Assessment 2 knuth book

Although the growth permitted by this bound is

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where 1 = log c bounds the lengths of the coefficients of Fa and F~. Although the growth permitted by this bound is slightly faster than linear in m~, the difference is rarely discernible since the second term is usually small compared to the first. If the coefficients of F~ and F2 are polynomials in y~, - • • , yn over the integers, with degree at most ej in yj, then clearly the coefficients of T~ have degree at most 2miej (20) in Yi, for j = 1, -.. , n. Thus, the growth in degree is strictly linear in m~. If the polynomial coefficients of F~ and F2 have at most t terms each and have integer coefficients bounded in magnitude by c, then the integer coefficients of the poly- nomial coefficients of T~ are bounded in magnitude by Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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486 w.s. BROWN (2mlc2t 2 )m~. (21) This generalization of (18) follows easily from the results of [9]. Taking the loga- rithm, we see that these integer coefficients are bounded in length by m~[21 ~- log (2m~) + 2 log t], (22) which is a generalization of (19). In the case of a primitive PRS, (15) and (16) imply that F~ I Ti, so the bounds (18) and (20) on the coefficients of T~ also apply to the coefficients of F~. When the coefficients of F~ and T~ are polynomials, it may happen in rare cases (see Section 5.2) that one or more integer coefficients of polynomial coefficients of F~ are larger than any integer coefficient of any polynomial coefficient of T~. Nevertheless, we conjecture that no integer coefficient of any polynomial coefficient of F~ can ever exceed the bound (21). To justify the reduced PRS algorithm (Section 3.4), it is shown in [7] and [8] that T~ I F~ for i = 3, • • • , k, and therefore all coefficients of F~ are in 9. In the case of a normal reduced PRS, it is further shown that Fi = ±Ti, i = 3, ... , k, (23) and therefore the bounds (18)-(22) apply to the coefficients of F~ in this case as well. 3.6 THE SUBRESVLTANT PRS ALGORITmL If we could choose the ~ in such a way as to satisfy (23) even in abnormal cases, then no coefficient GCD's would be needed to compute the F~, and yet the bounds (18)-(22) would apply to their coefficients. Fortunately this can be done. It is shown in [8] that F~ = T~ for i = 3, ... , ]c provided that we choose ~ = (- 1)~+1, (24) ~ = --fi_2~b~ '-2, i = 4, ... , k, where (25) ( 4" ~i--3.Ll--~i--3 ~ = ~--ji-2j ~i-1 , i = 4, ... ,k. At the present time it is not known whether or not these equations imply ~, ~ E ~" In any event, 6~ and 3~ belong to the quotient field 5: of ~, and they yield the PRS, F1, F2, T3, ... , Tk in 9[x]. It is possible to eliminate 6~ from (24), and write ~ explicitly as a product of powers of f2, • • " , fi-2, and (- 1). The resulting formula is given in [7]. In the case of a normal PRS, note that the subresultant PRS algorithm (24) and the reduced PRS algorithm (13) agree up to signs as required by (23); in fact, the signs also agree if ~1 is odd. It is natural to wonder how close a subresultant PRS is likely to be to the cor- responding primitive PRS. In the example (4), the subresultant PRS is 1, 0, 1, 0, --3, --3, 8, 2, --5 3, 0, 5, 0, --4, --9, 21 15, 0, --3, 0, 9 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 487 65, 125, --245 9326, -- 12300 260708 (26)
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