  We present a multiplication algorithm to recursively construct vertices for
the Schroedinger functional in the abelian background field case. The algorithm
is suited for automatic perturbative calculations with a variety of actions. As
first applications, we derive ratios of the lambda parameters between the
lattice scheme (improved gauge actions including six link loops) and the
$\bar{\rm MS}$ scheme, and one-loop results for the Schroedinger functional
coupling with a lattice $T=L \pm a$, which is motivated by considering
staggered fermions.
