Evaluating the action of a matrix function on a vector, that is $x=f(\mathcal M)v$, is an ubiquitous task in applications. When $\mathcal M$ is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating $x$ when $f(z)$ is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is equivalent, completely monotonic) and $\mathcal M$ is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case $\mathcal M=I \otimes A - B^T \otimes I$, and $v$ obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of $x$. Pole selection strategies with explicit convergence bounds are given also in this case.

中文翻译：

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Stieltjes 矩阵函数的有理 Krylov：收敛和极点选择

评估矩阵函数对向量的作用，即 $x=f(\mathcal M)v$，是应用程序中无处不在的任务。当 $\mathcal M$ 很大时，通常依赖于 Krylov 投影方法。在本文中，当 $f(z)$ 是 Cauchy-Stieltjes 或 Laplace-Stieltjes（或等效的完全单调）且 $\ mathcal M$ 是一个正定矩阵。依靠与分析泛型情况相同的工具，我们接下来关注 $\mathcal M=I \otimes A - B^T \otimes I$ 的情况，$v$ 得到向量化一个低秩矩阵；例如，这可以应用于求解二维张量网格上的分数扩散方程。我们看到了如何利用张量化的 Krylov 子空间来利用 Kronecker 结构，并介绍了 $x$ 的数值近似值的误差分析。在这种情况下也给出了具有显式收敛边界的极点选择策略。