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A(x) - x m- n ·lc(a) . b(x) and m':= deg(c) . 2 Polynomials Then m' < m. For m' < n we can set q' := 0, r := lc(b)m-n . c and we get lc(b)m-n . c(x) = q'(x) . b(x) + r(x). For m' ::: n we can use the induction hypothesis on c and b, yielding ql ,rl such that lc(b)m'-n+1 . b + rl and (rl =0 or deg(rd < deg(b)) . Now we can multiply both sides by lc(b)m-m'-I and we get lc(b)m-n . c(x) = q'(x) . b(x) + r(x), where r =0 or deg(r) < deg(b) . 3). 3). Then ql ·b+rl = q2 ·b+r2, and (ql -q2)·b = r2-rl. Forql #- q2 we would have deg«ql-q2)·b) ::: deg(b) > deg(rl-r2), which is impossible.
For smaller inputs the constant of the Schonhage-Strassen algorithm determines the practical complexity. A similar phenomenon can be observed in factorization of polynomials, where in practice the Beriekamp-Hensel algorithm is preferred to the theoretically better Lenstra-Lenstra-Lovasz algorithm. , the value of n for which f(m) < gem), for m > n. See Fig. 7. 6 Bibliographic notes There are several general articles and books on computer algebra and related topics. Some of them are Akritas (1989), Boyle and Caviness (1990), Buchberger et al.
T1tLGCD(/I, 12, k) is O(min(/I, 12) . (max(ll, 12) - k + 1)). If d = 1, then the computing time for the gcd in QF_SUMC is roughly 4n 2 , whereas the gcd in QF _SUMH takes time roughly n 2 . SO QF_SUMH is faster than QF _SUMC by a factor of 4. Now let us assume that d =j:. 1, k = n/2, and e = 1. In this case the computation time for the gcd in QF_SUMC is 2n(2n - n/2) = 3n 2 . The times for the gcd computations in QF _SUMH are n (n - n /2) = n 2 /2 and (n /2) (3n /2) = 3n 2 /4. So in this case QF _SUMH is faster than QF _SUMC by a factor of 12/5.
Advances in Computers, Vol. 13