Increasing the reliability of adaptive quadrature using explicit interpolants

ACM Transactions on Mathematical Software

Volumn 37, Issue 3, 2010, Pages

  Gonnet, Pedro ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

Adaptive quadrature; Error estimation; Interpolation; Orthogonal polynomials

Indexed keywords

ADAPTIVE QUADRATURE; BATTERY TESTS; ERROR ESTIMATION; GAUTSCHI; INTEGRATION ERROR; INTERPOLANTS; ORTHOGONAL POLYNOMIAL;

INTERPOLATION; ORTHOGONAL FUNCTIONS;

ALGORITHMS;

EID: 77957578733     PISSN: 00983500     EISSN: 15577295     Source Type: Journal    
DOI: 10.1145/1824801.1824804     Document Type: Article

References (60)

1. BAUER, F. L., RUTISHAUSER, H., AND STIEFEL, E. 1963. New aspects in numerical quadrature. In Proceedings of the Experimental Arithmetic, High Speed Computing and Mathematics, N. C. Metropolis, A. H. Taub, J. Todd, and C. B. Tompkins, Eds. American Mathematical Society, Providence, RI, 199-217.
2. BERNTSEN, J. AND ESPELID, T. O. 1991. Error estimation in automatic quadrature routines. ACM Trans. Math. Softw. 17, 2, 233-252.
3. BJÖ RCK, Å . AND PEREYRA, V. 1970. Solution of Vandermonde systems of equations. Math. Comput. 24, 112, 893-903.
4. CASALETTO, J., PICKETT, M., AND RICE, J. 1969. A comparison of some numerical integration programs. SIGNUM Newslett. 4, 3, 30-40.
5. CLENSHAW, C. W. AND CURTIS, A. R. 1960. A method for numerical integration on an automatic computer. Numer. Math. 2, 197-205.
6. DAVIS, P. J. AND RABINOWITZ, P. 1984. Numerical Integration 2nd Ed. Blaisdell Publishing Company, Waltham, MA.
7. DE BOOR, C. 1971. CADRE: An algorithm for numerical quadrature. In Mathematical Software, J. R. Rice, Ed. Academic Press, New York, 201-209.
8. DE DONCKER, E. 1978. An adaptive extrapolation algorithm for automatic integration. SIGNUM Newsl. 13, 2, 12-18.
9. EATON, J. W. 2002. GNU Octave Manual. Network Theory Limited.
10. ESPELID, T. O. 2007. Algorithm 868: Globally doubly adaptive quadrature-ReliableMatlab codes. ACM Trans. Math. Softw. 33, 3, 21.
11. FAVATI, P., LOTTI, G., AND ROMANI, F. 1991. Interpolatory integration formulas for optimal composition. ACM Trans. Math. Softw. 17, 2, 207-217.
12. FORSYTHE, G. E.,MALCOLM, M. A., AND MOLER, C. B. 1977. Computer Methods for Mathematical Computations. Prentice-Hall, Inc., Englewood Cliffs, NJ 07632.
13. GALASSI, M., DAVIES, J., THEILER, J., GOUGH, B., JUNGMAN, G., ALKEN, P., BOOTH, M., AND ROSSI, F. 2009. GNU Scientific Library Reference Manual 3rd Ed. Network Theory Ltd.
14. GALLAHER, L. J. 1967. Algorithm 303: An adaptive quadrature procedure with random panel sizes. Comm. ACM 10, 6, 373-374.
15. GANDER, W. AND GAUTSCHI, W. 1998. Adaptive quadrature-Revisited. Tech. rep. 306, Department of Computer Science, ETH Zurich, Switzerland.
16. GANDER, W. AND GAUTSCHI, W. 2001. Adaptive quadrature-Revisited. BIT 40, 1, 84-101.
17. GAUTSCHI, W. 1975. Norm estimates for inverses of Vandermonde matrices. Numer. Math. 23, 337-347.
18. GAUTSCHI, W. 1997. Numerical Analysis, An Introduction. Birkhäuser Verlag, Basel.
19. GENTLEMAN, W. M. 1972. Implementing Clenshaw-Curtis quadrature, II computing the cosine transformation. Comm. ACM 15, 5, 343-346.
20. GOLUB, G. H. AND WELSCH, J. H. 1969. Calculation of Gauss quadrature rules. Math. Comput. 23, 106, 221-230.
21. GONNET, P. G. 2009a. Adaptive quadrature re-revisited. Ph.D. thesis, ETH Z̈ urich, Switzerland.
22. GONNET, P. G. 2009b. Efficient construction, update and downdate of the coefficients of interpolants based on polynomials satisfying a three-term recurrence relation. arxiv:1003.4628v2 [cs.NA].
23. HENRIKSSON, S. 1961. Contribution no. 2: Simpson numerical integration with variable length of step. BIT Numer. Math. 1, 290.
24. HIGHAM, N. J. 1988. Fast solution of Vandermonde-like systems involving orthogonal polynomials. IMA J. Numer. Anal. 8, 473-486.
25. HIGHAM, N. J. 1990. Stability analysis of algorithms for solving confluent Vandermonde-like systems. SIAM J. Matrix Anal. Appl. 11, 1, 23-41.
26. HILLSTROM, K. E. 1970. Comparison of several adaptive Newton-Cotes quadrature routines in evaluating definite integrals with peaked integrands. Comm. ACM 13, 6, 362-365.
