Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

ON A STUDY OF ERROR BOUNDS OF TRAPEZOIDAL RULE

  • Hahm, Nahmwoo (Department of Mathematics, Incheon National University) ;
  • Hong, Bum Il (Department of Applied Mathematics, Kyung Hee University)
  • Received : 2014.02.07
  • Accepted : 2014.02.18
  • Published : 2014.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, through a direct computation with subintervals partitioning [0, 1], we compute better a posteriori bounds for the average case error of the difference between the true value of $I(f)=\int_{0}^{1}f(x)dx$ with $f{\in}C^r$[0, 1] minus the composite trapezoidal rule and the composite trapezoidal rule minus the basic trapezoidal rule for $r{\geq}3$ by using zero mean-Gaussian.

Keywords

Acknowledgement

Supported by : Kyung Hee University

References

  1. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975
  2. N. Hahm and B. I. Hong, An Error of Composite Trapezoidal Rule, J. of Appl. Math. & Computing, 13 (2003), 365-372.
  3. B. I. Hong, N. Hahm and M. Yang, An Error Bounds of Trapezoidal Rule on Subintervals Using Zero-mean Gaussian, J. of Korea information processing Soc., 12-A (2005), 391-394.
  4. M. Yang and B. I. Hong, A Study of Average Error Bound of Trapezoidal Rule, Honam Math. J., 30(1) (2008), 581-587. https://doi.org/10.5831/HMJ.2008.30.3.581
  5. H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 463 (1975), 1-109. https://doi.org/10.1007/BFb0082008
  6. E. Novak, Deterministic and Stochastic Error Bound in Numerical Analysis, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1349, 1988.
  7. A. V. Skorohod, Integration in Hilbert Space, Springer-Verlag, New York, 1974
  8. A. V. Suldin, Wienner Measure and Its Applications to Approximation Methods, I, Izv. Vyssh. Ucheb. Zaved. Mat., 13 (1959), 145-158.
  9. A. V. Suldin, Wienner Measure and Its Applications to Approximation Methods, II, Izv. Vyssh. Ucheb. Zaved. Mat., 18 (1960), 165-179.
  10. J. F. Traub, G. W. Wasilkowski and H. Wozniakowski, Information-Based Com-plexity, Academic Press, New York, 1988.
  11. N. N. Vakhania, Probability Distributed on Linear Spaces, North-Holland, New York, 1981
  12. G. W. Wasikowski, On a Posteriori Upper Bounds for Approximating Linear Functionals in a Probabilistic Setting, J. of Complexity, 57 (1992), 424-433.