  The characterization of the solvability of matrix versions of truncated
Stieltjes-type moment problems led to the class of $\alpha$-Stieltjes
non-negative definite sequences of complex $q \times q$ matrices. In [21], a
parametrization of this class was introduced, the so-called $\alpha$-Stieltjes
parametrization. The main topic of this first part of the paper is the
construction of a Schur-type algorithm which produces exactly the
$\alpha$-Stieltjes parametrization.
