Symbolic Computation: A Formula is Worth 10000 Numbers

  • J. H. Davenport
Conference paper
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 83)


Hamming [29] remarked that “The purpose of computing is insight, not numbers”, and the aim of all computation must be to help the user obtain insight into his problem. Insight is rarely achieved by large quantities of numbers alone, and one very common way of increasing the insight obtained is to use graphics — “A picture is worth ten thousand numbers”. In this paper we present another means of gaining insight — the use of symbolic manipulation (also known as Computer Algebra) — “a formula is worth 10000 numbers”. There are three ways in which symbolic manipulation can help solve problems:
  1. a)

    It may provide a complete answer;

  2. b)

    It may provide a symbolic approximate answer;

  3. c)

    It may assist in the provision of a numeric answer.

This distinction is somewhat vague, since a symbolic approximate answer is an exact answer to an approximation problem (unlike the case in numerical analysis), and the assistance in providing numerical answers comes from the previous two areas. We will look at these three areas, give some examples of major uses of each, and discuss, where possible, applications particularly related to control theory. Where there seems a technique of wide applicaibility for which we do not know a reference in control theory however, we have not hesitated to quote details from elsewhere.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aldins,J, Kinoshita,T, Brodsky,S.J. & Dufner,A.3., Photon-Photon Scatter-ing Contribution to the Sixth Order Magnetic Moment of the Muon. Phys. Rev. Lett. 23(1969) pp. 441–443.Google Scholar
  2. [2]
    Karlhede,A, A Computer-aided Complete Classification of Geometries in General Relativity. First Results. Phys. Lett. 8011 (1980) pp. 229–231.Google Scholar
  3. [3]
    Aman,J.E. & Karlhede,A, An Algorithmic Classification of Geometries in General Relativity. Proc. SYMSAC el (ed. P.S. Wang ), ACM, New Yor 1981, pp. 79–84.Google Scholar
  4. [4]
    d’Inverno,R.A., & Ouartic Equations and Algorithms for Riemann Tensor Classification. Proc. EUROSAM 84 (Springer Lecture Notes in Computer Science 174 ) pp. 47–58.Google Scholar
  5. [5]
    Andrews,G.E., Plane Partitions aII): The Weak MacDonald Conjecture. Inventiones Math. 53 (1979) pp. 193–225.CrossRefGoogle Scholar
  6. [6]
    Barton,D.R, & Fitch,J.P, A Review of Algebraic Manipulative Programs and their Application. Comp. J. 15 (1972) pp. 362–381CrossRefGoogle Scholar
  7. [7]
    Barton,D, & Zahar,R.V.M., The automatic solution of systems of ordinary differential equations by the method of Taylor series. The best computer papers of 1971, Auerbach, New York, 1971. See also Comp. J. 14 (1971) pp. 243–248.CrossRefGoogle Scholar
  8. [8]
    ConnP11,E.H. & Wright,D, The Jacobian Conjecture: Reduction of T1F’grP Formal Expansion of the Inverse. Bull. (N.S.) A.M.S. 7 (1982) pp. 297Google Scholar
  9. [9]
    Baur,W, & Strassen,V., The Complexity of Partial Derivatives. Theor. Comp. Sci. 22 (1983) pp. 317–330.Google Scholar
  10. [10]
    Buchberger,B, Ein algOrithmisches Kriterium +Ur die L6sbarkeit eines alqebraischen Gleichungssystem. Aeg. Math. 4 (1970) pp. 374–383.Google Scholar
  11. [11]
    Buchberger,S, A Survey on the Method of Groebner bases for Solving Protdems in Connection with Systems of Multi-variate Pcdynomials. Proc. 2nd. RIKEN Symp. Symbolic & Algebraic Computation (ed N. Inada & T, Soma), World Scientific Publ„ Singapore, 1985, pp, 69–83.Google Scholar
  12. [12]
    Buchberder,B., Collins,9. 8. Loos,R., Computer Pidebra. Computind Sup-plementurn Springer-verrlag, 1982.Google Scholar
  13. [13]
    Gentleman,M.W. Sonnet,G.H„ The Desidn of’ MAPLE: Es Compact, Portable, and POwerful Computer Aldabra System. Froc. EURO-CAL 87 (.8.:pringer Lecture Notes (Computer Science 162, 1987) PP. 101–115.Google Scholar
  14. [14]
    Collins,G,E„ The Calcul.,?„tior, of FolynorniEkl Resultants. ACM 18(1971) pp. 515-:532.Google Scholar
  15. [15]
    Collins,G.E., Quantifier Elimination -for Real Closed Fields by Cylindrical F:r1Q8brai.0 Decomposition. Froc. 2nd. GI Con+. Automata Theory & Formal LariQuageE:F.,pr-3.1–1CIEr Lecture Notes 15 Computer Science =7, 1975 ) pp. 174–187.Google Scholar
  16. [16]
    Collins,S.E., Loos,R., RE-al Zeros, of Polynomials. C122, pp. 87–94.Google Scholar
  17. [17]
    Coppersmith,D, ET!,.veriport,J,H., An Application of Factoring. J, Es„r’mbolic. Computation iti9e5) pp. 241–247:Google Scholar
