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Methods of Numerical Integration

The double-exponential transformation in numerical analysis

A review is presented of the most effective methods for the computation of Sommerfeld integral tails. Such integrals, which are often oscillatory, singular and divergent, commonly arise in layered media Green functions. The mathematical foundations of various pertinent methods are discussed in detail and their performance is illustrated by relevant numerical examples. The associated algorithms are given in pseudocode for their easy computer implementation.

https://tandfonline.com/doi/pdf/10.1080/09205071.2015.1129915

Efficient computation of Sommerfeld integral tails – methods and algorithms

This article is mainly concerned how the double exponential formula for numerical integration was discovered and how it has been developed thereafter. For evaluation of I = 1 −1 f (x)dx H. Takahasi and M. Mori took advantage of the optimality of the trapezoidal formula over (−∞, ∞) with an equal mesh size and proposed in 1974 x = φ(t) = tanh( π 2 sinh t) as an optimal choice of variable transformation which transforms the original integral to I = ∞ −∞ f (φ(t))φ � (t)dt. If the trapezoidal formula is applied to the transformed integral and the resulting infinite summation is properly truncated a quadrature formula I ≈ h n=−n f (φ(kh))φ � (kh) is obtained. Its error is expressed as O(exp(−CN/log N )) as a function of the number N (= 2n +1 ) of function evaluations. Since the integrand decays double exponentially after the transformation it is called the double exponential (DE) formula. It is also shown that the formula is optimal in the sense that there is no quadrature formula obtained by variable transformation whose error decays faster than O(exp(−CN/log N )) as N becomes large. Since the paper by Takahasi and Mori was published the DE formula has gradually come to be used widely in various fields of science and engineering. In fact we can find papers in which the DE formula is successfully used in the fields of molecular physics, fluid dynamics, statistics, civil engineering, financial engineering, in particular in the field of the boundary element method, and so on. The DE transformation has turned out to be also useful for evaluation of indefinite integrals, for solution of integral equations and for solution of ordinary differential equations, so that the scope of its applications is expected to spread also in the future.

/pdf/discovery-of-the-double-exponential-transformation-and-its-4qp8sssumy.pdf

Discovery of the Double Exponential Transformation and Its Developments

This paper develops algorithms for the pricing of discretely sampled barrier, lookback, and hindsight options and discretely exercisable American options. Under the Black-Scholes framework, the pricing of these options can be reduced to evaluation of a series of convolutions of the Gaussian distribution and a known function. We compute these convolutions efficiently using the double-exponential integration formula and the fast Gauss transform. The resulting algorithms have computational complexity of O(nN), where the number of monitoring/exercise dates is n and the number of sample points at each date is N, and our results show the error decreases exponentially with N. We also extend the approach and provide results for Merton's lognormal jump-diffusion model.

/pdf/a-double-exponential-fast-gauss-transform-algorithm-for-1y1qblh929.pdf

A Double-Exponential Fast Gauss Transform Algorithm for Pricing Discrete Path-Dependent Options

The double exponential formula for oscillatory functions over the half infinite interval

A robust double exponential formula for Fourier-type integrals

In this paper, we propose a new and efficient method that is applicable for the computation of the Fourier transform of a function which may possess a singular point or slowly converge at infinity. The proposed method is based on a generalization of the method of the double exponential (DE) formula; the DE formula is a powerful numerical quadrature proposed by H. Takahasi and M. Mori in 1974 [1]. Although it is a widely applicable formula, it is not effective in computing the Fourier transform of a slowly decreasing function. Actually it is not very efficient even if one wants to compute the value of a Fourier transform at a particular point, i.e., a Fourier-type integral. To conquer this weakness at least for a Fourier-type integral, M. Mori and the author proposed a new DE formula in 1991 [2]. See also [3] for a further improvement. The method proposed there is effective for Fourier-type integrals, but it is still weak in the computation of the Fourier transform. Here we propose another DE formula which is applicable to the computation of the Fourier transform. One point in the new method proposed here is that it makes use of fixed sampling points even if we change the point where the Fourier integral is evaluated. In this paper we propose the new method and illustrate the efficiency of the new method in several concrete examples through the comparison with the older methods.

/pdf/a-double-exponential-formula-for-the-fourier-transforms-2zaxuvlnn5.pdf

Takuya Ooura

Papers

The double exponential formula for oscillatory functions over the half infinite interval

A robust double exponential formula for Fourier-type integrals

A double exponential formula for the Fourier transforms

A continuous Euler transformation and its application to the Fourier transform of a slowly decaying fuction

A generalization of the continuous Euler transformation and its application to numerical quadrature