Arithmetic in Basic Algebraic Domains

  • G. E. Collins
  • M. Mignotte
  • F. Winkler
Part of the Computing Supplementa book series (COMPUTING, volume 4)


This chapter is devoted to the arithmetic Operations, essentially addition, multiplication, exponentiation, division, gcd calculation and evaluation, on the basic algebraic domains. The algorithms for these basic domains are those most frequently used in any Computer algebra system. Therefore the best known algorithms, from a computational point of view, are presented. The basic domains considered here are the rational integers, the rational numbers, integers modulo m, Gaussian integers, polynomials, rational functions, power series, finite fields and P-adic numbers. Bounds on the maximum, minimum and average Computing time (t +, t -, t*) for the various algorithms are given.


Great Common Divisor Rational Integer Modular Exponentiation Average Computing Time Modular Arithmetic 
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. [1]
    Aho, A. V., Hopcroft, J. E., Ullman, J. D.: The Design and Analysis of Computer Algorithms. Reading, Mass.: Addison-Wesley 1974.MATHGoogle Scholar
  2. [2]
    Berlekamp, E. R.: Algebraic Coding Theory. New York: McGraw-Hill 1968.MATHGoogle Scholar
  3. [3]
    Bonneau, R. J.: Polynomial Operations Using the Fast Fourier Transform. Cambridge, Mass., MIT Dept. of Math., 1974.Google Scholar
  4. [4]
    Brent, R. P.: Fast Multiple Precision Evaluation of Elementary Functions. J. ACM 23, 242–251 (1976).CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    Brown, W. S., Hyde, J. P., Tague, B. A.: The ALP AK System for Nonnumerical Algebra on a Digital Computer-Ii: Rational Functions of Several Variables and Truncated Power Series with Rational Function Coefficients. Bell Syst. Tech. J. 43, No. 1, 785–804 (1964).Google Scholar
  6. [6]
    Caviness, B. F.: A Lehmer-Type Greatest Common Divisor Algorithm for Gaussian Integers. SIAM Rev. 15, No. 2, Part 1, 414 (April 1973).Google Scholar
  7. [7]
    Caviness, B. F., Collins, G. E.: Algorithms for Gaussian Integer Arithmetic. SYMSAC 1976, 36–45.Google Scholar
  8. [8]
    Collins, G. E.: Computing Time Analyses of Some Arithmetic and Algebraic Algorithms. Univ. of Wisconsin, Madison, Comp. Sci. Techn. Rep. 36 (1968).Google Scholar
  9. [9]
    Collins, G. E.: The SAC-1 Rational Function System. Univ. of Wisconsin, Madison, Comp. Sci. Techn. Rep. 135 (1971).Google Scholar
  10. [10]
    Collins, G. E.: Computer Algebra of Polynomials and Rational Functions. Am. Math. Mon. 80, No. 2, 725–755 (1973).CrossRefMATHGoogle Scholar
  11. Collins, G. E.: Lecture Notes in Computer Algebra. Univ. of Wisconsin, Madison.Google Scholar
  12. [12]
    Collins, G. E., Musser, D. R.: Analysis of the Pope-Stein Division Algorithm. Inf. Process. Lett. 6, 151–155 (1977).CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    Fateman, R. F.: On the Computation of Powers of Sparse Polynomials. Stud. Appl. Math. LIII, No. 2, 145–155 (1974).MathSciNetGoogle Scholar
  14. [14]
    Fateman, R. F.: Polynomial Multiplication, Powers and Asymptotic Analysis: Some Comments. SIAM J. Comput. 3, No. 3, 196–213 (1974).CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    Gentleman, W. M.: Optimal Multiplication Chains for Computing a Power of a Symbolic Polynomial. Math. Comput. 26, No. 120 (1972).Google Scholar
  16. [16]
    Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers, 5th ed. Oxford: Clarendon Press 1979.MATHGoogle Scholar
  17. [17]
    Henrici, P.: A Subroutine for Computations with Rational Numbers. J. ACM 3, 6–9 (1956).CrossRefMathSciNetGoogle Scholar