  18. [18]
    Davenport,J.H., Integration: Form-al and Numeric Approaches, Froc, “Tools, Method’s Landuages for Scientific:le Endineerind Calculation” (ed. B. Ford, J.C. Fault F. Thomasset, North-Holland, 1’384, pp. 417–427Google Scholar
  19. [19]
    Davenport,J.H., and NumeriL pui ti on IntepralS. TO appear in Froc. 1985 IBM Deutschland Symp. on Accurate Scientific Corr,putation::Springer. LECt.LtrE r’.10t.LEE Computer Science).Google Scholar
  20. [20]
    Davenport,J,H,, Computer Algebra for Cylindrical Ploe.braic F2,ecomposition. TRITp-NA-s5:11, KTH, Stockholm, Sept. 1985.Google Scholar
  21. [21]
    Dacvenport,:l.H., Real Roots of Polynomials. Submitted to B.I.TGoogle Scholar
  22. [22]
    Fitch,J.R., Litilis.:ations du Calcul Forme“ Froc. Techniques de Calcul Formel pour l’Hutomatidue, l’Hnalyse. de.s S’../St:IMEE Er_ 1E. Tr.:-..atement du IMHb, Grenoble, 1.7.-2u Dec. 1987,,to be repubiished by Editions CNRS).Google Scholar
  23. [27]
    Solvino Alcletraic Problems.. with REDUCE. J. a.,-,T1bolic C7…:FilpUt_E,.- tion 1(1’9’25) pp. 211–227Google Scholar
  24. [24]
    Fitch.J.R: Moore,p,m,p_ The 474,,tomatic Derivation of Periodic Solutions to a. Class of We.akly Dif-Ferential Equations, PrOC. msAc s w=m, Nev., York, pp. -2,79–244.Google Scholar
  25. [25]
    Fliess,M., Fonctionne.11es causales non lingaires et indgterming-es non commutatives. Bull. Soc. Math. France 10’3T, 1981 ) pp. 7–40.Google Scholar
  26. [26]
    Frick,i, The Computer Aldebra System SHEEP: what it cam amd cannot do in General Rslati.yity. Preprint., Institute of Theoratical Pft::/sics, sit,, of Stoci,:holm, 1977Google Scholar
  27. [27]
    Gebauer,R. Note on “A System o-F Polynomial Equations”, SIG-spri Bulletin 18,:19R4)Google Scholar
  28. [28]
    Gramain,F. Re,:essat,E., Sur une question de A. Douad,./ et P. Mousse.C..R. Hcad. Sc. Faris 8,5r. I 302(19086) pp. 199–201.Google Scholar
  29. [29]
    Hamming,R.N., Nuffierical ried.-Iods tor Scientists and Enpineers. rlcbr - Hill, New York, 1962.Google Scholar
  30. [30]
    Hearm,A.C,, Scientific Applioations Cuiliputstion. Computer- Sci-ence and E.cientific Cc..mputing, Academic Press, 1976, pp. 27Google Scholar
  31. [31]
    Hearn,A.C., The Per.-ui,a1 Aldebra Machine. Prop.;FIE’ 50, Elsevier-North Holland, Ne,,4 Yo.,r1:., 1980, pp. 621–628.Google Scholar
  32. [32]
    Jenks,R.D., A Primer! 11 to Nev.! SCRATCHFAC. Proc. EUROSAM 84 r.Springer Lecture t..c.,tea.. in Computer Science 174, 1994 ) pp. 127–147.Google Scholar
  33. [33]
    Lamnahhi-I agarrigue,m, Lamnabhi,M., Algebraic Comp.utation of the Ptatistics o-F sorne Nonlinear Stochastic Differential Equations. Proc. EUROCAL 873 (Springer Lecture Notes in Computer Science 162, 1983 ) pp. 55–67.CrossRefGoogle Scholar
  34. [34]
    Moses,J., Solutio,-, of a System of Polynomial Equations b’)., Comm. ACM 9 (1968) pp. 634–627.Google Scholar
  35. [35]
    Norman,A.C., TAYLOR Users.’ Manual. University Ser,,ice, 1973.Google Scholar
  36. [36]
    Norman,A.C., Expanding the solutions of inplicit ferential equations. CcJimp. J. 19 (1976) pp. 63–6e.Google Scholar
  37. [37]
    Norman,A.C„ Private Communication, March 1977.Google Scholar
  38. [38]
    Paveiie, R,A., Utilisation du Calcul Formel pourGoogle Scholar
  39. [39]
    Paksan’ R,A., Utilisation du Calcul Formel pourGoogle Scholar
  40. [40]
    Paksan’ R,A., Utilisation du Calcul Formel pour cl’equations polynomiales (Applications Cy.cle, Paris IX, 27 mars 1’986.Google Scholar
  41. [41]
    Pcidwartz..1,T. Sharir,M., On the “Piano Movers” Proble-m. II. General Techniques for Computing Topological Properties of Reail Algebraic Math. 4 (1983) pp. 298–351.Google Scholar
  42. [42]
    Stoutemyer,D.R., A PrE.-,.(iew of the !“…i.ext IEM-PC Version of. muMath. Proc. EURrinAL 85 I (Springer Lecture Notes…in Computer Science 203) pp. 72–4L1.Google Scholar
  43. [43]
    Nalter,E., of State Space Models. Sprinder Lecture Notes in Biomathe.matics 46, 1982.Google Scholar
  44. [44]
    Nang,P.S.. Chang,T.Y.P. van Hulzen,J.A., Code Generation and Optimization for Finite Element Analysis. Proc. EUROSAM 54 (Springer Lecture Notes in Computer science 174 ) pp. 227–247.Google Scholar
  45. [45]
    Nolfe,F.S.,Checking the-. Calculation of Gradients. ACM TOMS 6(1982) pp. 227–17 4 O.Google Scholar
  46. [46]
    Zassenhaus,H., Hs-nsel Fectbrizaticn Numbe.r Theory icidx-.9!., 291–211:Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1986

Authors and Affiliations

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

Personalised recommendations