SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 370-400, January 2021.
We study a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks (PDNs), which are formulated as semi-explicit differential algebraic equations (DAEs). The solution can be decomposed to a sum of two projections, one in the range of the system operator and the other in its null space. The range projection can be computed with a shift-and-invert Krylov subspace method. The other projection can be computed with the algebraic equations. Differing from the ordinary Arnoldi method, the orthogonality in the Krylov subspace is replaced with the semi-inner product induced by the positive semidefinite system operator. With proper adjustment, numerical ranges of the Krylov operator lie in the right half-plane, and we obtain theoretical convergence analysis for the modified Arnoldi algorithm in computing phi-functions. Last, simulations on RLC networks are demonstrated to validate the effectiveness of the Arnoldi algorithm.

中文翻译：

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结构正交化的Arnoldi算法

SIAM数值分析学报，第59卷，第1期，第370-400页，2021年1月。
我们在时域仿真大型电力输送网络（PDN）中研究了矩阵指数的稳定保留Arnoldi算法，该网络被公式化为半显式微分代数方程（DAE）。该解决方案可以分解为两个投影的总和，一个投影在系统操作员的范围内，另一个投影在其零空间内。范围投影可以使用移位和反转Krylov子空间方法来计算。另一个投影可以用代数方程来计算。与普通的Arnoldi方法不同，Krylov子空间中的正交性由正半定系统算子引起的半内积代替。经过适当的调整，Krylov算子的数值范围位于右半平面，并获得了改进的Arnoldi算法在计算phi函数时的理论收敛性分析。最后，在RLC网络上的仿真被证明可以验证Arnoldi算法的有效性。