  18. [18]
    Henrici, P.: Automatic Computations with Power Series. J. ACM 3, 10–15 (1956).CrossRefMathSciNetGoogle Scholar
  19. [19]
    Horowitz, E.: The Efficient Calculation of Powers of Polynomials. 13th Annual Symp. on Switching and Automata Theory (IEEE Comput. Soc.) (1972).Google Scholar
  20. [20]
    Horowitz, E.: A Sorting Algorithm for Polynomial Multiplication. J. ACM 22, No. 4, 450–462 (1975).CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    Johnson, S. C.: Sparse Polynomial Arithmetic. EUROSAM 1974, 63–71.Google Scholar
  22. Karatsuba, A., Ofman, Yu.: Dokl. Akad. Nauk SSSR 145, 293–294 (1962) [English translation: Multiplication of Multidigit Numbers on Automata. Sov. Phys., Dokl. 7, 595–596 (1963)].Google Scholar
  23. [23]
    Knuth, D. E.: The Art of Computer Programming, Vol. 2, 2nd ed. Reading, Mass.: Addison- Wesley 1981.MATHGoogle Scholar
  24. [24]
    Knuth, D. E.: The Analysis of Algorithms. Proc. Internat. Congress Math. (Nice, 1970), Vol. 3, pp. 269–274. Paris: Gauthier-Villars 1971.Google Scholar
  25. [25]
    Lehmer, D. H.: Euclid’s Algorithm for Large Numbers. Am. Math. Mon. 45, 227–233 (1938).CrossRefMathSciNetGoogle Scholar
  26. [26]
    Mahler, K.: Introduction to p-adic Numbers and Their Functions. Cambridge Univ. Press 1973.MATHGoogle Scholar
  27. [27]
    Matula, D. W.: Fixed-Slash and Floating-Slash Rational Arithmetic. Proc. 3rd Symp. on Comput. Arith. (IEEE), 90–91 (1975).Google Scholar
  28. [28]
    Matula, D. W., Kornerup, P.: A Feasibility Analysis of Binary Fixed-Slash and Floating-Slash Number Systems. Proc. 4th Symp. on Comput. Arith. (IEEE), 29–38 (1978).Google Scholar
  29. [29]
    Matula, D. W., Kornerup, P.: Approximate Rational Arithmetic Systems: Analysis of Recovery of Simple Fractions Düring Expression Evaluation. EUROSAM 1979, 383–397.Google Scholar
  30. [30]
    Moenck, R. T.: Studies in Fast Algebraic Algorithms. Ph.D. Thesis, Univ. of Toronto, 1973.Google Scholar
  31. [31]
    Moenck, R. T.: Fast Computations of GCD’s. Proc. 5th Symp. on Theory of Comput. (ACM), 142–151 (1973).Google Scholar
  32. [32]
    Moenck, R. T.: Practical Fast Polynomial Multiplication. SYMSAC 1976, 136–148.Google Scholar
  33. [33]
    Pollard, J. M.: The Fast Fourier Transform in a Finite Field. Math. Comput. 25, 365–374 (1971).CrossRefMATHMathSciNetGoogle Scholar
  34. Pope, D. A., Stein, M. L.: Multiple Precision Arithmetic. Commun. ACM 3, 652–654 (1960).CrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    Schönhage, A.: Schnelle Berechnung von Kettenbruchentwicklungen. Acta Inf. 1, 139–144 (1971).CrossRefMATHGoogle Scholar
  36. [36]
    Schönhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971).CrossRefMATHGoogle Scholar
  37. [37]
    Shaw, M., Traub, J. F.: On the Number of Multiplications for the Evaluation of a Polynomial and Some of its Derivatives. J. ACM 21, No. 1, 161–167 (1974).CrossRefMATHMathSciNetGoogle Scholar
  38. [38]
    Strassen, V.: The Computational Complexity of Continued Fractions. SYMSAC 1981, 51–67.Google Scholar
  39. [39]
    van der Waerden, B. L.: Modern Algebra, Vol. 1. New York: Frederick Ungar 1948.Google Scholar

Copyright information

© Springer-Verlag/Wien 1983

Authors and Affiliations

  • G. E. Collins
    • 1
  • M. Mignotte
    • 2
  • F. Winkler
    • 3
  1. 1.Computer Science DepartmentUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Centre de Calcul de l’EsplanadeUniversite Louis PasteurStrasbourg CédexFrance
  3. 3.Institut für MathematikJohannes-Kepler-Universität LinzLinzAustria

Personalised recommendations