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Assessment 2 knuth book

Journal of the association for computing machinery

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Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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Euclid's Algorithm and Computation of Polynomial GCD's 497 (A3) In Algorithms C and 5/[, it is assumed that the integer-length (Section 5.2) of an exact quotient of polynomials does not exceed the integer-length of the dividend. In Algorithm C, it is further assumed that the integer-length of a sum or difference of polynomials is the maximum of the integer-lengths of the summands, and that the integer-length of a product does not exceed the sum of the integer- lengths of the factors. (A4) In Algorithm M, it is assumed that the integer-lengths of the given poly- nomials are small compared to the largest single-word integer fl so that the supply of single-word primes will not be exhausted. Similarly it is assumed that the number of terms in the GCD is small compared to ~, so that the required number of primes can be bounded in a simple and realistic way. After introducing some basic concepts in Section 5.2, we shall discuss integer operations in Section 5.3, polynomial operations in Section 5.4, Algorithm C in Section 5.5, Algorithm P in Section 5.6, and Algorithm M in Section 5.7. Finally we compare Algorithms C and 5I in Section 5.8. 5.2 BASIC CONCEPTS, Let f and g be real functions defined on a set S. If there is a positive real number c such that If(x)l < clg(x)l for all x ~ S, we write f ,~ g, and say that f is dominated by g. If f ~ g and g ,~ f, we write f ~ g, and say that f and g are eodominant. Finally, if f ~ g but f ~-~ g, we write f < g, and say that. f is strictly dominated by g. Clearly codominance is an equivalence relation among the real functions on S, while strict dominance defines a partial ordering among the resulting codominance classes. In the author's opinion, this notation and terminology (from [10]) are significantly superior to the traditional "little-oh" and "big-oh" notation [11, p. 5]. Since the purpose of the analysis is to provide insight, not detail, we shall con- sider only a minimum number of independent parameters. Every polynomial F which is considered in the analysis will be characterized either by its dimension vector (1, d), defined below, or by its degree in the main variable together with a single dimension vector for all its coefficients. First we define the length of a nonzero integer to be the logarithm (to some fixed base such as 2 or 10) of its magnitude. Next we define the integer-length of a nonzero polynomial F ~ Z[x~, ... , x~] to be the maximum of the lengths of its nonzero coefficients; this will be denoted by il(F). Finally, for nonzero F C Z[Xl, ... , x,] we define the dimension vector (l, d) by the relations 1 = il(F) and d = max (Oi (F)). For integers it is clear that the length of a product is the sum of the lengths of the factors. For polynomials, the integer-length of a product may be smaller or larger than the sum of the integer-lengths of the factors, because of the additive combina- tion of terms in polynomial multiplication. For example, if A (x) = x - 1, then A (x)2 = x 2 _ 2x + 1, and il(A 2) = log 2 > 2 il (A) = 0. On the other hand, letting B(x) = x s+2S+3x 6+4x 5+5x 4+4x 3+3x 2+ 2x+ 1, we have A (x )B (x ) =
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