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Regularized Normalization Methods for Solving Linear and Nonlinear Eigenvalue Problems

Author

Listed:
  • Chein-Shan Liu

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Chung-Lun Kuo

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Chih-Wen Chang

    (Department of Mechanical Engineering, National United University, Miaoli 36063, Taiwan)

Abstract

To solve linear and nonlinear eigenvalue problems, we develop a simple method by directly solving a nonhomogeneous system obtained by supplementing a normalization condition on the eigen-equation for the uniqueness of the eigenvector. The novelty of the present paper is that we transform the original homogeneous eigen-equation to a nonhomogeneous eigen-equation by a normalization technique and the introduction of a simple merit function, the minimum of which leads to a precise eigenvalue. For complex eigenvalue problems, two normalization equations are derived utilizing two different normalization conditions. The golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues, and simultaneously, we can obtain precise eigenvectors to satisfy the eigen-equation. Two regularized normalization methods can accelerate the convergence speed for two extensions of the simple method, and a derivative-free fixed-point Newton iterative scheme is developed to compute real eigenvalues, the convergence speed of which is ten times faster than the golden section search algorithm. Newton methods are developed for solving two systems of nonlinear regularized equations, and the efficiency and accuracy are significantly improved. Over ten examples demonstrate the high performance of the proposed methods. Among them, the two regularization methods are better than the simple method.

Suggested Citation

  • Chein-Shan Liu & Chung-Lun Kuo & Chih-Wen Chang, 2023. "Regularized Normalization Methods for Solving Linear and Nonlinear Eigenvalue Problems," Mathematics, MDPI, vol. 11(18), pages 1-24, September.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3997-:d:1243918
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    References listed on IDEAS

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    1. Ioannis K. Argyros & Stepan Shakhno, 2019. "Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions," Mathematics, MDPI, vol. 7(2), pages 1-12, February.
    2. Chein-Shan Liu & Jiang-Ren Chang & Jian-Hung Shen & Yung-Wei Chen & Xian-Ming Gu, 2023. "A New Quotient and Iterative Detection Method in an Affine Krylov Subspace for Solving Eigenvalue Problems," Journal of Mathematics, Hindawi, vol. 2023, pages 1-17, March.
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