  In this thesis we propose a novel implementation of IDRstab that avoids
several unlucky breakdowns of current IDRstab implementations and is further
capable of benefiting from a particular lucky breakdown scenario. IDRstab is a
very efficient short-recurrence Krylov subspace method for the numerical
solution of linear systems.
  Current IDRstab implementations suffer from slowdowns in the rate of
convergence when the basis vectors of their oblique projectors become linearly
dependent.
  We propose a novel implementation of IDRstab that is based on a successively
restarted GMRES method. Whereas the collinearity of basis vectors in current
IDRstab implementations would lead to an unlucky breakdown, our novel IDRstab
implementation can strike a benefit from it in that it terminates with the
exact solution whenever a new basis vector lives in the span of the formerly
computed basis vectors.
  Numerical experiments demonstrate the superior robustness of our novel
implementation with regards to convergence maintenance and the achievable
accuracy of the numerical solution.
