New conformal map for the trapezoidal formula for infinite integrals of unilateral rapidly decreasing functions. (English) Zbl 1459.65030
Summary: While the trapezoidal formula can attain exponential convergence when applied to infinite integrals of bilateral rapidly decreasing functions, it is not capable of this in the case of unilateral rapidly decreasing functions. To address this issue, F. Stenger [Handbook of Sinc numerical methods. With CD-ROM. Boca Raton, FL: CRC Press (2011; Zbl 1208.65143)] proposed the application of a conformal map to the integrand such that it transforms into bilateral rapidly decreasing functions. T. Okayama and S. Hanada [Math. Comput. Simul. 186, 3–18 (2021; Zbl 1540.65097)] modified the conformal map and provided a rigorous error bound for the modified formula. This paper proposes a further improved conformal map, with two rigorous error bounds provided for the improved formula. Numerical examples comparing the proposed and existing formulas are also given.
MSC:
| 65D30 | Numerical integration |
| 65D32 | Numerical quadrature and cubature formulas |
| 65G20 | Algorithms with automatic result verification |
Software:
Sinc-PackReferences:
| [1] | Schwartz, C., Numerical integration of analytic functions, J. Comput. Phys., 4, 19-29 (1969) · Zbl 0208.41101 |
| [2] | Stenger, F., Integration formulae based on the trapezoidal formula, J. Inst. Math. Appl., 12, 103-114 (1973) · Zbl 0262.65011 |
| [3] | Stenger, F., Numerical Methods Based on Sinc and Analytic Functions (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0803.65141 |
| [4] | Stenger, F., Handbook of Sinc Numerical Methods (2011), CRC Press: CRC Press Boca Raton, FL · Zbl 1208.65143 |
| [5] | Okayama, T.; Hanada, S., A modified Stenger’s quadrature formula for infinite integrals of unilateral rapidly decreasing functions and its theoretical error bound, Math. Comput. Simulation (2020) |
| [6] | R. Hara, T. Okayama, Explicit error bound for Muhammad-Mori’s SE-Sinc indefinite integration formula over the semi-infinite interval, in: Proceedings of the 2017 International Symposium on Nonlinear Theory and Its Applications, 2017, pp. 677-680. |
| [7] | Okayama, T.; Machida, K., Error estimate with explicit constants for the trapezoidal formula combined with Muhammad-Mori’s SE transformation for the semi-infinite interval, JSIAM Lett., 9, 45-47 (2017) · Zbl 07037424 |
| [8] | Okayama, T.; Shintaku, Y.; Katsuura, E., New conformal map for the Sinc approximation for exponentially decaying functions over the semi-infinite interval, J. Comput. Appl. Math., 373, Article 112358 pp. (2020) · Zbl 1524.65058 |
| [9] | Okayama, T.; Matsuo, T.; Sugihara, M., Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration, Numer. Math., 124, 361-394 (2013) · Zbl 1281.65020 |
| [10] | Stein, E. M.; Shakarchi, R., Complex Analysis (2003), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1020.30001 |
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