Geodesic continued fractions and LLL

We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to dd real numbers α1,…,αdα1,…,αd. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter tt as t↓0t↓0. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in tt