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Double exponential formulas for numerical indefinite integration. (English) Zbl 1038.65018
Summary: We derive a formula for indefinite integration of analytic functions over \((-1,s)\) where \(-1 < s < 1\), by means of the double exponential transformation and the Sinc method. The integrand must be analytic on \(-1 < x < 1\) but may have a singularity at the end points \(x = \pm 1\). The error of the formula behaves approximately as exp\((-c_1N/{\log} c_2N)\) where \(N\) is the number of function evaluations of the integrand. This error term shows a much faster convergence to zero when \(N\) becomes large than that of the known formula by S. Haber [Math. Comput. 60, No. 201, 279–296 (1993; Zbl 0795.65008)]. Also we derive efficient double exponential formulas for numerical evaluation of indefinite integrals over \((0,s), 0<s<{\infty}\) and over \((-{\infty},s)\), \(-\infty < s < +\infty\). Several numerical examples indicate high efficiency of the formulas.

MSC:
65D30 Numerical integration
41A55 Approximate quadratures
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References:
[1] Haber, S., Two formulas for numerical indefinite integration, Math. comp., 60, 279-296, (1993) · Zbl 0795.65008
[2] M. Mori, M. Muhammad, A method for numerical evaluation of indefinite integrals by the double exponential transformation, Proceedings of 2002 Symposium on Applied Mathematics, Ryukoku University, Ohtsu, December 19-21, 2002 (in Japanese).
[3] Sugihara, M., Optimality of the double exponential formula – functional analysis approach, Numer. math., 75, 379-395, (1997) · Zbl 0868.41019
[4] Takahasi, H.; Mori, M., Double exponential formulas for numerical integration, Publ. res. inst. math. sci., 9, 721-741, (1974) · Zbl 0293.65011
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