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While preconditioning is a long-standing concept to accelerate iterativemethods for linear systems, generalizations to matrix functions are still intheir infancy. We go a further step in this direction, introducing polynomialpreconditioning for Krylov subspace methods which approximate the action of thematrix square root and inverse square root on a vector. Preconditioning reducesthe subspace size and therefore avoids the storage problem together with -- fornon-Hermitian matrices -- the increased computational cost per iteration thatarises in the unpreconditioned case. Polynomial preconditioning is anattractive alternative to current restarting or sketching approaches since itis simpler and computationally more efficient. We demonstrate this for severalnumerical examples.

Polynomial Preconditioning for the Action of the Matrix Square Root and Inverse Square Root