An Implementation of Rational Functions in PCL

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The main aim of our work with PCL, a kind of "object-oriented" expansion of COMMON-LISP, is to test whether it is suitable for the implementation of our Grobner bases package. Criteria for this are speed and the facility of generic programming. For this purpose we implemented the domain of Rational Functions . For arithmetic in this field it is necessary to compute the greatest common divisor (gcd) of two multivariate polynomials, which was the main effort of this work. 1 Introduction In this paper a generic implementation of Rational Functions is described. Arithmetic for the occurring polynomials is independent of the domain of coefficients. Special attention was focussed on the gcd of polynomials as simplification-algorithm for Rational Functions. The modular approach goes back to [Knuth 69], [Brown 71] and [Collins 72]. In our implementation we follow the description given in [Davenport 88] and [Winkler 88] for computing the gcd of polynomials. The gcd is computed modulo several ...

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... We c o d e d G R OBNER partially in PCL, but the speed of the resulting programs was not satisfactory. H o wever, this experiment showed the amount o f work that must be done for re-coding algebraic algorithms if such algorithms are not available in the language (see Windsteiger, 1990b]). C Advantages: ...
This paper shows one possibility how to achieve some aspects of object-oriented programming in C. Since this should be used in Algebraic Computation, the most important feature one desires to have, is the possibility to use the same names for algorithms that perform the same actions. For example the multiplication algorithm should always be called
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Computer algebra is an alternative and complement to numerical mathematics. Its importance is steadily increasing. This volume is the first systematic and complete treatment of computer algebra. It presents the basic problems of computer algebra and the best algorithms now known for their solution with their mathematical foundations, and complete references to the original literature. The volume follows a top-down structure proceeding from very high-level problems which will be well-motivated for most readers to problems whose solution is needed for solving the problems at the higher level. The volume is written as a supplementary text for a traditional algebra course or for a general algorithms course. It also provides the basis for an independent computer algebra course.
This paper examines the computation of polynomial greatest common divisors by various generalizations of Euclid's algorithm. The phenomenon of coefficient growth is de- scribed, and the history of successful efforts first to control it and then to eliminate it is re- lated. The recently developed modular algorithm is presented in careful detail, with special atten- tion to the case of multivariate polynomials. The computing times for the classical algorithm and for the modular algorithm are analyzed, and it is shown that the modular algorithm is markedly superior. In fact, in the multivariate ease, the maximum computing time for the modular algorithm is strictly dominated by the maximum computing time for the first pseudo-division in the classical algorithm.
  • B Buchberger
  • G E Collins
  • R Loos
Buchberger 83] B. Buchberger, G.E. Collins, R. Loos: Computer-Algebra { Symbolic and Algebraic Computation, 2nd ed., Springer, 1983.
  • G E Collins
Collins 72] G.E. Collins: The SAC-1 Polynomial GCD and Resultant System, Univ. of Wisconsin Comp. Sci. Dept. Tech. Report No.145, 1-93, 1972.