On a certain quadrature formula

Iri, Masao; Moriguti, Sigeiti; Yoshimitsu, Tetsuo

doi:10.1016/0377-0427(87)90034-3

Cited by 73 publications

(45 citation statements)

References 2 publications

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“…The IMT-rule At the RIMS symposium in November 1969 another interesting research result was reported. It was on a new quadrature formula based on a variable transformation by Masao Iri, Sigeiti Moriguti and Yoshimitsu Takasawa [8]. It is for an integral over (0, 1) of an analytic function f (x) which may have end-point singularity:…”

Section: §23 Optimality Of the Trapezoidal Formulamentioning

confidence: 99%

“…Later in 1987 a special issue of Journal of Computational and Applied Mathematics devoted to numerical integration was published whose editors were Mori and Robert Piessens. The original paper in Japanese by Iri, Moriguti and Takasawa [8] was translated into English by the authors themselves and republished in the special issue [9]. §3.…”

Section: §23 Optimality Of the Trapezoidal Formulamentioning

confidence: 99%

“…There Mori met Robert Piessens and they agreed to publish a special issue of Journal of Computational and Applied Mathematics on numerical integration. The special issue was published in 1987 in which the English translation [9] of the original paper [8] on the IMT-rule can be found.…”

Section: Numerical Analysis Group In Tsukubamentioning

confidence: 99%

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Discovery of the Double Exponential Transformation and Its Developments

Mori¹

2005

Publ. Res. Inst. Math. Sci.

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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 = If the trapezoidal formula is applied to the transformed integral and the resulting infinite summation is properly truncated a quadrature formula I ≈ h n k=−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.

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Section: §23 Optimality Of the Trapezoidal Formulamentioning

confidence: 99%

Section: §23 Optimality Of the Trapezoidal Formulamentioning

confidence: 99%

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Discovery of the Double Exponential Transformation and Its Developments

Mori¹

2005

Publ. Res. Inst. Math. Sci.

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“…In particular, contour plots in the complex plane are presented in [172] comparing best rational approximations to e t on (−∞, 0] with approximations derived from the trapezoidal rule on hyperbolic, parabolic, and Talbot Hankel contours as discussed in section 15. Methods of this kind were introduced in the 1960s [143,148] and developed much further in the 1970s by Mori, Takahasi, and other Japanese researchers [85,128,131,161,162,163].…”

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confidence: 99%

The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

434

359

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Abstract.It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

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“…Before we end this section, we would like to comment on the use of exponential type variable transformations ψ(t), such as the tanh transformation of Sag and Szekeres [10], the IMT transformation of Iri, Moriguti, and Takasawa [3], and the double exponential transformation of Mori [9]. These transformations have the property that…”

Section: Introductionmentioning

confidence: 99%

Variable transformations and Gauss–Legendre quadrature for integrals with endpoint singularities

Sidi

2009

Math. Comp.

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Abstract. Gauss-Legendre quadrature formulas have excellent convergence properties when applied to integrals 1 0 f (x) dx with f ∈ C ∞ [0, 1]. However, their performance deteriorates when the integrands f (x) are in C ∞ (0, 1) but are singular at x = 0 and/or x = 1. One way of improving the performance of Gauss-Legendre quadrature in such cases is by combining it with a suitable variable transformation such that the transformed integrand has weaker singularities than those of f (x). Thus, if x = ψ(t) is a variable transformation that maps [0, 1] onto itself, we apply Gauss-Legendre quadrature to the transformed integral 1 0 f (ψ(t))ψ (t) dt, whose singularities at t = 0 and/or t = 1 are weaker than those of f (x) at x = 0 and/or x = 1. In this work, we first define a new class of variable transformations we denote S p,q , where p and q are two positive parameters that characterize it. We also give a simple and easily computable representative of this class. Next, by invoking some recent results by the author concerning asymptotic expansions of Gauss-Legendre quadrature approximations as the number of abscissas tends to infinity, we present a thorough study of convergence of the combined approximation procedure, with variable transformations from S p,q . We show how optimal results can be obtained by adjusting the parameters p and q of the variable transformation in an appropriate fashion. We also give numerical examples that confirm the theoretical results.

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On a certain quadrature formula

Cited by 73 publications

References 2 publications

Discovery of the Double Exponential Transformation and Its Developments

Discovery of the Double Exponential Transformation and Its Developments

The Exponentially Convergent Trapezoidal Rule

Variable transformations and Gauss–Legendre quadrature for integrals with endpoint singularities

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