  We develop a semistability algorithm for vector bundles which are given as a
kernel of a surjective morphism between splitting bundles on the projective
space over an algebraically closed field K. This class of bundles is a
generalization of syzygy bundles. We show how to implement this algorithm in a
computer algebra system. Further we give applications, mainly concerning the
computation of Tannaka dual groups of stable vector bundles of degree 0 on the
projective space and on certain smooth complete intersection curves. We also
use our algorithm to close an open case left in a recent work of L. Costa, P.
Macias Marques and R. M. Miro-Roig regarding the stability of the syzygy bundle
of general forms. Finally, we apply our algorithm to provide a computational
approach to tight closure. All algorithms are implemented in the computer
algebra system CoCoA.
