Back
 OJAppS  Vol.2 No.4 B , December 2012
Interval Integration Revisited
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.
Cite this paper: nullGaldino, S. (2012) Interval Integration Revisited. Open Journal of Applied Sciences, 2, 108-111. doi: 10.4236/ojapps.2012.24B026.
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/

 
 
Top