27. KAHANER, D. K. 1971. 5.15 Comparison of numerical quadrature formulas. In Mathematical Software, J. R. Rice, Ed. Academic Press, New York and London, 229-259.
28. KROMMER, A. R. AND ÜBERHUBER, C. W. 1998. Computational Integration. SIAM, Philadelphia, PA.
29. KRONROD, A. S. 1965. Nodes and Weights of Quadrature Formulas-Authorized Translation from the Russian. Consultants Bureau, New York.
30. KUNCIR, G. F. 1962. Algorithm 103: Simpson's rule integrator. Comm. ACM 5, 6, 347.
31. LAURIE, D. P. 1983. Sharper error estimates in adaptive quadrature. BIT 23, 258-261.
32. LAURIE, D. P. 1992. Stratified sequences of nested quadrature formulas. Quaestiones Mathematicae 15, 364-384.
33. LYNESS, J. N. 1970. Algorithm 379: SQUANK (Simpson quadrature used adaptively - noise killed). Comm. ACM 13, 4, 260-262.
34. LYNESS, J. N. AND KAGANOVE, J. J. 1977. A technique for comparing automatic quadrature routines. Comput. J. 20, 2, 170-177.
35. MALCOLM, M. A. AND SIMPSON, R. B. 1975. Local versus global strategies for adaptive quadrature. ACM Trans. Math. Softw. 1, 2, 129-146.
36. MCKEEMAN, W. M. 1962. Algorithm 145: Adaptive numerical integration by Simpson's rule. Comm. ACM 5, 12, 604.
37. MCKEEMAN, W. M. 1963. Algorithm 198: Adaptive integration and multiple integration. Comm. ACM 6, 8, 443-444.
38. MCKEEMAN, W. M. AND TESLER, L. 1963. Algorithm 182: Nonrecursive adaptive integration. Comm. ACM 6, 6, 315.
39. MORRIN, H. 1955. Integration subroutine - Fixed point. Tech. rep. 701 Note #28, U.S. Naval Ordnance Test Station, China Lake, California.
40. NINHAM, B. W. 1966. Generalised functions and divergent integrals. Numer. Math. 8, 444-457.
41. NINOMIYA, I. 1980. Improvements of adaptive Newton-Cotes quadrature methods. J. Inform. Porc. 3, 3, 162-170.
42. O'HARA, H. AND SMITH, F. J. 1968. Error estimation in Clenshaw-Curtis quadrature formula. Comput. J. 11, 2, 213-219.
43. O'HARA, H. AND SMITH, F. J. 1969. The evaluation of definite integrals by interval subdivision. Comput. J. 12, 2, 179-182.
44. OLIVER, J. 1972. A doubly-adaptive Clenshaw-Curtis quadrature method. Comput. J. 15, 2, 141-147.
45. PATTERSON, T. N. L. 1973. Algorithm 468: Algorithm for automatic numerical integration over a finite interval. Comm. ACM 16, 11, 694-699.
46. PIESSENS, R. 1973. An algorithm for automatic integration. Angewandte Inf. 9, 399-401.
47. PIESSENS, R., DE DONCKER-KAPENGA, E., ÜBERHUBER, C. W., AND KAHANER, D. K. 1983. QUADPACK A Subroutine Package for Automatic Integration. Springer-Verlag, Berlin.
48. RALSTON, A. AND RABINOWITZ, P. 1978. A First Course in Numerical Analysis. McGraw-Hill Inc., New York.
49. RICE, J. R. 1975. A metalgorithm for adaptive quadrature. J. ACM 22, 1, 61-82.
50. ROBINSON, I. 1979. A comparison of numerical integration programs. J. Comput. Appl. Math. 5, 3, 207-223.
51. ROWLAND, J. H. AND VAROL, Y. L. 1972. Exit criteria for Simpson's compound rule. Math. Comput. 26, 119, 699-703.
52. RUTISHAUSER, H. 1976. Vorlesung über Numerische Mathematik. Birkhäuser Verlag, Basel.
53. SCHWARZ, H. R. 1997. Numerische Mathematik. B. G. Teubner, Stuttgart.
54. STIEFEL, E. 1961. Einf Ührung in Die Numerische Mathematik. B. G. Teubner Verlagsgesellschaft, Stuttgart.
55. SUGIURA, H. AND SAKURAI, T. 1989. On the construction of high-order integration formulae for the adaptive quadrature method. J. Comput. Appl. Math. 28, 367-381.
56. THE MATHWORKS. 2003. MATLAB 6.5 Release Notes. The Mathworks, Natick, MA.
57. TREFETHEN, L. N. 2008. Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 1, 67-87.
58. VENTER, A. AND LAURIE, D. P. 2002. A doubly adaptive integration algorithm using stratified rules. BIT 42, 1, 183-193.
59. VILLARS, D. S. 1956. Use of the IBM 701 computer for quantum mechanical calculations II. overlap integral. Tech. rep. 5257, U.S. Naval Ordnance Test Station, China Lake, CA.
60. WALDVOGEL, J. 2009. Towards a general error theory of the trapezoidal rule. In Approximation and Computation, W. Gautschi and G. Mastroianni and Th.M. Rassias, To appear. Preprint Eds. Springer. http://www.math.ethz.ch/waldvoge/ Projects/nisJoerg.pdf.