  We define certain higher-dimensional Dedekind sums that generalize the
classical Dedekind-Rademacher sums, and show how to compute them effectively
using a generalization of the continued-fraction algorithm.
  We present two applications. First, we show how to express special values of
partial zeta functions associated to totally real number fields in terms of
these sums via the Eisenstein cocycle introduced by the second author. Hence we
obtain a polynomial-time algorithm for computing these special values. Second,
we show how to use our techniques to compute certain special values of the
Witten zeta-function, and compute some explicit examples.
