结构正交化Arnoldi算法
Arnoldi algorithms with structured orthogonalization
摘要：
研究了大规模电力传输网络（PDN）时域仿真中矩阵指数保持稳定的Arnoldi算法，该算法可表示为半显式微分代数方程（DAE）。解可以分解为两个投影的和，一个在系统算子的范围内，另一个在其零空间内。距离投影可以用一种平移-逆Krylov子空间方法来计算。另一个投影可以用代数方程计算。与通常的Arnoldi方法不同，该方法用半正定系统算子诱导的半内积代替Krylov子空间中的正交性。通过适当的调整，Krylov算子的数值域位于右半半平面内，得到了修正Arnoldi算法计算φ函数的理论收敛性分析.最后，通过RLC网络的仿真验证了结构正交化Arnoldi算法的有效性。
英文摘要：
We study a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks (PDN), 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 one 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 semi-definite 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. Lastly, simulations on RLC networks are demonstrated to validate the effectiveness of the Arnoldi algorithm with structured-orthogonalization.
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