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Assessment 2 knuth book

By assumption a3 the dimension vectors for 0 and for

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By assumption (A3), the dimension vectors for 0 and for the co- efficients of G are bounded by (1, d) ; hence the dimension vectors for the coefficients of 0 and for cont(G) are bounded by (2l, 2 d). Now by the argument used in Step (1), cont(G) can be computed in time d.C(v -- 1, 2l, 2d). Also, by the argument used in Step (2), each division of this content into a coefficient of C~ can be performed in time l 2 (d --b 1)2~-22v. Hence, CT(v,l,d) £ d.C(v- 1,21,2d) + l ~(d + 1)2v-12v. (76) Step (8). Here (1, d) bounds the dimension vectors of c and of the coefficients of G. By (66), the time to maltiply one of these coefficients by c is at most 2 (d + 1)2v-2; hence C8 • l~ (d + 1)2~-1. (77) Total Time. Summing these bounds, we find C(v, l, d) • d.C(v - 1, 21, 2d) -+- 12(d + 1)4v22v3~, (78) for v > 0. Starting with the formula C (0, l) £ 12, (79) which follows from (63), we can now prove by induction on v that C(v, l, d) £ 12(g -¢- 1)4~22~23~. (80) 5.6 ALGORITHM P. In analyzing the computing time for Algorithm P, we shall use the notation of Section 4.5. Let F~' and F2' be the given nonzero polynomials in Zp[Xl, • • • , x~], and let d bound the components of their degree vectors. Let P (v, d) denote the maximum computing time for Algorithm P, and let Pi (v, d) denote the time for the ith step. We shall omit the analyses of several steps which obviously make no contribution to the final bound. Step (1). Since F~' and F~' each have at most (d + 1)~-1 terms, and since the time to compute a GCD in the coefficient domain ~ = Zp[x] by Algorithm U is at most d 2 [by (70)], we have P~ .~ (d + 1) ~+~. (81) Step (2). By (67), the time required to divide Cl or c~ into a coefficient of Fi' or F( is dominated by (d -t- 1 )5. Since there are at most 2 (d + 1 )~'-~ such divisions, we have P2 £ (d + 1)~+~. (82) Step (7). Let ~ denote either 0 or any coefficient of F~ or F2. Thus ~ C Z~,[x~], and 0~. (~) < d. Since the time to map ~ into ~ mod(x~ - b) = ~ (b) ~ Zp, either by division or by Horner's rule [1, p. 423], is dominated by d + 1, the time to map 0 and all of the coefficients of F1 and F2 into Zp is dominated by (d + 1 )". Since this 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 501 dominates the time required for the ensuing multiplications, we have P7 ,~ (d + 1)v. (83) Step (8). Here we invoke Algorithm P recursively, and then multiply the re- sulting GCD and cofactors by ~ and --' g , respectively. Thus Ps(v,d) ~P(v-- 1,~/) + (d+ 1)v-1. (84) Step (11). Here each of the coefficients of G*, H**, and H2* must be extended from degree n - 2 in x~ to degree n - 1. By (71), the required time for each suell extension is codominant with n; hence Pu ~ n (d + 1)~-1. (85) Step (1~). Recall that G* = (~ = (O/g)G. BythesamereasoningasinStep(1), eont(G*) can be computed in time (d + l) v+'. By ~he same reasoning as in Step (2), the ensuing divisions of this content into G*, and of g into H,* and H2*, can also be performed in time (d + 1)v+l. Hence P14 ,~ (d + 1)v+~. (86) Step (15). By (66), each multiplication of a coefficient of G by c can be per- formed in time (d + 1)2. Hence the time to compute G' is bounded by (d + 1)~+~.
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