On a Simultaneous Approach to the Even and Odd Truncated Matricial   Stieltjes Moment Problem I: An $\alpha$-Schur-Stieltjes-type algorithm for   sequences of complex matrices
  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. 
Bernd Fritzsche and Bernd Kirstein and Conrad M\"adler
math.CV