大規模ネットワークシステムにおける固有値の効率的かつ信頼性の高い計算 [in Japanese] Efficient and Highly-Reliable Calculation of Eigenvalues in Large-Scale Network Systems [in Japanese]
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An investigation is performed on a method for calculating eigenvalues in large-scale network systems. The proposed method uses two calculation techniques; numerical solution of a differential equation model to set up appropriate initial guesses of dependent variables for a root-finding method and numerical differentiation of a given function by a complexstep method to obtain highly-accurate numerical derivatives. For comparison, a finite-difference method is also used for numerical differentiation. The results reveal that regarding steady-state dependent variable values for differential equations models, both numerical differentiation methods provide calculated values within machine accuracy. Regarding the matrix values used for constituting characteristic equations, on the other hand, the complex-step method provides calculated values within machine accuracy, whereas the finite-difference method provides calculated values with 12<sup>-</sup>13 significant digits of accuracy, which results in a low accuracy of eigenvalues. Although the accuracies of eigenvalues calculated by the complex-step method may be lowered slightly by matrix operation, most of them are fundamentally within machine accuracy and highly accurate. In conclusion, the proposed method provides highly reliable eigenvalues and is useful to analyze large-scale network systems.
Eco-Engineering 27(2), 35-42, 2015
The Society of Eco-Engineering