  We discuss a proposal for a continued fraction-like algorithm to determine
simultaneous rational approximations to $d$ real numbers
$\alpha_1,\ldots,\alpha_d$. It combines an algorithm of Hermite and Lagarias
with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with
parameter $t$ as $t\downarrow0$. The new idea in this paper is that checking
the LLL-conditions consists of solving linear equations in $t$.
