Algorithm 691: Improving QUADPACK automatic integration routines

ABSTRACT

Two automatic adaptive integrators from QUADPACK (namely, QAG, and QAGS) are modified by substituting the Gauss-Kronrod rules used for local quadrature with recursive monotone stable (RMS) formulas. Extensive numerical tests, both for one-dimensional and two-dimensional integrals, show that the resulting programs are faster, perform less functional evaluations, and are more suitable

REFERENCES

• 1: CASALETTO, J., PICKET, M., AND RICE, J. R. A comparison of some numerical integration programs, Signum Newsletter 4 (1969), 30-40.
• 2: DAvis, P. J., AND RABINOWITZ, P. Methods of Numerical Integration. Academic Press, New York, 1984.
• 3: DEBooR, C.. CADRE: An algorithm for numerical quadrature. In Mathematical Software, J. R. Rice, Ed., Academic Press, New York, 1971, pp. 417-449.
• 4: Jack J. Dongarra , Eric Grosse, Distribution of mathematical software via electronic mail, ACM SIGNUM Newsletter, v.20 n.3, p.45-47, July 1985 [doi>10.1145/1057947.1057951]
• 5: ENGELS, H. Numertcal Quadrature and Cubature. Academic Press, New York, 1980.
• 6: Graeme Fairweather , Pat Keast, An investigation of Romberg quadrature, ACM Transactions on Mathematical Software (TOMS), v.4 n.4, p.316-322, December 1978 [doi>10.1145/356502.356477]
• 7: Paola Favati , Grazia Lotti , Francesco Romani, Interpolatory integration formulas for optimal composition, ACM Transactions on Mathematical Software (TOMS), v.17 n.2, p.207-217, June 1991 [doi>10.1145/108556.108571]
• 8: P. A. Fox , A. D. Hall , N. L. Schryer, Algorithm 528: Framework for a Portable Library [Z], ACM Transactions on Mathematical Software (TOMS), v.4 n.2, p.177-188, June 1978 [doi>10.1145/355780.355789]
• 9: W. Morven Gentleman, Algorithm 424: Clenshaw-Curtis quadrature [D-1], Communications of the ACM, v.15 n.5, p.353-355, May 1972 [doi>10.1145/355602.355603]
• 10: GENZ, A. C., AND MALIK, A. A. Remarks on Algorithm 6: An adaptive algorithm for numerical integrations over an N-dimensional rectangular region. J. Comput. Appl. Math. 6 (1980), 295-302.
• 11: KAHANER, D.K. Comparison of numerical quadrature formulas. In Mathematical Software, J. R. Rice, Ed., Academic Press, New York, 1971, pp. 229-259.
• 12: David K. Kahaner , Ottis W. Rechard, TWODQD an adaptive routine for two-dimensional integration, Journal of Computational and Applied Mathematics, v.17 n.1-2, p.215-234, Feb. 1987 [doi>10.1016/0377-0427(87)90048-3]
• 13: David K. Kahaner , Mark B. Wells, An Experimental Algorithm for N-Dimensional Adaptive Quadrature, ACM Transactions on Mathematical Software (TOMS), v.5 n.1, p.86-96, March 1979 [doi>10.1145/355815.355821]
• 14: LYNESS, J. N. When not to use an automatic quadrature routine. SIAM Rev. 25 (1983), 63-87.
• 15: LYNESS, J. N., AND KAGANOVE, J. J. A technique for comparing automatic quadrature routines. Comp. J. 20 (1975), 170-177.
• 16: J. N. Lyness , J. J. Kaganove, Comments on the Nature of Automatic Quadrature Routines, ACM Transactions on Mathematical Software (TOMS), v.2 n.1, p.65-81, March 1976 [doi>10.1145/355666.355671]
• 17: LYNESS, J. N., AND NINHAM, B. W. Numerical quadrature and asymptotic expansions. Math. Comput. 21 (1967), 162-178.
• 18: PIESSENS, R. An algorithm for automatic integration. Angewandte Informatik 9 (1973), 399-401.
• 19: PIE~gEN~, R, ~ AL. QUADPACK: A Sulgroutlne Pac}~age for At~torr~ati~ l~tcgratioT~. Springer-Verlag, Berlin, 1983.
• 20: ROBINSON, I. A comparison of numerical integration programs, J. Comput. Appl. Math. 5 (1979), 207-223.
• 21: SHANKS, D. Nonlinear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34 (1955), 1-42.
• 22: WYNN, P. On a device for computing the em(Sn) transformation. Math. Comput. 10 (1956), 91-96.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.4 Quadrature and Numerical Differentiation
          Subjects: Adaptive and iterative quadrature

Additional Classification:
  G. Mathematics of Computing

General Terms:
Algorithms

Keywords:
integration, interpolatory quadrature, program testing