Extension of a class of periodizing variable transformations for numerical integration

Mathematics of Computation

Volumn 75, Issue 253, 2005, Pages 327-343

  Sidi, Avram ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Asymptotic expansions; Euler Maclaurin expansions; Numerical integration; Sinm transformation; Trapezoidal rule; Variable transformations

Indexed keywords

EID: 85009726455     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-05-01773-4     Document Type: Article

References (26)

1. M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Number 55 in Nat. Bur. Standards Appl. Math. Series. U.S. Government Printing Office, Washington, D.C., 1964.MR0167642 (29:4914)
2. K.E. Atkinson. An Introduction to Numerical Analysis. Wiley, New York, second edition, 1989.MR1007135 (90m:65001)
3. P.J. Davis and P. Rabinowitz. Methods of Numerical Integration. Academic Press, New York, second edition, 1984.MR0760629 (86d:65004)
4. D. Elliott. Sigmoidal transformations and the trapezoidal rule. J. Austral. Math. Soc., Series B (E), 40:E77–E137, 1998.MR1658346 (99j:65030)
5. W. Gautschi. Algorithm 726: ORTHPOL – a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Transactions on Mathematical Software, 20:21-62, 1994.
6. M. Hill and I. Robinson. d2lri: A nonadaptive algorithm for two-dimensional cubature. J. Comp. Appl. Math., 112:121-145, 1999.MR1728455
7. M. Iri, S. Moriguti, and Y. Takasawa. On a certain quadrature formula. Kokyuroku of Res. Inst. for Math. Sci. Kyoto Univ., 91:82-118, 1970.
8. Japanese. English translation in J. Comp. Appl. Math., 17:3-20, 1987.MR0884257 (88j:65057)
9. P.R. Johnston. Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals. Intern. J. Numer. Methods Engrg., 47:1709-1730, 2000.MR1750249
10. N.M. Korobov. Number-Theoretic Methods of Approximate Analysis. GIFL, Moscow, 1963. In Russian.MR0157483 (28:716)
11. D.P. Laurie. Periodizing transformations for numerical integration. J. Comp. Appl. Math., 66:337-344, 1996.MR1393741
12. D. Levin. Development of non-linear transformations for improving convergence of sequences. Intern. J. Computer Math., B3:371-388, 1973.MR0359261 (50:11716)
13. M. Mori. An IMT-type double exponential formula for numerical integration. Publ. Res. Inst. Math. Sci. Kyoto Univ., 14:713-729, 1978.MR0527197 (81c:65012)
14. A. Ralston and P. Rabinowitz. A First Course in Numerical Analysis. McGraw-Hill, New York, second edition, 1978.MR0494814 (58:13599)
15. I. Robinson and M. Hill. Algorithm 816: r2d2lri: An algorithm for automatic two-dimensional cubature. ACM Trans. Math. Software, 28:75-100, 2002.
16. T.W. Sag and G. Szekeres. Numerical evaluation of high-dimensional integrals. Math. Comp., 18:245-253, 1964.MR0165689 (29:2969)
17. D. Shanks. Nonlinear transformations of divergent and slowly convergent sequences. J. Math. and Phys., 34:1-42, 1955.MR0068901 (16:961e)
18. A. Sidi. A new variable transformation for numerical integration. In H. Brass and G. Hammerlin, editors, Numerical Integration IV, number 112 in ISNM, pages 359-373, Birkhauser, Basel, 1993.MR1248416 (94k:65032)
19. 3. Preprint. Computer Science Dept., Technion-Israel Institute of Technology, 2003.
20. A. Sidi. Practical Extrapolation Methods: Theory and Applications. Number 10 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2003.MR1994507 (2004e:65005)
21. A. Sidi. Euler-Maclaurin expansions for integrals with endpoint singularities: a new perspective. Numer. Math., 98:371-387, 2004.MR2092747
22. m variable transformations. Part I: Smooth integrands. Appl. Math. Comp., 2005. In press. First published electronically on April 11, 2005.
23. m variable transformations to numerical integration over surfaces of spheres. J. Comp. Appl. Math., 2005. In press. First published electronically on April II, 2005.
24. I.H. Sloan and S. Joe. Lattice Methods in Multiple Integration. Clarendon Press, Oxford, 1994.MR 1442955 (98a:65026)
25. P. Verlinden, D.M. Potts, and J.N. Lyness. Error expansions for multidimensional trapezoidal rules with Sidi transformations. Numer. Algorithms, 16:321-347, 1997.MR1617168 (99h:65039)
26. n) transformation. Mathematical Tables and Other Aids to Computation, 10:91-96, 1956.MR0084056 (18:801e)