Odd-flavor Hybrid Monte Carlo Algorithm for Lattice QCD

We discuss hybrid Monte Carlo algorithms for odd-flavor lattice QCD simulations. The algorithms include a polynomial approximation which enables us to simulate odd-flavor QCD in the framework of the hybrid Monte Carlo algorithm. In order to make the algorithms exact, the correction factor to the polynomial approximation is also included in an economical, stochastic way. We test the algorithms for $n_f=1$, 1+1 and 2+1 flavors and compare results with other algorithms.