  By closely following a construction by Ganelius, we construct Faber rational
functions that allow us to derive tight and explicit bounds on Zolotarev
numbers. We use our results to bound the singular values of matrices, including
complex-valued Cauchy matrices and Vandermonde matrices with nodes inside the
unit disk. We construct Faber rational functions using doubly-connected
conformal maps and use their zeros and poles to supply shift parameters in the
alternating direction implicit method.
