This work is concerned with the computation of the action of a matrix function f(A), such as the matrix exponential or the matrix square root, on a vector b. For a general matrix A, this can be done by computing the compression of A onto a suitable Krylov subspace. Such compression is usually computed by forming an orthonormal basis of the Krylov subspace using the Arnoldi method. In this work, we propose to compute (non-orthonormal) bases in a faster way and to use a fast randomized algorithm for least-squares problems to compute the compression of A onto the Krylov subspace. We present some numerical examples which show that our algorithms can be faster than the standard Arnoldi method while achieving comparable accuracy.

翻译：本文研究矩阵函数f(A)（如矩阵指数或矩阵平方根）作用于向量b时的计算问题。对于一般矩阵A，可通过将A压缩到合适的Krylov子空间上实现该计算。这类压缩通常采用Arnoldi方法构造Krylov子空间的标准正交基来完成。本文提出一种更快的非标准正交基构造方法，并利用快速随机化算法求解最小二乘问题，从而实现A到Krylov子空间的压缩。数值算例表明，本文算法在保持相当精度的同时，计算速度可优于标准Arnoldi方法。