The Fréchet derivative $L_f(A,E)$ of the matrix function $f(A)$ plays an important role in many different applications, including condition number estimation and network analysis. We present several different Krylov subspace methods for computing low-rank approximations of $L_f(A,E)$ when the direction term $E$ is of rank one (which can easily be extended to general low rank). We analyze the convergence of the resulting methods both in the Hermitian and non-Hermitian case. In a number of numerical tests, both including matrices from benchmark collections and from real-world applications, we demonstrate and compare the accuracy and efficiency of the proposed methods.

中文翻译：

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使用 Krylov 子空间方法计算矩阵函数的 Fréchet 导数的低秩近似

Fréchet 导数 ${升}_F(\mathrm{一种},乙)$ 矩阵函数的 $F(\mathrm{一种})$在许多不同的应用中扮演着重要的角色，包括条件数估计和网络分析。我们提出了几种不同的 Krylov 子空间方法来计算${升}_F(\mathrm{一种},乙)$ 当方向项 $乙$等级为 1（可以很容易地扩展到一般的低等级）。我们分析了 Hermitian 和非 Hermitian 情况下所得方法的收敛性。在许多数值测试中，包括来自基准集合和实际应用的矩阵，我们展示并比较了所提出方法的准确性和效率。