×

Discovery of the double exponential transformation and its developments. (English) Zbl 1098.41031

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 \(I= \int_{-1}^1 f(x)\,dx\) H. Takahasi and M. Mori took advantage of the optimality of the trapezoidal formula over \((-\infty,\infty)\) with an equal mesh size and proposed in 1974 \(x= \varphi(t)= \tanh(\frac\pi 2\sinh t)\) as an optimal choice of variable transformation which transforms the original integral to \(I = \int_{-\infty}^\infty f(\varphi(t))\varphi'(t)\,dt\). If the trapezoidal formula is applied to the transformed integral and the resulting infinite summation is properly truncated a quadrature formula \(I= \approx h\sum_{k=-n}^n f(\varphi(kh))\varphi'(kh)\) 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.

MSC:

41A55 Approximate quadratures
65-03 History of numerical analysis
65D30 Numerical integration

Software:

QUADPACK
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Aihie, V. U. and Evans, G. A., A comparison of the error function and the tanh trans- formation as progressive rules for double and triple singular integrals, J. Comput. Appl. Math., 30 (1990), 145-154. · Zbl 0708.65024
[2] Aimi, A. and Diligenti, M., Hypersingular kernel integration in 3D Galerkin boundary element method, J. Comput. Appl. Math., 138 (2002), 51-72. · Zbl 0995.65123
[3] Bailey, D. H. and Li, X. S., A comparison of three high-precision quadrature schemes, Proceedings of the 5th Conference on Real Numbers and Computers, September 3-5, 2003, Ecole normale superieure de Lyon, Lyon, France.
[4] Davis, P. J. and Rabinowitz, P., Methods of Numerical Integration, Academic Press, 1st edition 1975, 2nd edition 1984, Japanese translation by M. Mori, Nippon Computer Kyokai, 1981. · Zbl 0304.65016
[5] Haber, S., Two formulas for numerical indefinite integration, Math. Comp., 60 (1993), 279-296. · Zbl 0795.65008
[6] Higashimachi, T., Okamoto, N., Ezawa, Y., Aizawa, T. and Ito, A., Interactive struc- tural analysis system using the advanced boundary element method, Proceedings of the Fifth International Conference on Boundary Elements, Hiroshima, November 1983, Eds. Brebbia, C. A., Futagami, T. and Tanaka, M., A Computational Mechanics Center Publication, Springer-Verlag, 1983, 847-856.
[7] Horiuchi, K. and Sugihara, M., Sinc-Galerkin method with the double exponential trans- formation for the two point boundary problems, Tech. Rep., 99-05, Dept. of Mathemat- ical Engineering, University of Tokyo, 1999.
[8] Iri, M., Moriguti, S. and Takasawa, Y., On a certain quadrature formula, (in Japanese), Kokyuroku RIMS, Kyoto Univ., 91 (1970), 82-119. · Zbl 0616.65023
[9] , On a certain quadrature formula, J. Comput. Appl. Math., 17 (1987), 3-20 (translation of the original paper in Japanese [8]).
[10] Kobayashi, K., Okamoto, H. and Zhu, J., Numerical computation of water and solitary waves by the double exponential transform, J. Comp. Appl. Math., 152 (2003), 229-241. · Zbl 1063.76075
[11] Koshihara, T. and Sugihara, M., A numerical solution for the Sturm-Liouville type eigenvalue problems employing the double exponential transformation (in Japanese), Proceedings of 1986 Annual Meeting of the Japan Society for Industrial and Applied Mathematics, 1986, 136-137.
[12] Kunihiro, N., Hayami, K. and Sugihara, M., Automatic numerical integration for the boundary element method using variable transformation and its error analysis (in Japanese), Trans. Japan Soc. Industr. Appl. Math., 5 (1995), 101-119. 933
[13] Lund, J. and Bowers, K. L., Sinc Methods for Quadrature and Differential Equations, SIAM, 1992. · Zbl 0753.65081
