Advertisement

On the integration of numeric and algebraic computations

  • Gianfranco Mascari
  • Alfonso Miola
Conference paper
  • 135 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 307)

Abstract

This paper studies the properties of computations on numeric and algebraic expressions in order to capture their relations and differences. An analysis of the two kinds of computations from the point of view of program semantics is made. The possibility of integrating these two kinds of computations in a unified computational framework is discussed. The use of p-adic arithmetic, as a possible useful interpretation of algebraic expression, is considered to allow exact mathematical computations in that framework.

Keywords

Operational Semantic Program Scheme Algebraic Computation Algebraic Semantic Program Semantic 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ALB80]
    R. ALBRECHT: Roundings and Approximations in Ordered Sets, Computing Suppl. 2, 17–31 (1980).Google Scholar
  2. [ADJ77]
    ADJ: Initial algebra semantics and continuous algebras, JACM 24 68–95 (1977).Google Scholar
  3. [A-M79]
    G. AUSIELLO, G.F. MASCARI: On the Design of Algebraic Data Structures with the Approach of Abstract Data Types, Proc. Of EUROSAM 79, Springer Verlag (1979).Google Scholar
  4. [A-N80]
    A. ARNOLD, M. NIVAT: Metric interpretations of infinite trees and semantics of non deterministic programs, Theoretical Computer Science 11 181–205 (1980).Google Scholar
  5. [BCR86]
    H.J. BOEHM, R. CARTWRIGHT, M. RIGGLE, O'DONNELL: Exact Real Arithmetic: A Case Study in Higher Order Programming. Proc. Lisp Conference (1986).Google Scholar
  6. [BDG84]
    S.R. BALZAC, and others: The SCRATCHPAD II, IBM Research Report (1984).Google Scholar
  7. [BER82]
    M. BERGMAN: Algebraic specifications: a constructive methodology in logic programming, in Proc. EUROCAM '82, L.N.C.S. vol. 144, Springer Verlag (1982).Google Scholar
  8. [BGY80]
    R.P. BRENT, F.G. GUSTAVSON, D.Y.Y. YUN: Fast solution of Toeplitz systems of equations and computation of Pade approximants, J. of Algorithms, 1, (1980).Google Scholar
  9. [BRO81]
    W.S. BROWN: A simple but Realistic Model of Floating-Point Computation, ACM Transactions of Mathematical Software, Vol. 7, n. 4 (1981).Google Scholar
  10. [B-H79]
    W.S. BROWN, A.C. HEARN: Applications of Symbolic Algebraic Computation. Comp. Comm. 17 (1979).Google Scholar
  11. [C-L83]
    B. COURCELLE, F. LAVANDIER: A class of program schemes based on tree rewriting systems, Proc. of CAAP — 83, LNCS n., L'Aquila (1983).Google Scholar
  12. [C-N76]
    B. COURCELLE, M. NIVAT: Algebraic families of interpretations, 17th FOCS 137–146 (1976).Google Scholar
  13. [C-V85]
    M.M. CERIMELE, M. VENTURINI ZILLI: Effective Numerical Approximations by Intervals, Freiburger Interwall-Berichte 85/4, 1–24 (1985).Google Scholar
  14. [E-M85]
    E. ENGELER, R. MÄDER: Scientific Computation: The Integration of Symbolic, Numeric and Graphic Computation, L.N.C.S. 203 (1985).Google Scholar
  15. [ERS82]
    A.P. ERSHOV: Mixed computation: Potential applications and problems for study, Theoretical Computer Sciernce 18, 41–67 (1982).Google Scholar
  16. [G-K84]
    R.T. GREGORY, E.V. KRISHNAMURTHY: Methods and Applications of Error-Free Computation, Springer Verlag (1984).Google Scholar
  17. [G-Y79]
    F.G. GUSTAVSON, D.Y.Y. YUN: Fast algorithms for rational Hermite approximation and solution of Toeplitz systems, IEEE Trans. Circuits and Systems, CAS 26, n. 9 (1979).Google Scholar
  18. [HEN08]
    K. HENSEL: Theorie der Algebraischen Zahlen, Teubner, Leipzig-Stuttgart (1908).Google Scholar
  19. [JEN79]
    R. JENKS: MODLISP: An Introduction, Proc. of EUROSAM '79 Conference, Marseille, June 1979, Ed. by Ed, Ng, LNCS, n. 72, 1979.Google Scholar
  20. [KEM81]
    P. KEMP: Symbolic-numeric interface (abstract), SIGSAM Bulletin 15, 2, 6, ACM (1981).Google Scholar
  21. [KOB77]
    N. KOBLITZ: p-adic Numbers, p-adic Analysis and Zeta Functions, Springer Verlag (1977).Google Scholar
  22. [K-G83]
    P. KORNERUP, R.T. GREGORY: Mapping integers and Hensel-codes onto Farey fractions, DAIMI PB 149, Comp. Sc. Dept. of Aarhus University, Denmark (1982). BIT 23, 9–20 (1983).Google Scholar
  23. [K-M80]
    U. KULISCH, W,L, MIRANKER: Computer arithmetic in theory and practice, Academic Press (1980).Google Scholar
  24. [LIP76]
    J.D. LIPSON: Newton's Method: A Great Algebraic Algorithm, Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation (1976).Google Scholar
  25. [LOO74]
    R. LOOS: Toward a Formal Implementation of Computer Algebra, Proc. of EUROSAM '74, ACM SIGSAM Bulletin, Vol., 8, n. 3 (1974).Google Scholar
  26. [McS78]
    R,J, McELIECE, J.B. SKEARER: A property of Euclid's algorithm and an application to Pade approximation, SIAM J. Appl. Math. 34, n. 4 (1978).Google Scholar
  27. [MIO83]
    A. MIOLA: A unified view of approximate rational arithmetics and rational interpolation, Proc. of IEEE ARITH-6 Conference, Aarhus (1983).Google Scholar
  28. [MIO84]
    A. MIOLA: Algebraic Approach to p-adic conversion of Rational Numbers, IPL 18 (1984).Google Scholar
  29. [M-T85]
    W.L. MIRANKER, R. A. TOUPIN (Eds.): Accurate Scientific Computations. L.N.C.S. 235 (1985).Google Scholar
  30. [M-Y74]
    A. MIOLA, D.Y.Y. YUN: Computational aspects of univariate polynomial GCD, Proc. ACM EUROSAM '74, SIGSAM Bulletin (8), (1974).Google Scholar
  31. [NGE79]
    E.W. NG: Symbolic-Numeric Interface: a Review, Proc. of EUROSAM '79, Springer Verlag (1979).Google Scholar
  32. [S-V86]
    M. SMYTH, S. VICKERS: The Domain of p-adic Integers. Report of the 2nd British Theoretical Computer Science Colloquium, Bulletin of the EATCS Number 30, (1986).Google Scholar
  33. [YUN75]
    D.Y.Y. YUN: Hensel meets Newton-Algebraic constructions in an analytic setting, IBM Tech. Report RC 5538 (1975).Google Scholar
  34. [YUN76]
    D.Y.Y. YUN: Algebraic Algorithms using p-adic Constructions, Proc. 1976 ACM, Symposium on Symbolic and Algebraic Computation (1976).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Gianfranco Mascari
    • 1
  • Alfonso Miola
    • 2
  1. 1.Istituto per le Applicazioni del CalcoloConsiglio Nazionale delle RicercheRomaItaly
  2. 2.Dipartimento di Informatica e SistemisticaUniversità di Roma “La Sapienza”RomaItaly

Personalised recommendations