Complexity of an extended lattice reduction algorithm

We consider the complexity of a Lenstra Lenstra Lovasz lattice reduction algorithm ([LLL]) in which the vectors are allowed to be linearly dependent and in which one also asks for the matrix of the transformation from the given generators to the reduced basis. The main problem will be to show that the entries of the transformation matrix remain bounded through the algorithm, with a reasonable bound. Here the difficulty is of course that due to the dependence of the generators the transformation is not determined by the basis. To remedy this we work with two inner products and apply the LLL methods to both.