytic functions (Murota et al., 2023).

2. Numerical Integration and Quadrature Methods

The double-exponential formula, especially in the form known as tanh-sinh quadrature, constitutes a widely used method for numerical integration, particularly for functions with endpoint singularities or on infinite intervals. The transformed integral

$$I = \int_a^b f(x)dx = \int_{-\infty}^{\infty} f(\phi(t))\phi'(t)dt$$

is computed with the trapezoidal rule over a truncated range, exploiting the rapid decay of the integrand outside a finite region.

Key properties:

• Error estimates: The discretization error behaves as $O(h^2)$ with exponential suppression of truncation error (Khatibi et al., 2017).
• Endpoint singularities: The DE transform regularizes weak singularities, allowing robust and simple quadrature schemes where standard rules would fail (Khatibi et al., 2017, Murota et al., 2023).
• Practical tuning: The mesh size $h$ and number of terms $N$ can be chosen so that the error is below a prescribed tolerance, with the error estimate often being independent of $N$ beyond a cutoff, ensuring stability and efficiency (Khatibi et al., 2017).

Besides the archetypal tanh-sinh quadrature, the DE approach is integral to modern high-performance algorithms for oscillatory and Fourier-type integrals, such as those arising in computational chemistry for three-centre nuclear attraction integrals, where it is combined with "S transformations" and careful parameter management to efficiently evaluate highly oscillatory and slowly decaying integrals (Lovrod et al., 2019).

3. Function Approximation and Sinc Methods

The combination of DE transformations with Sinc approximation forms the basis of DE–Sinc methods, which are particularly effective for approximating analytic functions with endpoint singularities on finite intervals (Murota et al., 2023, Adcock et al., 2013, Adcock et al., 2015). The mapping clusters Sinc nodes near the endpoints, counteracting the effects of singular behavior.

• Convergence rates: With appropriate parameter choices, DE–Sinc schemes achieve nearly exponential convergence rates, far surpassing single exponential (SE) methods which only attain $O(\exp(-c \sqrt{N}))$ convergence.
• Resolution improvement: Modern parametrized DE maps allow near-optimal points-per-wavelength for oscillatory functions, approaching the theoretical minimum imposed by Fourier analysis (Adcock et al., 2015).
• Endpoint singularities: The mapping regularizes the function's behavior, permitting the accurate representation of functions like $x^{\alpha}$ or $\log x$ by standard discretizations (Adcock et al., 2013, Adcock et al., 2015).

This approach is widely used in the numerical resolution of singular Sturm-Liouville problems, spectral methods for fractional PDEs, and the eigenvalue problems for quantum oscillators (Gaudreau et al., 2014, Gaudreau et al., 2014, Rieder, 2020). In these contexts, DE–Sinc collocation produces symmetric, positive-definite generalized eigenvalue systems, and enables the accurate computation of large numbers of eigenvalues with high precision and controlled effort.

4. Parametrized and Generalized DE Transformations

Recent developments emphasize the design and theoretical analysis of parametrized versions of both exponential and double-exponential maps (Adcock et al., 2013, Adcock et al., 2015, Kyoya et al., 2019). These adopt free parameters (e.g., $\alpha$ and $L$) to optimize the trade-off between convergence rate and resolution power, enabling practitioners to "trade exponential convergence" for near-optimal points-per-wavelength and efficient computations at moderate accuracy.

Further, the convergence and analytic properties of the DE method degrade in proximity to singularities off the real axis. To address this, conformal mappings based on the localization of singularities—constructed via Schwarz–Christoffel transformations—are employed to steer the image of the real line away from problematic points, thereby maximizing the domain of analyticity and restoring the DE error bounds (Kyoya et al., 2019).

A typical generalized mapping is

$$H_{\mathrm{new}}(z) = C \int_0^z \frac{\prod_{j=1}^m \cosh(z-a_j)}{\prod_{j=1}^{m-1} \cosh(z-b_j)}dz + D,$$

with the constants and parameters chosen to maintain the desired asymptotic growth and singularity avoidance.

