我們將有系統地研究兩種用於解二次特徵值問題(QEPs)的演算法，包含線性化方法與多項式 Jacobi-Davidson (JD) 方法。這些特徵值問題在計算科學和工程中有重要的應用，像是聲學中的噪音控制、結構工程中的穩定性分析和電子工程。在線性化方法中，QEP被線性化為伴隨的廣義特徵值問題 (GEVP)，且解決了所得到的GEVP。另一方面，JD 方法是直接去找目標特徵值。我們使用一個 Matlab-based 的工具， a collection of nonlinear eigenvalue problems (NLEVP) 產生大量具有差異性值的矩陣來做數值實驗，並用 robustness， accuracy 和 efficiency 來評估效率問題。

We numerically investigate the numerical performance of two solution algorithms for the quadratic eigenvalue problems (QEP's), namely the linearization approach and the polynomial Jacobi-Davidson method. Such eigenvalue computations play an important role and highly-demanded in many computational sciences and engineering applications, such as the noise control in the acoustical design, stability analysis in the structural engineering, and electronic engineering. In the linearization approach, the QEP is linearized as a companion generalized eigenvalue problems (GEVP's), and then a variety of linear eigensolvers are solved the resulting GEVP's. On the other hand, the polynomial Jacobi-Davidson method targets the eigenvalue of interests directly without any transformation. The evaluation metrics are the robustness, accuracy, and efficiency. To draw the conclusion for more general situations, we conduct intensive numerical experiments for a large number of test cases generated by a collection of Nonlinear Eigenvalue Problem (NLEPV), with a various problem size and different coefficient matrices properties.

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