2022/12/01 by Zhongxiao Jia, Jia, Zhongxiao, Qingqing Zheng +1
Computer Science · Mathematics · #15A18 #15A22 #65F15 #93B20 #93B60 #Advanced Optimization Algorithms Research #Applied mathematics #Boundary value problem #Convergence (economics) #Eigenvalues and eigenvectors #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical analysis #Mathematics #Matrix Theory and Algorithms #Nonlinear system #Norm (philosophy) #Numerical Analysis (math.NA) #Physics #Rayleigh–Ritz method #Ritz method #Subspace topology
paper · pdf · doi:10.48550/arxiv.2212.00302
openalex publication_date 2022/12/01 · openalex created_date 2022/12/13 · openalex updated_date 2026/08/05
We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair (λ*,x*) of a given analytic regular nonlinear eigenvalue problem (NEP). In terms of the deviation ε of x* from a given subspace W, we establish a priori convergence results on the Ritz value, the Ritz vector and the refined Ritz vector. The results show that, as ε→ 0, there exists a Ritz value that unconditionally converges to λ_* and the corresponding refined Ritz vector does so too but the Ritz vector converges conditionally and it may fail to converge and even may not be unique. We also present an error bound for the approximate eigenvector in terms of the computable residual norm of a given approximate eigenpair, and give lower and upper bounds for the error of the refined Ritz vector and the Ritz vector as well as for that of the corresponding residual norms. These results nontrivially extend some convergence results on these two methods for the linear eigenvalue problem to the NEP. Examples are constructed to illustrate the main results.