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

Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions