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.

翻译：尽管预条件技术是加速线性系统迭代方法的经典概念，但其在矩阵函数领域的推广仍处于起步阶段。本文在这一方向上迈出进一步探索，针对Krylov子空间方法中近似计算矩阵平方根及逆平方根对向量作用的问题，引入多项式预条件技术。预条件方法能缩减子空间规模，从而避免存储问题——对于非Hermitian矩阵而言，还能规避无预条件情形下每步迭代计算量增加的问题。多项式预条件作为当前重启法或草图法的替代方案，因其操作更简便、计算效率更高而具有显著优势。我们通过多个数值算例验证了该方法的有效性。