  We present an algorithm for the solution of Sylvester equations with
right-hand side of low rank. The method is based on projection onto a block
rational Krylov subspace, with two key contributions with respect to the
state-of-the-art. First, we show how to maintain the last pole equal to
infinity throughout the iteration, by means of pole reodering. This allows for
a cheap evaluation of the true residual at every step. Second, we extend the
convergence analysis in [Beckermann B., An error analysis for rational Galerkin
projection applied to the Sylvester equation, SINUM, 2011] to the block case.
This extension allows to link the convergence with the problem of minimizing
the norm of a small rational matrix over the spectra or field-of-values of the
involved matrices. This is in contrast with the non-block case, where the
minimum problem is scalar, instead of matrix-valued. Replacing the norm of the
objective function with an easier to evaluate function yields several adaptive
pole selection strategies, providing a theoretical analysis for known
heuristics, as well as effective novel techniques.
