Assessment 2 knuth book

2 d and similarly using assumption a3 large random

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2) d, and similarly [using assumption (A3)] large random problems we find that d' > d, 1 ) (~ --l-- 2)~+I (d --k 1 )2,-2, 1) (d + 1)2~-2. (69) For polynomials in Zp[x], assumption (A4) permits us to restrict our attention to single-word primes; hence all coefficient operations can be performed in some fixed amount of time. For Algorithm U, let dl = O(Fi) and d2 = 0(F2), and suppose dl >_ d2 > 0. Then, by essentially the same argument that led to (63), T~ (U) ~ d, (< - d3), (70) where da is the degree of the GCD. For the Chinese remainder algorithm, we again use the notation of Section 4.8. Let d~ and d2 be the degrees of m~ and m2, respec- tively. By (70), the time to compute c in Step (1) is codominant with did2. Since Steps (2) and (3) can also be performed within this bound, we have Tp (CRA) ~ &d2. (71) 5.5 ALGORITmr C. In analyzing the computing time for Algorithm C, we shall use the notation of Section 2.4, and we shall assume that Step (4) is performed by means of the subresultant PRS algorithm (Section 3.6). Let F~' and F2' be the given nonzero polynomials in v variables, and let (1, d) bound their dimension vectors. Let C(v, l, d) denote the maximum computing time for Algorithm C, and let C~(v, l, 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 this step involves at most 2d + 1 GCD's of polynomial co- efficients, we have C~(v, l, d) ~ d.C(v - 1, l, d). (72) ! Step (2). By (67), the time required to divide Cl or c2 into a coefficient of F~ or F2' ' " ~ 2 1)2~-2. is dominated bv 1 (4 + Since there are at most, 2 (d + 1 ) such divisions, we have C~ ,~, 12 (d + 1 )2~-1. (73) Step (4). Clearly most of the work in this step is in the pseudo-divisions which yield prem(Fi_2, Fi_~) for i = 3, • • • , k. As in Section 3.1, let dl denote the degree of F~ in the main variable. Then by assumption (A1), we have d~ = d2 -t- 2 - i, for i = 3, • • • , k, and therefore k < d2 -4- 2. Furthermore, by (20) the degree of F~ in any auxiliary variable cammt exceed d(d~ -4- d2) _< 2d 2. Similarly, by (22), ignoring the logarithm terms in accordance with assumption (A3), the integer- length of Fi cannot exceed l(d~ -4- d2) < 2ld. Hence we can bound the time for a single pseudo-division by replacing l by 2hl, d~ by d, 5 by 1, and d by 2 d 2 in (69). Multiplying the result by d, which bounds the number of pseudo-divisions, and applying some obvious simplifying inequalities, we obtain C4 ,~, l 2 (d + 1 )4~22~3~. (74) Step (6). To bound the time for computing fk/O, we can replace v by v - 1, 12 by l, d2 by d, 13 by 21d, and da by 2d 2 in (67); the result is dominated Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971

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500 W.S. BROWN by 12 (d + 1 )3v-22v. To bound the time for dividing this quantity into a coefficient of Fk, we can replace v by v - 1, 12 by 21d, d2 by 2 d ~, 13 by l, and d3 by d in (67), with the same result. Multiplying by d + 2, which bounds the number of such divisions, we find C6 £ t ~ (d + 1)3v-12v. (75) Step (7).
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