We present an exact version of the local bosonic algorithm for the sim
ulation of dynamical quarks in lattice QCD. This version is based on a
non-hermitian polynomial approximation of the inverse of the quark ma
trix. A Metropolis test corrects the systematic errors. Two variants o
f this test are presented. For both of them, a formal proof is given t
hat this Monte Carlo algorithm converges to the right distribution. Si
mulation data are presented for different lattice parameters. The dyna
mics of the algorithm and its scaling in lattice volume and quark mass
are investigated.