Pedro Gonnet
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Abstract
We present two new adaptive quadrature routines. Both routines differ from previously published algorithms in many aspects, most significantly in how they represent the integrand, how they treat nonnumerical values of the integrand, how they deal with improper divergent integrals, and how they estimate the integration error. The main focus of these improvements is to increase the reliability of the algorithms without significantly impacting their efficiency. Both algorithms are implemented in MATLAB and tested using both the “families” suggested by Lyness and Kaganove and the battery test used by Gander and Gautschi and Kahaner. They are shown to be more reliable, albeit in some cases less efficient, than other commonly-used adaptive integrators.
- }}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.Google Scholar
- }}Berntsen, J. and Espelid, T. O. 1991. Error estimation in automatic quadrature routines. ACM Trans. Math. Softw. 17, 2, 233--252. Google ScholarDigital Library
- }}Björck, Å
. and Pereyra, V. 1970. Solution of Vandermonde systems of equations. Math. Comput. 24, 112, 893--903.Google Scholar
- }}Casaletto, J., Pickett, M., and Rice, J. 1969. A comparison of some numerical integration programs. SIGNUM Newslett. 4, 3, 30--40. Google ScholarDigital Library
- }}Clenshaw, C. W. and Curtis, A. R. 1960. A method for numerical integration on an automatic computer. Numer. Math. 2, 197--205.Google ScholarDigital Library
- }}Davis, P. J. and Rabinowitz, P. 1984. Numerical Integration 2nd Ed. Blaisdell Publishing Company, Waltham, MA.Google Scholar
- }}de Boor, C. 1971. CADRE: An algorithm for numerical quadrature. In Mathematical Software, J. R. Rice, Ed. Academic Press, New York, 201--209.Google Scholar
- }}de Doncker, E. 1978. An adaptive extrapolation algorithm for automatic integration. SIGNUM Newsl. 13, 2, 12--18. Google ScholarDigital Library
- }}Eaton, J. W. 2002. GNU Octave Manual. Network Theory Limited.Google Scholar
- }}Espelid, T. O. 2007. Algorithm 868: Globally doubly adaptive quadrature---Reliable Matlab codes. ACM Trans. Math. Softw. 33, 3, 21. Google ScholarDigital Library
- }}Favati, P., Lotti, G., and Romani, F. 1991. Interpolatory integration formulas for optimal composition. ACM Trans. Math. Softw. 17, 2, 207--217. Google ScholarDigital Library
- }}Forsythe, G. E., Malcolm, M. A., and Moler, C. B. 1977. Computer Methods for Mathematical Computations. Prentice-Hall, Inc., Englewood Cliffs, NJ 07632. Google ScholarDigital Library
- }}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. Google ScholarDigital Library
- }}Gallaher, L. J. 1967. Algorithm 303: An adaptive quadrature procedure with random panel sizes. Comm. ACM 10, 6, 373--374. Google ScholarDigital Library
- }}Gander, W. and Gautschi, W. 1998. Adaptive quadrature---Revisited. Tech. rep. 306, Department of Computer Science, ETH Zurich, Switzerland.Google Scholar
- }}Gander, W. and Gautschi, W. 2001. Adaptive quadrature---Revisited. BIT 40, 1, 84--101.Google ScholarCross Ref
- }}Gautschi, W. 1975. Norm estimates for inverses of Vandermonde matrices. Numer. Math. 23, 337--347.Google ScholarDigital Library
- }}Gautschi, W. 1997. Numerical Analysis, An Introduction. Birkhäuser Verlag, Basel. Google ScholarDigital Library
- }}Gentleman, W. M. 1972. Implementing Clenshaw-Curtis quadrature, II computing the cosine transformation. Comm. ACM 15, 5, 343--346. Google ScholarDigital Library
- }}Golub, G. H. and Welsch, J. H. 1969. Calculation of Gauss quadrature rules. Math. Comput. 23, 106, 221--230.Google Scholar
- }}Gonnet, P. G. 2009a. Adaptive quadrature re-revisited. Ph.D. thesis, ETH Zürich, Switzerland.Google Scholar
- }}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}.Google Scholar
- }}Henriksson, S. 1961. Contribution no. 2: Simpson numerical integration with variable length of step. BIT Numer. Math. 1, 290.Google Scholar
- }}Higham, N. J. 1988. Fast solution of Vandermonde-like systems involving orthogonal polynomials. IMA J. Numer. Anal. 8, 473--486.Google ScholarCross Ref
- }}Higham, N. J. 1990. Stability analysis of algorithms for solving confluent Vandermonde-like systems. SIAM J. Matrix Anal. Appl. 11, 1, 23--41. Google ScholarDigital Library
- }}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. Google ScholarDigital Library
