Assessment 2 knuth book

Since the same reasoning holds for h and h we have

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Since the same reasoning holds for H( and H(, we have P15 ,~ (d + 1 )'+~. (87) Total Time. By assumption (A2) no unlucky b-values will occur. Hence if F, and F2 are relatively prime, only one b-value will be needed; otherwise the required number is ~=o+1 = max (0~ (fi,), O~ (/Tz)) + 1 < 2 d + 1. (88) Summing the preceding bounds, with Steps (6)-(12) weighted by (88), we obtain P(v,d)~d.P(v- 1, d) + (d+ 1)~+t, (89) for v > 1. Starting with the formula P(1, d) ~ d~, (90) which follows from (70), we can now prove by induction on v that P(v, d) ~ (d + 1) ~+~. (91) 5.7 ALGORITHM M. In analyzing the computing time for Algorithm M, we shall use the notation of Section 4.3. Let F~' and F' 2 be the given nonzero poly- nomials in Z[xl, • • • , x~J, and let (1, d) bound their dimension vectors. Let M(v, l, d) denote the maximum computing time for Algorithm 5'I, and let Mi(v, l, d) denote the time for the ith step. In view of the similarity of Algorithm M to Algorithm P, we shall merely present the results of the analyses of the sig- nificant steps: Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971

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502 w.s. BROWN Mi,~ 12(d + 1)~', M~ ~ 1 ~ (d + 1 )v, MT,~l(d+ 1) ~, Ms~P(v,d)+ (d+ 1)~,~, (d+l)'+', Mll ~ n(d + l)V, M14~l 2(d+ 1)~, M15 ,~ 12 (c/ -t- 1 )~. (92) Total Time. By assumption (A2) no unlucky primes will occur. Hence if F1 and F2 are relatively prime, only a single prime will be needed; otherwise the re quired number is < log~u , (93) where ~ is the final value of u* in Step (13). Clearly u need not exceed 2c2t, where t is the number of terms in (~, and where c bounds the magnitudes of the coefficients of (~, flit, and /t~. It follows immediately that ~ < 2 logs c -t- logs t + logs '2. But by assumption (A3), logs c < 21, and by assumption (A4), log~ t < 1. Hence < 4l + 2. (94) Summing (92) with Steps (6)- (12) weighted by (94), we obtain M(v, l, d) £ 12(d + 1) ~ ÷ l(d + 1) ~+1. (95) When F~ and F2 are relatively prime (RP), it suffices to sum Steps (1)-(9) and (15); thus we find the smaller bound 211(Re)(V, l, d) £ 12(d + 1) ~ + (d + 1) ~'+~. (96) 5.8 COMPARISON. From (80) and (95), we see that the bound on M is strictly dominated by the bound on C. We shall now prove that M is strictly dominated by C in the region v >_ 2. Let F~ and F2 be random polynomials in v variables subject only to the constraints imposed by the dimension vector (l, d), and let C_ be the maximum computing time to obtain their pseudo-remainder F3. Then by (69), C_ dominates 12 (d + 1)2~-~. On the other hand, by (95), M is strictly dominated by M+ = 12(d + 1) "+~. v 2 Since C_/M+ = (d -I- 1) , it follows that M < M+ < C_ ~ C in the region v _> 2. This completes the proof. 6. Summary and conclusions In attempting to generalize Euclid's algorithm to the case of univariate or multi- variate polynomials with integer or rational coefficients, one immediately encounters the problem of coefficient growth. This is most serious when the growth is allowed to go unchecked, as in the Euclidean PRS algorithm, or when the algorithm is recur- sively applied to inflated polynomial coefficients, as in the multivariate case of the primitive PRS algorithm.
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