  While preconditioning is a long-standing concept to accelerate iterative
methods for linear systems, generalizations to matrix functions are still in
their infancy. We go a further step in this direction, introducing polynomial
preconditioning for Krylov subspace methods which approximate the action of the
matrix square root and inverse square root on a vector. Preconditioning reduces
the subspace size and therefore avoids the storage problem together with -- for
non-Hermitian matrices -- the increased computational cost per iteration that
arises in the unpreconditioned case. Polynomial preconditioning is an
attractive alternative to current restarting or sketching approaches since it
is simpler and computationally more efficient. We demonstrate this for several
numerical examples.
