  Polynomial preconditioning can improve the convergence of the Arnoldi method
for computing eigenvalues. Such preconditioning significantly reduces the cost
of orthogonalization; for difficult problems, it can also reduce the number of
matrix-vector products. Parallel computations can particularly benefit from the
reduction of communication-intensive operations. The GMRES algorithm provides a
simple and effective way of generating the preconditioning polynomial. For some
problems high degree polynomials are especially effective, but they can lead to
stability problems that must be mitigated. A two-level "double polynomial
preconditioning" strategy provides an effective way to generate high-degree
preconditioners.
