Symbolic and numeric manipulation of integrals

  • J. H. Davenport
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 235)


In this paper, we look at various techniques of integration. Computers have obviously revolutionised the theory and practice of numeric integration, but they have equally revolutionised the field of symbolic integration. We describe the current state of symbolic integration, and discuss the impact of this art on numeric integration.


Closed Form Computer Algebra Algebraic Function Elliptic Integral Springer Lecture Note 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Acton, 1970.
    Acton,F.S., Numerical Methods that Work. Harper & Row, 1970.Google Scholar
  2. Baur & Strassen, 1983.
    Baur, W., & Strassen, V., The Complexity of Partial Derivatives. Theor. Comp. Sci. 22(1983) pp. 317–330.Google Scholar
  3. Cherry, 1983.
    Cherry,G.W., Algorithms for Integrating Elementary Functions in Terms of Logarithmic Integrals and Error Functions. Ph.D. Thesis, U. of Delaware. August 1983.Google Scholar
  4. Cherry & Caviness, 1983.
    Cherry,G.W. & Caviness, B.F., Integration in Finite Terms with Special Functions: A Progress Report. Proc. EUROSAM 84 (Springer Lecture Notes in Computer Science 174) pp. 351–358.Google Scholar
  5. Davenport, 1981.
    Davenport,J.H., On the Integration of Algebraic Functions. Springer Lecture Notes in Computer Science 102, Springer Verlag, 1981.Google Scholar
  6. Davenport, 1982.
    Davenport,J.H., On the Parallel Fisch Algorithm (1). Proc. EUROCAM 82 (Springer Lecture Notes in Computer Science 144), pp. 144–157.Google Scholar
  7. Davenport, 1984.
    Davenport, J.H., Intégration algorithmique des fonctions élémentairement transcendantes sur une courbe algébrique. Annales de l'Institut Fourier 34(1984) pp.271–276.Google Scholar
  8. Fateman, 1981.
    Fateman,R.J., Computer Algebra and Numeric Integration. Proc. SYMSAC 81, ACM, New York, pp. 228–232.Google Scholar
  9. Fitch, 1981.
    Fitch,J.P., User-based Integration Software. Proc. SYMSAC 81, ACM, New York, pp. 245–248.Google Scholar
  10. Hardy, 1916.
    Hardy,G.H., The Integration of Functions of a Single Variable, 2nd. ed. C.U.P., 1916.Google Scholar
  11. Hearn, 1980.
    Hearn,A.C., The Personal Algebra Machine. Proc. IFIP 80. Elsevier North-Holland, 1980, pp. 621–628.Google Scholar
  12. Hulshof & van Hulzen, 1984.
    Hulshof,B.J.A. & van Hulzen, J.A., Automatic Error Cumulation Control. Proc. EUROSAM 84 (Springer Lecture Notes in Computer Science 174) pp. 260–271.Google Scholar
  13. Liouville, 1835.
    Liouville, J., Mémoire sur l'intégration d'une classe de fonctions transcendantes, Crelle's J. 13(1835) pp. 93–118.Google Scholar
  14. Lovelace, 1844.
    Lovelace,A.A., Sketch of the Analytic Engine Invented by Charles Babbage (L.F. Menabrea, trans. A.A. Lovelace) Translator's Note A. Taylor's Scientific Memoirs 3(1844). (The relevant passage is quoted by D.E. Knuth in The Art of Computer Programming Vol. I. Addison-Wesley, 1983, p. 1).Google Scholar
  15. Lyness, 1977.
    Lyness, J.N., An Interface Problem in Numerical Software. Proc. Sixth Manitoba Conf. Numerical Methods and Computing (1976), Congr. Numerantium XVIII, 1977, pp. 251–263.Google Scholar
  16. Lyness, 1983.
    Lyness, J.N., When Not to Use an Automatic Quadrature Routine. SIAM Review 25(1983) pp. 63–87.Google Scholar
  17. Moses, 1971.
    Moses, J., Symbolic Integration, the Stormy Decade. Comm. A.C.M. 14(1971) pp. 548–560.Google Scholar
  18. Nolan, 1953.
    Nolan,J., Analytic Differentiation on a Digital Computer. M.A. Thesis, M.I.T. Dept. of Math., May 1953.Google Scholar
  19. NAG, 1982.
    The NAG Library Manual, Mark 9. Numerical Algorithms Group, Limited. Oxford, 1982.Google Scholar
  20. [Norman & Moore, 1977]
    Norman,A.C. & Moore,P.M.A., Implementing the New Risch Integration Algorithm. Proc. 4th. Int. Colloq. Adv. Computing Methods in Theor. Physics, Marseilles, 1977, pp. 99–110.Google Scholar
  21. Rall, 1981.
    Rall,L.B., Automatic Differentiation — Techniques and Examples. Springer Lecture Notes in Computer Science 120, Springer Verlag, 1981.Google Scholar
  22. Risch, 1969.
    Risch, R.H., The Problem of Integration in Finite Terms. Trans A.M.S. 139(1969) pp. 167–189.Google Scholar
  23. Trager, 1979.
    Trager,B.M., Integration of Simple Radical Extensions. Proc. EUROSAM 79 (Springer Lecture Notes in Computer Science 79) pp. 408–414.Google Scholar
  24. Trager, 1984.
    Trager,B.M., Integration of Algebraic Functions. Ph.D. Thesis, M.I.T. Dept. of EE&CS, August 1984.Google Scholar
  25. van der Waerden, 1949.
    van der Waerden, B.L., Modern Algebra, Frederick Ungar, New York, 1949.Google Scholar
  26. Waldvogel, 1983.
    Waldvogel,J., Private Communication, June 3rd, 1983.Google Scholar
  27. Wang et al., 1984
    Wang,P.S., Chang,T.Y.P. & van Hulzen, J.A., Code Generation and Optimization for Finite Element Analysis. Proc. EUROSAM 84 (Springer Lecture Notes in Computer Science 174) pp. 237–247.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • J. H. Davenport
    • 1
  1. 1.School of MathematicsUniversity of BathBathEngland

Personalised recommendations