All Title Author
Keywords Abstract


Interval Integration Revisited

DOI: 10.4236/ojapps.2012.24B026, PP. 108-111

Keywords: Interval computing, self-validating methods, numerical integration & interval arithmetic

Full-Text   Cite this paper   Add to My Lib

Abstract:

We present an overview of approaches to selfvalidating?one-dimensional integration quadrature formulas and?a verified numerical integration algorithm with an adaptive?strategy. The new interval integration adaptive algorithm delivers?a desired integral enclosure with an error bounded by a specified?error bound. The adaptive technique is usually much more?efficient than Simpson’s rule and narrow interval results can?be reached.

References

[1]  G. F. Corliss and L. B. Rall: Adaptive, self-validating numerical quadrature. SIAM J. Sci. Statist. Comput. 8(5):831–47,(1985)
[2]  R. Kelch. Ein adaptives Verfahren zur numerischen Quadratur mit automatischer Ergebnisverifikation. PhD thesis, Universit?t Karlsruhe,(1989).
[3]  U. Storck. Numerical integration in two dimensions with automatic result verification. In E. Adams and U. Kulisch, editors, Scientific Computing with Automatic Result Verification,Academic Press, New York, etc., 187–224, (1993).
[4]  J. C. Burkill: Functions of intervals. Proceedings of the London Mathematical Society, 22:375-446, (1924)
[5]  R. C. Young: The algebra of many-valued quantities. Math. Ann. 104:260-290, (1931)
[6]  T . Sunaga: Theory of an Interval Algebra an d its Application to Numerical Analysis. Gaukutsu Bunken Fukeyu-kai, Tokyo, (1958)
[7]  R. E. Moore: Interval Arithmetic and Automatic Error Analysis in Digital Computing. PhD thesis, Stanford University, October, (1962).
[8]  R.E. Moore: Interval Analysis. Prentice Hall, Englewood Clifs, NJ, USA,(1966)
[9]  G.F. Kuncir: Algorithm 103: Simpson’s rule integrator. Communications of the ACM 5(6): 347, (1962)
[10]  J.N. Lyness: Notes on the adaptive Simpson quadrature routine. Journal of the ACM 16 (3): 483–495, (1969)
[11]  P. Gonnet: Adaptive quadrature re-revisited. ETH Zürich Thesis Nr.18347 (2009).http://dx.doi.org/10.3929/ethz-a-005861903
[12]  S.M. Rump: INTLAB - INTerval LABoratory. Tibor Csendes, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 77–104, (1999), http://www.ti3.tu-harburg.de/rump/
[13]  Douglas N. Arnold: A Concise Introduction to Numerical Analysis. Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455 (2001). http://www.ima.umn.edu/~arnold/

Full-Text

comments powered by Disqus