Numerical integration in a symbolic context

ABSTRACT

Techniques for numerical integration within a symbolic computation environment are discussed. The goal is to develop a fully automated numerical integration code that handles infinite intervals of integration and that handles various types of integrand singularities. Such a code should also be able to compute to arbitrarily high precision. For the case of an analytic integrand on a finite interval, a Clenshaw-Curtis quadrature routine is used. A concept of general (non-Taylor) series expansions forms the basis of techniques for identifying transformations that may yield an analytic integrand. For the case when no transformation is successful, the general series expansion is used to represent the integrand and it is directly integrated to move beyond the singular point. The latter technique relies on a powerful symbolic integrator that can express integrals in terms of special functions.

REFERENCES

• Cha83: Bruce W. Char , Keith O. Geddes , W. Morven Gentleman , Gaston H. Gonnet, The design of maple: A compact, portable and powerful computer algebra system, Proceedings of the European Computer Algebra Conference on Computer Algebra, p.101-115, March 28-30, 1983
• Cha85: B.W. Char, K.O. Geddes, G.H. Gonnet, and S.M. Watt, Maple User'8 Guide, WATCOM Publications Ltd., Waterloo, Ontario, Canada (1985). Incorporates the Maple tutorial "First Leaves", and "Maple Reference Manual, 4th edition".
• Don75: Elise de Doncker and Robert Piessens, A bibliography on automatic integration, Report TW 26, Katholieke Universiteit Leuven, (August 1975).
• Don75a: Elise de Doncker and Robert Piessens, Automatic computation of integrals with singular integrand, over a finite or aa infinite range, Report 7W 22, Katholieke Universiteit Leuven, (June 1975).
• Fat81: Richard J. Fateman, Computer algebra and numerical integration, Proceedings of the fourth ACM symposium on Symbolic and algebraic computation, p.228-232, August 05-07, 1981, Snowbird, Utah, United States [doi>10.1145/800206.806401]
• Ged79: K. O. Geddes, Remark on “Algorithm 424: Clenshaw-Curtis Quadrature [O1]”, ACM Transactions on Mathematical Software (TOMS), v.5 n.2, p.240, June 1979 [doi>10.1145/355826.355838]
• Gen72: 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]
• Gro82: Numerical Algorithms Group, Subroutine Library, NAG USA Inc, 1131 Warren Avenue, Downers Grove, ill. 60515 (1982).
• IMS79: IMSL, Inc., IMSL Library, Edition 7, International Mathematical and Statistical Libraries, Inc., Houston, Texas (1979).
• Ris69: R.H. Risch, The problem of integration in finite terms, Trans. Am. Math. Soc. 139 pp. 167-189 (1969).
• Wan71: P.S. Wang, Evaluation of Definite Integrals by Symbolic Manipulation, M.I.T., Cambridge, Mass. (1971). (Ph.D thesis).

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

Additional Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION

General Terms:
Algorithms, Design