5. Applications in Computational Linear Algebra

DE-based quadrature has been extended to matrix function evaluation, notably for computing the matrix logarithm and matrix exponential (Tatsuoka et al., 2019, Tatsuoka et al., 2023). The key approach is to reformulate the matrix function as a contour integral with a DE-suited representation—e.g., for the matrix logarithm,

$\log(A) = (A-I)\int_0^1 (t(A-I) + I)^{-1}dt,$

with the integrand mapped to the real line via

$u = \tanh(\sinh(x)).$

Similarly, for the matrix exponential,

$$\exp(A) = \frac{2}{\pi}\int_0^{\infty} x\sin(x)(x^2I + A^2)^{-1}dx,$$

which is then transformed and discretized using DE quadrature tailored for oscillatory integrals.

Key algorithmic steps include:

• Construction of the integral representation with endpoint-variant or oscillatory integrands.
• Application of the DE transformation (with explicit management of the mesh size $h$ and truncation interval) to ensure double-exponential decay.
• Rigorous error analysis and adaptive selection of quadrature parameters for guaranteed accuracy, even in ill-conditioned or highly non-normal cases (Tatsuoka et al., 2019, Tatsuoka et al., 2023).

These algorithms provide improvements in convergence and stability over conventional Gauss–Legendre techniques, especially for matrices far from the identity or with eigenvalues near the negative real axis.

6. Application to Fractional Diffusion and Evolution Problems

In fractional PDEs, solution formulas often involve contour integrals in the complex plane (e.g., Riesz–Dunford representations). DE quadrature enables highly accurate discretization of these integrals, exploiting the transformation

$$z = \psi_{\theta, \sigma}(y) = \kappa\left[\cosh\left(\frac{\pi\sigma}{2} \sinh y\right) + i\theta\sinh\left(\frac{\pi}{2} \sinh y\right)\right],$$

enclosing the operator's spectrum and yielding near-exponential error decay with mesh size chosen as $k\sim \ln N/N$ (Rieder, 2020). Notable consequences:

• Robust performance across a range of problems without a-priori parameter tuning.
• Adaptivity to data smoothness: when data belong to Gevrey classes or exhibit higher regularity, the method improves automatically.
• Straightforward parallelization: each quadrature node involves independent elliptic or parabolic subproblems.

Numerical experiments demonstrate both asymptotic convergence rates and practical superiority over classical sinc or Balakrishnan-based quadrature for both elliptic and time-dependent fractional problems.

7. Historical Development and Limitations

The double-exponential transformation was developed intensively in Japan during the 1970s. The tanh-sinh quadrature is credited to Takahasi and Mori, with optimality results for Hardy classes rigorously established by Sugihara and others in subsequent decades (Murota et al., 2023). The approach has subsequently evolved to encompass a wide array of numerical algorithms in approximation theory, differential equations, and computational chemistry.

Limitations include:

• Suboptimal behavior for functions with very sharp internal features or non-analyticity away from endpoints; modified transformations or alternative acceleration strategies may be required (Khatibi et al., 2017).
• Loss of optimality when singularities are close to the real axis unless sophisticated conformal mappings are used (Kyoya et al., 2019).
• Need for careful parameter tuning in applications with oscillatory or slowly decaying integrands, typically addressed via adaptive mesh size selection algorithms (Tatsuoka et al., 2023).

Significant future directions focus on refining parametrized maps, extending the approach to broader problem classes (including highly non-smooth functions and higher-dimensional domains), and developing comprehensive error certification strategies.

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The double-exponential transformation provides the analytical and practical foundation for a broad spectrum of high-accuracy numerical methods, especially where traditional schemes falter due to singularities, endpoint effects, or oscillatory behavior. Its current theoretical generalizations, adaptive algorithms, and incorporation of conformal mapping techniques ensure it remains a central tool in modern scientific computation.