Comparing the Performance of Four Sinc Methods for Numerical Indefinite Integration
Asian Research Journal of Mathematics,
Aims/ Objectives: To compare the performance of four Sinc methods for the numerical approximation of indefinite integrals with algebraic or logarithmic end-point singularities.
Methodology: The first two quadrature formulas were proposed by Haber based on the sinc method, the third is Stengers Single Exponential (SE) formula and Tanaka et al.s Double Exponential (DE) sinc method completes the number. Furthermore, an application of the four quadrature formulas on numerical examples, reveals convergence to the exact solution by Tanaka et al.s DE sinc method than by the other three formulae. In addition, we compared the CPU time of the four quadrature methods which was not done in an earlier work by the same author.
Conclusion: Haber formula A is the fastest as revealed by the CPU time.
- indefinite integrals
How to Cite
Kreyszig E. Advanced Engineering Mathematics, eighth ed., John Wiley & Sons, Inc., New York. 1999.
Engels, H. Numerical Quadrature and Cubature, Computational Mathematics and Applications, Academic Press Inc., London; 1980.
Muhammad M, Mori M. Double exponential formulas for numerical indefinite integration. Journal of Comput. and Appl. Math. 2003;161:431-448.
Monegato G, Scuderi L. Numerical Integration of Functions with Endpoint Singularities and/or Complex Poles in 3D Galerkin Boundary Element Methods. Publ. RIMS, Kyoto Univ. 2005;41:869-895.
Keller P. A method for indefinite integration of oscillatory and singular functions. Numer Algor. 2007;46:219-251. https://doi.org/10.1007/s11075-007-9134-y.
Keller P, Wony P. On the convergence of the method for indefinite integration of oscillatory and singular functions; 2010.
Okayama T. Error estimates with explicit constantsfor Sinc quadrature and Sinc indefinite integration overinfinite intervals, Reliab. Comput. 2013;19:45–65.
Okayama T, Matsuo T, Sugihara M. Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration, Numer. Math. 2013;124:361– 394.
Okayama T. and Machida K. Error estimate with explicit constants for the trapezoidal formula combined with Muhammad–Moris SE transformation for the semi-infinite interval, JSIAM Lett. 2019;9:45-47.
Hara R, Okayama T. Explicit Error Bound for MuhammadMoris SE–Sinc Indefinite Integration Formula over the Semi-Infinite Interval. 2017 International Symposium on Nonlinear Theory and Its Applications, NOLTA2017, Cancun, Mexico; December 4–7, 2017. https://www.ieice.org/nolta/symposium/archive/2017/articles/5053.pdf.
Okayama T, Hamada R. Modified SE-Sinc approximation with boundary treatment over the semi-infinite interval and its error bound. JSIAM Letters. 2019;11:5–7.
Okayama T. Error estimates with explicit constants for the Sinc approximation over infinite intervals. Applied Mathematics and Computation. 2018;319:125–137.
Okayama T, Kurogi C. Improvement of selection formulas of mesh size and truncation numbers for the double-exponential formula JSIAM Letters 2020;12:13-16.
Asif M, et al. Legendre multi-wavelets collocation method for numerical solution of linear and nonlinear integral equations. Alexandria Engineering Journal. 2020;59(6):5099-5109.
Boas Jr. RP. Entire Functions, Academic Press, New York; 1954.
Shultz HS, Gearhart WB. The Function sinx . The College Mathematics Journal. 1990;21(2):90-
Lund J, Bowers KL. Sinc Methods for Quadrature and Differential Equations, SIAM Philadelphia; 1992.
Abramowitz M, Stegun IA. Handbook of Mathematical Functions. Dover Publications, Inc., New York, U.S. Government Printing Office, Washington, DC; 1964.
Takahasi H, Mori M. Double exponential formulas for numerical integration. Publications of the Research Institute for Mathematical Sciences, Kyoto University. 1974;4(3):721–741.
McNamee J, Stenger F, Whitney EL. Whittakers Cardinal Function in Retrospect, Mathematics of Computation. 1971;25(113):141–154.
Kearfott RB. A Sinc Approximation for the Indefinite Integral, Mathematics of Computation. 1983;41(164):559–572.
Stenger F. Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, New York; 1993.
Haber S. Two Formulas for Numerical Indefinite Integration, Mathematics of Computation. 1993;60(201):279–296.
Akinola RO. Numerical indefinite integration using the sinc method, Stellenbosch University, South Africa; Master’s Thesis; 2007.
Stenger F. Approximations Via Whittaker’s Cardinal Function. Journal of Approximation Theory. 1976;17:222–240.
Stenger F. Approximations Via Whittakers Cardinal Function, Journal of Approximation Theory. 1976;17:222240.
Lundin L, Stenger F. Cardinal-Type Approximation of a Function and its Derivatives, SIAM Journal of Mathematical Analysis. 1979;10(1):139–160.
Haber S. The Tanh Rule for Numerical Integration, SIAM Journal of Numerical Analysis. 1977;14(4):668–685.
Nurmuhammad A, Muhammad M, Mori M. Numerical solution of initial value problems based on the double exponential transformation. Publications of the Research Institute for Mathematical Scineces, Kyoto University. 2005;41:937–948.
Stenger F. Summary of Sinc Numerical Methods, Journal of Computation and Applied Mathematics. 2000;121:379420.
Tanaka K, Sugihara M, Murota K. Numerical indefinite integration by double exponential sinc method. Mathematics of Computation. 2004;74(250).
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