Assessment 2 knuth book

Journal of the association for computing machinery

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Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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482 w.s. BROWN Since the familiar process of polynomial division with remainder requires exact divisibility in the coefficient domain, it is usually impossible to carry it out for non- zero F1, F2 C 9[x]. However, the process of pseudo-division [1, p. 369] always yields a unique pseudo-quotient Q = pquo(F1, F2) and pseudo-remainder R = prem (F1, F2), such that f25+lF1 - QF2 = R and 0 (R) < 0 (F2), where f2 is the leading coefficient of F~, and ~ = 0 (F1) - 0 (F2). For nonzero F1, F2 C 9[x], we say that F1 is similar to F2 (F1 ~ F~) if there exist al, a~ E 9 such that alF1 = a2F2. Here al and a2 are called coe~cients of similarity. For nonzero F1, F2 C 9Ix] with 0 (F1) > 0 (F2), let F1, F2, • • • , Fk be a sequence of nonzero polynomials such that F~ ~ prem(F~_2, F~-i) for i = 3, ..., k, and prem (Fk-1, Fk) = 0. Such a sequence is called a polynomial remainder sequence (PRS). From the definitions, it follows that there exist nonzero a~, f~i C 9 and Q~ ~-~ pquo(F1, F2) such that ~,F, = o~iFi_2 - - Q,F~_i, O(F~) < 0(F~_i), i = 3, ..., k. (7) Because of the uniqueness of pseudo-division, the PRS beginning with F~ and F is unique up to similarity. Furthermore, it is easy to see that gcd(F1, F2) ~ gcd(F ,2 F3) ..... gcd(Fk_~, Fk) ~ Fk. Thus, the construction of the PRS yields t2he desired GCD to within similarity. 2.4 ALGORITHM C. A polynomial F C 9Ix] will be called primitive if its nonzero coefficients are relatively prime; in particular, all polynomials over a field are primi- tive. Since 9 is a unique factorization domain, it follows [6, pp, 74-77] that 9Ix] is a unique factorization domain whose units are the units of 9. Hence each polynomial F C 9[x] has a unique representation of the form F = cont(F)pp(F), where eont(F) is the (unit normal) GCD of the coefficients of F, and pp(F) is a primitive polyno- mial. We shall refer to cont(F) and pp(F) as the content and primitive part, re- spectively, of F. Let F~' and F2' be given nonzero polynomials in 9[x] with 0 (Fi') > 0 (F21), and let G' be their GCD. Also, let el = cont(F~'), c: = cont(F'2), c = gcd(c~, c2), F1 = pp(F1'), F2 = pp(F2'), and G = gcd(F1, F2). Now, if F~, F2, ..., F~is a PRS, it is easy to show that G = pp(Fk) and G' = cG. Because of coefficient growth, the coefficients of Fk are likely to be much larger than those of F~ and F2. However, since G divides both F~ and F~, the coefficients of G are usually smaller than those of F1 and F2. Thus, Fk is likely to have a very large content. Fortunately, most of this unwanted content can be removed without computing any GCD's involving coefficients of Fk. Let fl = lc (F1), f: = lc (F2), fk = lc (Fk), g = lc (G), and ~ = gcd(fl, f2). Since G divides both El and F2, it follows that g divides both fl and f2, and therefore g [ ~. Let G = (~/g)G. Clearly, G has ~ as its leading coefficient, and G as its primitive part, and it is easy to see that G = OFk/f~. In the case of the Euclidean PRS algorithm (Section 3.2), the reduced PRS al- gorithm (Section 3.4), and the subresultant PRS algorithm (Section 3.6), it can be shown (see [7] and [8] ) that 0 divides a subresultant (Section 3.5) which in turn divides Fk, and therefore 0 I f~. Hence, G = F~/(f~/O).
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