By Georgia. Wildlife Resources Division
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The continued fraction is given by the sequence of quotients in the Euclidean algorithm A2 = = At-I = Al QIA2 +A3 Q2A3 + A4 Qt-IAt . Up to now we have always been able to give explicit polynomial or rational formulas for the output in terms of the input. In this case, however, we know neither the number t nor the degrees of the quotients Qi in advance. Hence, the length of the output may vary even if the degrees of the input polynomials A I 18 Chapter 1. Introduction and A2 are fixed. Such a computation cannot be performed within the straightline model.
3 we discuss the problem of multiplying several polynomials. An obvious divide-and-conquer technique yields the upper bound 0 (M (n) log t) for the complexity of the problem of multiplying t polynomials whose degrees sum up to n. A more refined version which uses a Huffman coding technique gives an upper bound of order M(n)(l + H), where H is the entropy of the probability distribution corresponding to the degree sequence of the polynomials. In Sect. 4 it is shown that the arithmetic complexity of multiplication, inversion, and division of formal power series mod X n+ 1 is also bounded by O(M(n)), thus emphasizing the fundamental role of polynomial multiplication.
T(L) + E(£) < 2E(8) < 2A· T(L) + 2E(a) + 2E(y) + 2E(rp) + E(f3) + E(y-l) + E(rp-l). Adding two elements of A2L or multiplying an element of A2L by a power of X or by an element of P costs at most 2L arithmetic operations in A. As no 38 Chapter 2. Efficient Polynomial Arithmetic other scalar multiplications occur in the F FT algorithms for cp and its inverse, we obtain by Ihm. 6) E(cp) ::: 2L . (3A log(2A) - 2A + 1) and E(cp-I) ::: E(cp) + 2L ·2A. Now L . A = n, thus O(n logn) is an upper bound for both E(cp) and E(cp-I).
Alligator fact sheet by Georgia. Wildlife Resources Division