A Convergence Analysis of the Inexact Simplified Jacobi–Davidson Algorithm for Polynomial Eigenvalue Problems

被引：0
作者：
Tao Zhao
机构：
[1] Chinese Academy of Sciences,Shenzhen Institutes of Advanced Technology
来源：
Journal of Scientific Computing | 2018年 / 75卷
关键词：
Polynomial eigenvalue problem; Jacobi–Davidson algorithm; Convergence analysis; Inexact correction equation solver; Simple eigenvalue;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
The simplified Jacobi–Davidson (JD) method is a variant of the JD method without subspace acceleration. If the correction equation is solved approximately, the inexact simplified JD method is obtained. In this paper, we present a new convergence analysis of the inexact simplified JD method. The analysis applies to polynomial eigenvalue problems with simple eigenpairs. We first establish a relationship between the solution of the correction equation and the residual of the approximate eigenpair. From this relationship, we find the difference of two adjacent approximate eigenvalues bounded in terms of the residual norm of the approximate eigenpair. Then we prove the convergence of the inexact simplified JD method in terms of the residual norm of the approximate eigenpair. Depending on how accurately we solve the correction equation, the convergence rate of the inexact simplified JD may take several different forms. Numerical experiments confirm the convergence analysis.
引用
收藏
页码：1207 / 1228
页数：21
相关论文
共 72 条
  • [1] Asakura J(2010)A numerical method for polynomial eigenvalue problems using contour integral Jpn. J. Ind. Appl. Math. 27 73-90
  • [2] Sakurai T(2013)NLEVP: a collection of nonlinear eigenvalue problems ACM Trans. Math. Softw. 39 7:1-7:28
  • [3] Tadano H(2012)An integral method for solving nonlinear eigenvalue problems Linear Algebra Appl. 436 3839-3863
  • [4] Ikegami T(1999)A restricted additive Schwarz preconditioner for general sparse linear systems SIAM J. Sci. Comput. 21 792-797
  • [5] Kimura K(2012)Chebyshev interpolation for nonlinear eigenvalue problems BIT 52 933-951
  • [6] Betcke T(2007)Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem Electr. Trans. Numer. Anal. 28 40-64
  • [7] Higham NJ(2013)Algebraic domain decomposition methods for highly heterogeneous problems SIAM J. Sci. Comput. 35 C284-C302
  • [8] Mehrmann V(2008)Harmonic and refined Rayleigh–Ritz for the polynomial eigenvalue problem Numer. Linear Algebra Appl. 15 35-54
  • [9] Schröder C(2009)Controlling inner iterations in the Jacobi–Davidson method SIAM J. Matrix Anal. Appl. 31 460-477
  • [10] Tisseur F(2010)A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation J. Comput. Phys. 229 2932-2947
12345678