Addition Chains Using Continued Fractions
نویسندگان
چکیده
This paper introduces a new algorithm for the evaluation of monomials in two variables X " JJ~ based upon the continued fraction expansion of a/b. A method for fast explicit generation of addition chains of small length for a positive integer n is deduced from this Algorithm. As an illustration of the properties of the method, a Schoh-Brauer-like inequality p(N) s nb + k + p(n + l), is shown to be true whenever N is an integer of the form 2k(l + 2' +. . * + 2nb). Computer experimentation has shown that the length of the chains constructed are of optimal length for all integers up to 1000, with 29 exceptions for which the length is equal to the optimal length plus one. Q 1989 Academic press. I~C. Let n be a positive integer. The problem of how to evaluate most efficiently a monomial x " (considered as early as 1894) has been considered many times. For an history of this problem and further results, see Knuth [IS]. The main contribution of this paper is an efficient algorithm for the generation of short addition chains for ordered pairs (a, b) as a tool for the efficient evaluation of monomials x " y b in two variables. As a corollary, one obtains an easy method for the generation of addition chains for integers (see Olivos [O]). Surprisingly enough, this algorithm produces shorter chains than the binary (or n-ary) algorithm, without involving too many computation steps. Both concepts of addition chains are special cases of the concept of word chains (see [BB]) in a semi-group, introduced for the study of the efficient evaluation of monomials in non-commutative variables.
منابع مشابه
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عنوان ژورنال:
- J. Algorithms
دوره 10 شماره
صفحات -
تاریخ انتشار 1989