The Rayleigh–Ritz method, refinement and Arnoldi process for periodic matrix pairs

• Published in: Journal of computational and applied mathematics Vol. 235; no. 8; pp. 2626 - 2639
• Main Authors: Chu, Eric King-Wah, Fan, Hung-Yuan, Jia, Zhongxiao, Li, Tiexiang, Lin, Wen-Wei
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 15.02.2011; Elsevier
• Subjects: Approximation; Arnoldi process; Calculus of variations and optimal control; Deviation; Eigenvalues; Eigenvectors; Exact sciences and technology; Mathematical analysis; Mathematical models; Mathematics; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Periodic eigenvalues; Periodic matrix pairs; Rayleigh-Ritz method; Refinement; Ritz values; Sciences and techniques of general use; Vectors (mathematics); Periodic matrix pairs; Refinement; Ritz values; Arnoldi process; Periodic eigenvalues; Rayleigh–Ritz method; Rayleigh-Ritz method; Transcendental equation; Approximation; Arnoldi method; Refinement method; Eigenvector; Numerical linear algebra; Eigenvalue; Non linear equation; Numerical analysis; Applied mathematics; Algebraic equation; Periodic matrix
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2010.11.014

We extend the Rayleigh–Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for Rayleigh–Ritz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the Rayleigh–Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors.