An efficient Krylov subspace algorithm for computing actions of the $\varphi$ matrix function for large matrices is proposed. This matrix function is widely used in exponential time integration, Markov chains and network analysis and many other applications. Our algorithm is based on a reliable residual based stopping criterion and a new efficient restarting procedure. For matrices with numerical range in the stable complex half plane, we analyze residual convergence and prove that the restarted method is guaranteed to converge for any Krylov subspace dimension. Numerical tests demonstrate efficiency of our approach for solving large scale evolution problems resulting from discretized in space time-dependent PDEs, in particular, diffusion and convection-diffusion problems.

中文翻译：

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$\varphi$矩阵函数的Krylov子空间求值的残差概念

提出了一种有效的 Krylov 子空间算法，用于计算大矩阵的 $\varphi$ 矩阵函数的动作。这种矩阵函数广泛用于指数时间积分、马尔可夫链和网络分析等许多应用。我们的算法基于可靠的基于残差的停止标准和新的高效重启程序。对于稳定复半平面中数值范围的矩阵，我们分析了残差收敛性，并证明重新启动的方法保证收敛于任何 Krylov 子空间维度。数值测试证明了我们的方法在解决由空间时间相关 PDE 离散化引起的大规模演化问题，特别是扩散和对流扩散问题时的效率。