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Assessment 2 knuth book

In the multivariate case the linear coefficient

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worth the effort if the Euclidean PRS algorithm were the only alternative. In the multivariate case, the linear coefficient growth is, of course, compounded through each level of the recursion, but even so, the primitive PRS algorithm has demonstrated considerable practical utility. 3.4 THE REDUCED PRS ALGORITHM. Since the bulk of the work in the primitive PRS algorithm is in primitive part computations, we would like to find a way to avoid most of them and still reduce the coefficient growth sharply from that which occurs in the Euclidean PRS algorithm. Surprisingly, we can accomplish this to a significant extent by choosing ~3 = 1, ~i = ai-1, i = 4, ... , k. (13) This is Collins' reduced PRS algorithm, and is justified in [7] and [8] by a proof that is sketched in Section 3.5. In a normal reduced PRS, the coefficient growth is essentially linear (see Section 3.5). In an abnormal reduced PRS, the growth can be exponential, but of course not as badly so as in the corresponding Euclidean PRS. In the example (4), which is distinctly abnormal, the reduced PRS is 1, 0, 1, 0, -3, -3, 8, 2, -5 3, 0, 5, 0, -4, -9, 21 -15, 0, 3, 0, -9 585, 1125, -2205 - 18885150, 24907500 527933700. (14) 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 485 Notice that the coefficient sizes appear to be doubling at each step until the last. It is easy to show that the small size of F6 results from the fact that ~4 < ~3 • If all of the ~ were the same and greater than 1, then the growth would be uniformly ex- ponential. 3.5 SUBRESULTANTS. Let F~, F2, • • • , Fk be any PRS in 9Ix]. The major result of Collins [7] is that F, ~ Sd~ (Fi, F2) ~ Sd~_~-l(F~, F2), i = 3, ..., k, (15) where ~ denotes similarity (Section 2.3) and where, for 0 _< j < d:, Si(F~, F2) is a polynomial of degree at most j, each of whose coefficients is a determinant of order d~ + d2 - 2j with coefficients of F~ and F~ as its elements. In particular, So(F~, F2) is the classical resultant [5, Ch. 12] of F1 and F2 ; in general, following Collins, we shall call S~ (F~, F~) the jth subresultant of F~ and F2. The constants of similarity for (15) may be expressed as products of powers of a3, - • • , ai, ~3, • • • , ~i,f2, • • • ,f~, and (-1). Brown and Traub [8] rederive these results in somewhat greater generality; their treatment is brief and simple, and shows clearly how the subresultants arise. We shall now establish bounds on the coefficients of the subresultants Let T~ = Sji_,_i(F1 , F:), i = 3, "" , k. (16) 1 ml = ~(dl + d2 -+- 2) -- di-1 = [dl+ d~ -- 2(4i-1 - 1)], i = 3, ..., k. (17) This is an approximate measure of degree loss. As we proceed through the PRS, it increases monotonically from m3 > 1 to mk _< ½ (dl + d2). In a normal PRS, mi in- creases by 1 when i increases by 1; in general, the increment is ~i-~. Since each coefficient of Ti is a determinant of order 2mi with coefficients of F1 and F2 as its elements, our bounds depend simply on m~. If the coefficients of F1 and F2 are integers bounded in magnitude by c, then by Hadamard's theorem [1, p. 375] the coefficients of T~ are bounded in magnitude by (2m~J) m'. (18) Taking the logarithm (to the same base as the base of the number system), we see that the coefficients of Ti are bounded in length by mi[2l -4- log (2m~)], (19) where 1 = log c bounds the lengths of the coefficients of Fa and F~. Although the
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