A common way to approximate $F(A)b$ -- the action of a matrix function on a vector -- is to use the Arnoldi approximation. Since a new vector needs to be generated and stored in every iteration, one is often forced to rely on restart algorithms which are either not efficient, not stable or only applicable to restricted classes of functions. We present a new representation of the error of the Arnoldi iterates if the function $F$ is given as a Laplace transform. Based on this representation we build an efficient and stable restart algorithm. In doing so we extend earlier work for the class of Stieltjes functions which are special Laplace transforms. We report several numerical experiments including comparisons with the restart method for Stieltjes functions.

翻译：一种常见的近似计算$F(A)b$（矩阵函数作用于向量）的方法是使用Arnoldi逼近。由于每次迭代都需要生成并存储一个新向量，通常不得不依赖重启算法，但这些算法要么效率不高，要么不稳定，要么仅适用于特定类别的函数。我们针对函数$F$以拉普拉斯变换形式给出的情况，提出了Arnoldi迭代误差的新表示。基于该表示，我们构建了一种高效且稳定的重启算法。该工作是对Stieltjes函数（一类特殊的拉普拉斯变换）相关早期研究的扩展。我们报告了多个数值实验，包括与Stieltjes函数重启方法的对比结果。