A Krylov subspace recycling method for the efficient evaluation of a sequence of matrix functions acting on a set of vectors is developed. The method improves over the recycling methods presented in [Burke et al., arXiv:2209.14163, 2022] in that it uses a closed-form expression for the augmented FOM approximants and hence circumvents the use of numerical quadrature. We further extend our method to use randomized sketching in order to avoid the arithmetic cost of orthogonalizing a full Krylov basis, offering an attractive solution to the fact that recycling algorithms built from shifted augmented FOM cannot easily be restarted. The efficacy of the proposed algorithms is demonstrated with numerical experiments.

翻译：针对一组向量上的矩阵函数序列的高效求值，本文开发了一种Krylov子空间循环方法。该方法相较于[Burke等人，arXiv:2209.14163，2022]提出的循环方法有所改进，其优势在于采用了增广FOM逼近的闭式表达式，从而避免了数值求积过程。我们进一步将该方法扩展至随机化草图技术，以消除完整Krylov基正交化所需的算术开销，为基于移位增广FOM构建的循环算法难以直接重启的问题提供了具有吸引力的解决方案。数值实验验证了所提算法的有效性。