论文标题
用于矩阵函数近似的全局扩展理性Arnoldi方法
The global extended-rational Arnoldi method for matrix function approximation
论文作者
论文摘要
矩阵函数（例如$ f（a）v $）的数值计算，其中$ a $是$ n \ times n $大且稀疏的方形矩阵，$ v $是一个$ n \ times p $ block，带有$ p \ ll n $ and $ f $ and $ f $ and $ f $是非线性矩阵，是一个非线性矩阵，在各种应用程序中出现在各种应用程序中，例如网络分析（$ f（$ f（$ f），$ f（$ f（$ f（$ f）， $（f（t）= log（t））$，量子染色体动力学理论$（f（t）= t^{1/2}）$，电子结构计算等。在这项工作中，我们建议使用全局扩展理性的Arnoldi方法来计算此类表达式的近似值。派生的方法将初始问题投射到全球扩展理性的Krylov子空间$ \ MATHCAL {rk}^{ \ ldots，a^{m-1} v \}）$的低维度。给出了选择移位参数$ \ {s_1，\ ldots，s_m \} $的自适应过程。所提出的方法还应用于求解参数依赖性系统。提出了数值示例，以显示这些问题的全局扩展理性Arnoldi的性能。
The numerical computation of matrix functions such as $f(A)V$, where $A$ is an $n\times n$ large and sparse square matrix, $V$ is an $n \times p$ block with $p\ll n$ and $f$ is a nonlinear matrix function, arises in various applications such as network analysis ($f(t)=exp(t)$ or $f(t)=t^3)$, machine learning $(f(t)=log(t))$, theory of quantum chromodynamics $(f(t)=t^{1/2})$, electronic structure computation, and others. In this work, we propose the use of global extended-rational Arnoldi method for computing approximations of such expressions. The derived method projects the initial problem onto an global extended-rational Krylov subspace $\mathcal{RK}^{e}_m(A,V)=\text{span}(\{\prod\limits_{i=1}^m(A-s_iI_n)^{-1}V,\ldots,(A-s_1I_n)^{-1}V,V$ $,AV, \ldots,A^{m-1}V\})$ of a low dimension. An adaptive procedure for the selection of shift parameters $\{s_1,\ldots,s_m\}$ is given. The proposed method is also applied to solve parameter dependent systems. Numerical examples are presented to show the performance of the global extended-rational Arnoldi for these problems.
