  The idea of using polynomial methods to improve simple smoother iterations
within a multigrid method for a symmetric positive definite (SPD) system is
revisited. When the single-step smoother itself corresponds to an SPD operator,
there is in particular a very simple iteration, a close cousin of the Chebyshev
semi-iterative method, based on the Chebyshev polynomials of the fourth instead
of first kind, that optimizes a two-level bound going back to Hackbusch. A full
V-cycle bound for general polynomial smoothers is derived using the V-cycle
theory of McCormick. The fourth-kind Chebyshev iteration is quasi-optimal for
the V-cycle bound. The optimal polynomials for the V-cycle bound can be found
numerically, achieving an about 18% lower error contraction factor bound than
the fourth-kind Chebyshev iteration, asymptotically as the number of smoothing
steps goes to infinity. Implementation of the optimized iteration is discussed,
and the performance of the polynomial smoothers are illustrated with a simple
numerical example.
