Assessment 2 knuth book

10 setn n 1 11 if n 1 set q p g g h q h2 q otherwise

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(10) Setn= n+ 1. (11) If n = 1, set q = p, G* = G, H,* = /q, , H2* = ~q~ . Otherwise, update the quadruple (q, G*, H~*, H2*) to include (p, G, /7,, H~) by using the Chinese re- mainder algorithm (Section 4.8) (which in this ease is a form of interpolation [1, p. 430]) with moduli mi = q and m~ = Xv -- b to extend (53) (coefficient by co- efficient), and then replaeing q by q(x~ -- b) to extend (52). (12) Ifn < ~, go baek to Step (6). Otherwise, we now know thatn > ~ > ~,so (40) holds unless e > d. To exclude this unlikely possibility, it suffiees to prove the relations G'Hi* = ff~ and * * G H~ = ~0~, which hold modulo q by (33), (49), (52), and (53). (13) IfOv(G*) + O~(H,*) ~ ~,or 0,(G*) + O~(H~*) ~ ~, seth = 0 and go back to Step (6). Otherwise we have q] (G*H~* - ~,) and 0,, (q) = n > ~ >_ h >_ Ov (G'Hi* - 1~,), and therefore G'H,* = fla. Similarly, G*Hz* = fi~, and therefore (40) is established. H * (14) Set G pp(G*), g = le(G), H~ = H~ /g, H2 = ~ /g. (15) Set G' = cG, H; = (o/c)H~, H~' = (c~/c)H~, and return. 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 495 4.6 UNLUCKY b-VALUES. We shall call the integer b C Zp chosen in Step (6) of Algorithm P lucky if e = d, and unlucky otherwise. In Theorem 4 we shall prove that all of the unlucky b-values are roots (in Zp) of a polynomial o- (in 9 = Zp[x~]), which depends only on F~ and F~. Using this result, Theorem 5 bounds the total number u of unlucky b-values, thereby establishing the fact that the algorithm terminates. If b is chosen at random from the elements of Zv, the probability of its being un- lucky is u/p. If p is large (as suggested in Section 4.4), this probability is exceed- ingly small. Thus if F~ and F2 are relatively prime, we can expect to prove it with only a single b-value. Otherwise the expected number of b-values to determine the GCD and the cofactors is fi = p + 1 = max (0v(F1), 0v(/~2)) -t- ]. THEORE_U 4. Let F~ and F2 be given nonzero polynomials in 9[x~, ..., x,-1], where ~ = Zp[x~]. Let G = ged(F~, F2), and let di= O~(G). Also, let S}~) (F~, F:) denote the jth subresultant of F1 and F2 viewed as univa','iale polynomials in x~ , with eoeficients in ~[xl, -.. , xi_~, x~+~, ... , x~_~], and let ai be the content (in ~) of S(~) (F~ F2 ) viewed as a polynomial in 9Ix,, , x~_l]. (Here di is the degree of G di , " " " in X~, and not, as in Seclion 3.5, the degree of the ith polynomial in a PRS.) Finally, let v--1 = II,,~. (55) i=l Then every unlucky b-value is a root (in Zp) of a. PROOF. The proof is analogous to the proof of Theorem 1. THEOREM 5. Let u be the total number of unlucky b-values, let and let Finally, let V--1 m = ½ max (Oi(F~) + O~(F2)), (56) i=I e = max (0~ (F1), 0~ (F2)). (57) (ss) = 2me(v- 1). Then u < ~. PROOF. Let P = 1~ (xv -- b), where the product is taken over all unlucky b-values. Since P 1 cr by Theorem 4, we have u = 0v (P) < 0~ (~) = ~ 0v (~i). But by (20), 0" (gi) < 2me, sou < 2me(v - 1), aswas to be shown. 4.7 ALGORITHM U. Let F1 and F2 be given nonzero polynomials in Zv[x], where p is a fixed prime. Algorithm U computes G = gcd(F1, F2). Since Zp is a field, G must be monic in order to satisfy the requirement of unit normality. Also since Zp is a field, we may use the rational algorithm of Section 2.2. In the example (4) with p = 13, the monic PRS which mirrors (6) is 1, 0, 1, 0, --3, -3, --5, 2, --5 1, 0, 6, 0, 3, --3, --6 1,0,5,0, --2 1, --3 1. (59) Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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496 w.s. BROWN This proves that F1 and F~ are relatively prime in Zl~[x], and hence also in Z[x].
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