Assessment 2 knuth book

# 17 outline in section 2 we present the classical

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1.7 OUTLINE. In Section 2, we present the classical algorithm without specify- ing a method for constructing the required sequence of polyr~omials. Several al- gorithms for this purpose are discussed in Section 3, culminating in the subresultant PRS algorithm mentioned above. In Section 4, the modular algorithm is presented in careful detail, with special attention to the multivariate case. The required com- puting times for the classical algorithm (augmented by the subresultant PRS al- gorithm) and for the modular algorithm are then analyzed in Section 5. Finally, in Section 6 we review the highlights and present some tentative conclusions. 2. The Classical Algorithm 2.1 INTRODUCTION. The goal of this section is to obtain a straightforward gen- eralization of Euclid's algorithm, as presented in Section 1.4, to domains of uni-

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Euclid's Algorithm and Computation of Polynomial GCD' s 481 variate or multivariate polynomials. By viewing multivariate polynomials as poly- nomials in one variable, hereafter called the main variable, with coefficients in the domain of polynomials in the other variables, hereafter called the auxiliary variables, we may confine our attention without loss of generality to the univariate case. 2.2 THE RATIONAL ALGORITHM. For univariate polynomials F1 and F2 over a field, division yields a unique quotient Q and remainder R such that F1 = QF2 + R, O (R) < O (F2), (3) where 0 (F) denotes the degree of F, and 0 (0) = - ~. Thus Euclid's algorithm, as presented in Section 1.4, is directly applicable. As an example [1, pp. 370-371], if Fi(x) = x s+ x 6- 3x 4- 3x 3"4- 8x 2 + 2x-- 5, (4) F2(x) -~ 3x 6 + 5x 4 -- 4x 2 -- 9x + 21 are viewed as polynomials with rational coefficients, then the following sequence occurs (for brevity, we write only the coefficients): 1, 0, 1, 0, --3, --3, 8, 2, --5 3, 0, 5, 0, --4, --9, 21 -~, o, ~, o, -~ 117 -9, ~- 233150 102500. "-~591 , ---'~97 1988744821 (5) - 543589 ~-~5-'" It follows that F1 and F2 are relatively prime. To improve this procedure, we make each polynomial monic as soon as it is ob- tained, thereby simplifying the coefficients somewhat. In our example, we obtain the sequence 1,0,1,0,-3,-3,8,2,-5 1, 0, ~, 0, -~, -3,7 1,0, -], 0, ~ 1, 95 49 1, 615o -- T~-6--,gj 1. (6) Although this sequence minimizes the growth of coefficients, it requires integer GCD computations at each step in order to reduce the fractions to lowest terms. In general, if the coefficient domain is not a field, we could embed it in its field of quotients and then use the algorithm of Section 1.4, but we shall see that it is more efficient not to do so. 2.3 POLYNOMIAL REMAINDER SEQUENCES. Let 9 be a unique factorization do- main in which there is some way of finding GCD's, and let 9[x] denote the domain of" polynomials in x with coefficients in 9. Assuming that the terms of a polynomial F C 9[x] are arranged in the order of decreasing exponents, the first term is called the leading term, and its coefficient lc (F) is called the leading coe~cient.
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