- }}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.Google Scholar
- }}Krommer, A. R. and Ü
berhuber, C. W. 1998. Computational Integration. SIAM, Philadelphia, PA.Google Scholar
- }}Kronrod, A. S. 1965. Nodes and Weights of Quadrature Formulas---Authorized Translation from the Russian. Consultants Bureau, New York.Google Scholar
- }}Kuncir, G. F. 1962. Algorithm 103: Simpson’s rule integrator. Comm. ACM 5, 6, 347. Google ScholarDigital Library
- }}Laurie, D. P. 1983. Sharper error estimates in adaptive quadrature. BIT 23, 258--261.Google ScholarCross Ref
- }}Laurie, D. P. 1992. Stratified sequences of nested quadrature formulas. Quaestiones Mathematicae 15, 364--384.Google ScholarCross Ref
- }}Lyness, J. N. 1970. Algorithm 379: SQUANK (Simpson quadrature used adaptively -- noise killed). Comm. ACM 13, 4, 260--262. Google ScholarDigital Library
- }}Lyness, J. N. and Kaganove, J. J. 1977. A technique for comparing automatic quadrature routines. Comput. J. 20, 2, 170--177.Google ScholarCross Ref
- }}Malcolm, M. A. and Simpson, R. B. 1975. Local versus global strategies for adaptive quadrature. ACM Trans. Math. Softw. 1, 2, 129--146. Google ScholarDigital Library
- }}McKeeman, W. M. 1962. Algorithm 145: Adaptive numerical integration by Simpson’s rule. Comm. ACM 5, 12, 604. Google ScholarDigital Library
- }}McKeeman, W. M. 1963. Algorithm 198: Adaptive integration and multiple integration. Comm. ACM 6, 8, 443--444. Google ScholarDigital Library
- }}McKeeman, W. M. and Tesler, L. 1963. Algorithm 182: Nonrecursive adaptive integration. Comm. ACM 6, 6, 315. Google ScholarDigital Library
- }}Morrin, H. 1955. Integration subroutine -- Fixed point. Tech. rep. 701 Note #28, U.S. Naval Ordnance Test Station, China Lake, California.Google Scholar
- }}Ninham, B. W. 1966. Generalised functions and divergent integrals. Numer. Math. 8, 444--457.Google ScholarCross Ref
- }}Ninomiya, I. 1980. Improvements of adaptive Newton-Cotes quadrature methods. J. Inform. Porc. 3, 3, 162--170.Google Scholar
- }}O’Hara, H. and Smith, F. J. 1968. Error estimation in Clenshaw-Curtis quadrature formula. Comput. J. 11, 2, 213--219.Google ScholarCross Ref
- }}O’Hara, H. and Smith, F. J. 1969. The evaluation of definite integrals by interval subdivision. Comput. J. 12, 2, 179--182.Google ScholarCross Ref
- }}Oliver, J. 1972. A doubly-adaptive Clenshaw-Curtis quadrature method. Comput. J. 15, 2, 141--147.Google ScholarCross Ref
- }}Patterson, T. N. L. 1973. Algorithm 468: Algorithm for automatic numerical integration over a finite interval. Comm. ACM 16, 11, 694--699. Google ScholarDigital Library
- }}Piessens, R. 1973. An algorithm for automatic integration. Angewandte Inf. 9, 399--401.Google Scholar
- }}Piessens, R., de Doncker-Kapenga, E., Überhuber, C. W., and Kahaner, D. K. 1983. QUADPACK A Subroutine Package for Automatic Integration. Springer-Verlag, Berlin.Google Scholar
- }}Ralston, A. and Rabinowitz, P. 1978. A First Course in Numerical Analysis. McGraw-Hill Inc., New York.Google Scholar
- }}Rice, J. R. 1975. A metalgorithm for adaptive quadrature. J. ACM 22, 1, 61--82. Google ScholarDigital Library
- }}Robinson, I. 1979. A comparison of numerical integration programs. J. Comput. Appl. Math. 5, 3, 207--223.Google ScholarCross Ref
- }}Rowland, J. H. and Varol, Y. L. 1972. Exit criteria for Simpson’s compound rule. Math. Comput. 26, 119, 699--703.Google Scholar
- }}Rutishauser, H. 1976. Vorlesung über Numerische Mathematik. Birkhäuser Verlag, Basel.Google Scholar
- }}Schwarz, H. R. 1997. Numerische Mathematik. B. G. Teubner, Stuttgart.Google Scholar
- }}Stiefel, E. 1961. Einführung in Die Numerische Mathematik. B. G. Teubner Verlagsgesellschaft, Stuttgart.Google Scholar
- }}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. Google ScholarDigital Library
- }}The Mathworks. 2003. MATLAB 6.5 Release Notes. The Mathworks, Natick, MA.Google Scholar
- }}Trefethen, L. N. 2008. Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 1, 67--87. Google ScholarDigital Library
- }}Venter, A. and Laurie, D. P. 2002. A doubly adaptive integration algorithm using stratified rules. BIT 42, 1, 183--193.Google ScholarCross Ref
- }}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.Google Scholar
- }}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.Google Scholar
Index Terms
- Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants
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