论文标题
使用Krylov子空间方法计算矩阵函数的Fréchet衍生物的低级别近似值
Computing low-rank approximations of the Fréchet derivative of a matrix function using Krylov subspace methods
论文作者
论文摘要
矩阵函数$ f（a）$的fréchet衍生品$ l_f（a，e）$在许多不同的应用程序中起重要作用，包括条件号估计和网络分析。当方向项$ e $是等级第一时，我们提出了几种不同的Krylov子空间方法，用于计算$ l_f（a，e）$的低级别近似值（很容易扩展到一般的低级级别）。我们分析了最重要的特殊情况，即$ a $是Hermitian，而$ f $是指数，对数或stieltjes函数的融合。在许多数值测试中，包括来自基准收集的矩阵和现实世界应用的矩阵，我们演示和比较了所提出方法的准确性和效率。
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 method for the important special case that $A$ is Hermitian and $f$ is either the exponential, the logarithm or a Stieltjes function. 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.
