Nonlinear Functional Analysis and Applications

The Basic Sciences Research Institute, Kyungnam University (경남대학교 기초과학연구소)
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A TRIPLE MIXED QUADRATURE BASED ADAPTIVE SCHEME FOR ANALYTIC FUNCTIONS

  • Received : 2020.08.17
  • Accepted : 2021.04.10
  • Published : 2021.12.15
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Abstract

An efficient adaptive scheme based on a triple mixed quadrature rule of precision nine for approximate evaluation of line integral of analytic functions has been constructed. At first, a mixed quadrature rule SM1(f) has been formed using Gauss-Legendre three point transformed rule and five point Booles transformed rule. A suitable linear combination of the resulting rule and Clenshaw-Curtis seven point rule gives a new mixed quadrature rule SM10(f). This mixed rule is termed as triple mixed quadrature rule. An adaptive quadrature scheme is designed. Some test integrals having analytic function integrands have been evaluated using the triple mixed rule and its constituent rules in non-adaptive mode. The same set of test integrals have been evaluated using those rules as base rules in the adaptive scheme. The triple mixed rule based adaptive scheme is found to be the most effective.

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Acknowledgement

I would like to express my indebtedness to my teacher Dr.Rajani Ballav Dash, Department of Mathematics, Ravenshaw University, Cuttack for his guidance in preparing this paper. My special thanks to Dr.Dwiti Krushna Beher and Dr.Debasish Das for their logistic support.

References

  1. Kendall E. Atkinson, An introduction to numerical analysis,Wiley Student Edition 2, 2012.
  2. S. Conte and C.de Boor, Elementary numerical analysis, Mc-Graw Hill., 1980.
  3. R.N. Das and G. Pradhan, A mixed quadrature for approximate evaluation of real and definite integrals, Int. J. Math. Educ. Sci. Tech., 27(2) (1996), 279-283. https://doi.org/10.1080/0020739960270214
  4. D. Das and R.B. Dash, Applicaion of mixed quadrsture rules in the adaptive quadrature routines, Gen. Math. Notes, Copyright: ICSRS Publication., 18(1) (2013), 46-63.
  5. H.O. Hera and F.J. Smith, Error estimation in the Clenshaw-Curtis quadrature formula, The Comput. J., II(3) (1968), 213-219.
  6. F.G. Lether, On Birkhoff-Young quadrature of analytic functions, J. Comput. Appl. Math., 2 (1976), 81-92. https://doi.org/10.1016/0771-050X(76)90012-7
  7. S.K. Mohanty and R.B. Dash, A mixed quadrature rule for numerical integration of analytic functions, Bull. Pure and Appl. Sci., 27E(2) (2008), 373-376.
  8. S.K. Mohanty, A mixed quadrature rule using Clenshaw-Curtis five point rule modified by Richardson extrapolation, J. Ultra Sci. of Physical Sci., 32(2) (2020), 6-12.
  9. S.K. Mohanty, Dual mixed Gaussian quadrature based adaptive scheme for analytic functions, Annals of Pure and Appl. Math., 22(2) (2020), 83-92.
  10. M. Pal, Numerical analysis for scientists and engineers(Theory and C Programs), Narosa Publication, 2008.
  11. J. Davis Philip and R. Philip, Methods of numerical integration, 2nd Edition (Academic Press), Engineering Computation ,4th Edition, (New Age International Publisher), 2003.
  12. J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer International Edition, 3rd Edition. 2002.
  13. G. Walter and G. Walter, Adaptive quadrature Revisited, BIT Numerical Mathematics, 40(1) (2000), 84-101. https://doi.org/10.1023/A:1022318402393