[14] Maina, J. W. and Matsui, K., Development of software for elastic analysis of pavement structure due to vertical and horizontal surface loadings, Transportation Research Board 2004 Annual Meeting, to appear in J. Transportation Research Board.
[15] Mathworld: http://mathworld.wolfram.com/DoubleExponentialIntegration.html
[16] Matsuo, T., On an application of the DE transformation to a Sinc-type pseudospectral method (in Japanese), Proceedings of 1997 Annual Meeting of the Japan Society for Industrial and Applied Mathematics, 1997, 36-37.
[17] McNamee, J., Eror bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae, Math. Comp., 18 (1964), 368-381. · Zbl 0125.36202
[18] Momose, T. and Shida, T., Efficient formulas for molecular integrals over the Hiller- Sucher-Feinberg identity using Cartesian Gaussian functions: Towards the improvement of spin density calculation, J. Chem. Phys., 87 (1987), 2832-2846.
[19] Mori, M., Numerical analysis and theory of hyperfunction (in Japanese), Kokyuroku RIMS, Kyoto Univ., 145 (1972), 1-11.
[20] , Numerical Analysis (in Japanese), Kyoritsu Shuppan, Tokyo, 1973, 2002.
[21] , On the superiority of the trapezoidal rule for the integration of periodic analytic functions, Memoirs of Numerical Mathematics, 1 (1974), 11-19. · Zbl 0282.65022
[22] , Curves and Surfaces (in Japanese), Kyoiku Shuppan, Tokyo, 1974.
[23] , Complex Function Theory and Numerical Analysis (in Japanese), Chikuma Shobo, Tokyo, 1975.
[24] , An IMT-type double exponential formula for numerical integration (in Japanese), Kokyuroku, RIMS, Kyoto Univ., 310 (1977) 32-47.
[25] , An IMT-type double exponential formula for numerical integration, Publ. RIMS, Kyoto Univ., 14 (1978), 713-729. · Zbl 0402.65012
[26] , A method for evaluation of the error function of real and complex variable with high relative accuracy, Publ. RIMS, Kyoto Univ., 19 (1983), 1081-1094. · Zbl 0551.65008
[27] , Quadrature formulas obtained by variable transformation and the DE-rule, J. Comput. Appl. Math., 12 & 13 (1985), 119-130. · Zbl 0589.65019
[28] , Numerical Methods and FORTRAN 77 Programming (in Japanese), Iwanami Shoten, Tokyo, 1986, 168-186.
[29] , The double exponential formula for numerical integration over the half infinite interval, in: Numerical Mathematics Singapore 1988, International Series of Numerical Mathematics 86, Birkhäuser, Basel, 1988, 367-379. · Zbl 0669.65015
[30] , An error analysis of quadrature formulas obtained by variable transformation, in: M. Kashiwara and T. Kawai, eds., Algebraic Analysis Vol.1, Academic Press, Boston, 1988, 423-437. · Zbl 0702.41039
[31] , Developments in the double exponential formulas for numerical integration, Proceedings of the International Congress of Mathematicians, Kyoto 1990, Springer- Verlag, Tokyo, 1991, 1585-1594. · Zbl 0743.65016
[32] , Graph Processing Methods and FORTRAN 77 Programming (in Japanese), Iwanami Shoten, Tokyo, 1991.
[33] , Optimality of the double exponential transformation in numerical analysis (in Japanese), S\? ugaku, 50 (1998), 248-264. · Zbl 0935.65500
[34] , Optimality of the double exponential transformation in numerical analysis, Sugaku Expositions, 14 (2001), 103-123. (In this article Fig.3 and Fig.4 should be re- versed.) (translation of the original paper in Japanese [33]). · Zbl 0935.65500
[35] Mori, M. and Muhammad, M., Numerical indefinite integration by the double exponen- tial transformation (in Japanese), Trans. Japan Soc. Industr. Appl. Math., 13 (2003), 361-366.
[36] , Numerical iterated integration by the double exponential transformation (in Japanese), Trans. Japan Soc. Industr. Appl. Math., 13 (2003), 485-493.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.