By B.M.M. de Weger
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Extra resources for Algorithms for Diophantine Equations
4)), ord (L) > c + c Wm , p 1 2 j where c , c are small constants, and m is one of the variables. 1 2 j Moreover, the variables are bounded by a large constant N , that is 0 m m+1 explicitly known. We take m such that p is at least of size N , so 0 * that the lower bound for the shortest nonzero vector in G (or G ) is rmWN0 . 2). Therefore, + c Wm < m , 2 j so that we find a new upper bound for m , that is of the size of m , which j is about log N / log p . We repeat this procedure for all the m , in 0 j order to obtain a reduced upper bound for H .
Note that V is I (the unit matrix), in which case U’ C so B-1 0 B0WU . , v’ . We feed the algorithm with U and n U-1 as well. All manipulations in the algorithm done on the bi are also T done on the u , and the v’ are adjusted accordingly. This does not i i affect the computation time seriously. The algorithm now gives as output -1 matrices C , U’ and U’ , such that C is associated to a reduced basis, = B U U’ = BWU-1WU’ = B0WU’ not computed explicitly, unless = V . It follows that , is the matrix of the transformation from B0 to is known, then it is not much extra effort to compute C .
7 W3 /5W65 + ... 5. We also use the simple continued fraction algorithm in some instances. This we do as follows. Suppose we want to compute the continued fraction expansion of a real number y , that we have approximated by rational numbers that y 1 < y < y 2 for some small and y < y 1 such + e e . We can compute the continued fraction expansions of y 1 exactly. As far as they coincide, they coincide also with the 2 continued fraction expansion of y y , y 1 2 is needed so far that the y . If the continued fraction expansion of k th convergent with denominator known exactly, for a given (large) constant -2 as small as X .