Assessment 2 knuth book

Nomials is the product of the gcd of their contents

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nomials is the product of the GCD of their contents and the GCD of their primitive parts. 4.3 ALGORITHM 5I. Let Z denote the domain of integers, and let Zp denote the field of integers modulo a prime p. Let Fi' and F ' 2 be given nonzero polynomials in Z[Xl, ... , xo]. Algorithm M computes their GCD, G', and their cofactors, H~' = FI'/G' and H'2 = F2'/G', making essential use of Algorithm P (Section 4.5), which computes the GCD and cofactors of given nonzero polynomials in Zp[Xl, .." , Xv]. Each of these algorithms obtains the cofactors for the essential purpose of verifying the GCD; however, they are frequently a very welcome byproduct of the computa- tion. Let c~ = cont(F~'), c2 = cont(F2'), and c = gcd(o, c2). Also, let F1 = Fi'/O, F2 = F2'/c2, G = gcd (F~, F2), H~ = F~/G, and H2 = F2/G. Then G' = cG, H~' = (cl/c)H1, and H2' = (c~/c)H2. Let f~, f2, g, h~, and h2 be the leading coefficients of F~, F~, G, H1, and H2, re- spectively. As in Section 2.4, define ~ = gcd(f~, f~) and G = ((O/g)G. Also, define fil = oF~, F~ = OF2, tt~ = gH~, and/42 = gH2, so that fi~ = G/41 and F~ = ¢//~. Note thatpp(G) = G, lc(G) = g, lc(/41) = gh~ = fl, and lc(/4:) = gh~ = f~. Observe that fl, f2, and ¢ can easily be obtained at the beginning of the computation, while g cannot be known until the end. Let d = i) (G). Although d cannot be determined until the end of the computation, it is evident that d _< min (1) (F1), ~ (F~)). At any given time in the execution of Algorithm M, there is a set of odd primes pi, " " , p~ that have been used and not discarded. Furthermore, for i = 1, • • • , n, the algorithm has computed /~(~) = fi~ mod p~, (31) 1 P(~) F~ mod p~, 2 and three polynomials G(~), ~(~) ~(~) t/~ , and///: satisfying = • gcd (/~'~ , ,. e ) (32) and 1 ~(~) ~(~)~) (33) 2 ---- in Z~,[xl, ... , xv], where ~(1) = ~ rood pl. (34) Since the GCD in (32) is unit normal (in this case, monic) by definition, it follows that lc (~(i)) = ~)(i). Since G I F1 and G l F2 in Z[xl, ". , xv], we have Gv~ I F~", Gvi ] P~'), and therefore Gpi [ G (~) in Zp~[xl, ... , xv], where Gv~ = G mod pl. It follows immediately that Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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490 w.s. BROWN i) (G C~) ) >__ 0 (Gv~) = 0 (G) = d. The algorithm keeps only those pl for which ~) (~c,)) is minimal to date; hence, we always have 0 (G(o) = e _> d. If e > d, the algorithm will discover the fact (for proof, see Section 4.4) and start over. Eventually, e = d, and G (~) ~ G mod p~, H~) ~ /71 mod pl, (35) ~o ~_ /t2 mod p~, fori = 1, ...,n. Instead of preserving all of the quadruples (pl, G (~), H~), ~o), we maintain only the integer q = l~ P~, (36) i~l and the unique polynomials G*, Hi*, and H2* with integer coefficients of magnitude less than q/2, such that G* ~- G (~) mod pi, Hi* ------ H~') mod pi, (37) H2* ------ ~(~) • ~2 mod p~, for i = 1, • • • , n. Now as soon as e = d, we see from (35) and (37) that G* -= G mod q, Hi* --- /tl mod q, (38) H2* -= //2 mod q. When we also achieve q> ~ = 2 max [¢l, where ~k ranges over the coefficients of G,/tz, and/~, it follows that (39) G* ~ G, HI S -~ /~1, (40) H2* = IIi2. To obtain the final results, we then use the relations G = pp(G), H1 = ttl/g, (41) H2 = t72/g, and G' = cG, HI ' = (Cl/C)H1, H2'= (c2/c)H2, all of which were derived earlier. (42) 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 491 In order to guarantee that the coefficients of G*, Hi*, and H2* converge to the correct signed integers, we shall assume throughout that the integers modulo r
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