Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices
Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices
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DOI：
10.1137/140973463
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发表时间：
2014-12
期刊：
影响因子：
--
通讯作者：
A. Frommer;S. Güttel;M. Schweitzer
中科院分区：
文献类型：
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作者：
A. Frommer;S. Güttel;M. Schweitzer
大约$ f（a）{b} $ ---矩阵函数在向量上的作用---通过Krylov子空间方法，由于Arnoldi基础的存储要求或计算的增长，重新启动可能会重新启动。评估$ f $的复杂性在增长尺寸的黑森伯格矩阵上。近年来，文献中已经提出了许多重新启动方法，并且有关其稳定性和计算效率的算法进步已有很大的进步。但是，在哪些情况下可以保证可以保证这些方法的问题在很大程度上尚未得到答复。在本文中，我们考虑了stieltjes函数类别和相关类别的类别，这些函数包含（逆）平方根和矩阵对数等重要函数。对于这些类别的功能，我们提出了新的理论结果，这些结果证明了赫米尔人正确定矩阵$ a $ a $ a $ and tunsart的重新启动长度。我们还提出了对瓜兰的Arnoldi近似的修改...
To approximate $f(A){b}$---the action of a matrix function on a vector---by a Krylov subspace method, restarts may become mandatory due to storage requirements for the Arnoldi basis or due to the growing computational complexity of evaluating $f$ on a Hessenberg matrix of growing size. A number of restarting methods have been proposed in the literature in recent years and there has been substantial algorithmic advancement concerning their stability and computational efficiency. However, the question under which circumstances convergence of these methods can be guaranteed has remained largely unanswered. In this paper we consider the class of Stieltjes functions and a related class, which contain important functions like the (inverse) square root and the matrix logarithm. For these classes of functions we present new theoretical results which prove convergence for Hermitian positive definite matrices $A$ and arbitrary restart lengths. We also propose a modification of the Arnoldi approximation which guaran...