lobpcg.m, MATLAB implementation of the locally optimal block preconditioned conjugate gradient method: Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. We describe new algorithms of the locally optimal block preconditioned conjugate gradient (LOBPCG) method for symmetric eigenvalue problems, based on a local optimization of a three-term recurrence, and suggest several other new methods. To be able to compare numerically different methods in the class, with different preconditioners, we propose a common system of model tests, using random preconditioners and initial guesses. As the “ideal” control algorithm, we advocate the standard preconditioned conjugate gradient method for finding an eigenvector as an element of the null-space of the corresponding homogeneous system of linear equations under the assumption that the eigenvalue is known. We recommend that every new preconditioned eigensolver be compared with this “ideal” algorithm on our model test problems in terms of the speed of convergence, costs of every iteration, and memory requirements. We provide such comparison for our LOBPCG method. Numerical results establish that our algorithm is practically as efficient as the “ideal” algorithm when the same preconditioner is used in both methods. We also show numerically that the LOBPCG method provides approximations to first eigenpairs of about the same quality as those by the much more expensive global optimization method on the same generalized block Krylov subspace. We propose a new version of block Davidson’s method as a generalization of the LOBPCG method. Finally, direct numerical comparisons with the Jacobi-Davidson method show that our method is more robust and converges almost two times faster.

References in zbMATH (referenced in 93 articles , 1 standard article )

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  1. Li, Yingzhou; Lin, Lin: Globally constructed adaptive local basis set for spectral projectors of second order differential operators (2019)
  2. Ainsworth, Mark; Glusa, Christian: Hybrid finite element-spectral method for the fractional Laplacian: approximation theory and efficient solver (2018)
  3. Bergamaschi, Luca; Bozzo, Enrico: Computing the smallest eigenpairs of the graph Laplacian (2018)
  4. Duersch, Jed A.; Shao, Meiyue; Yang, Chao; Gu, Ming: A robust and efficient implementation of LOBPCG (2018)
  5. Klawonn, Axel; Kühn, Martin; Rheinbach, Oliver: Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems (2018)
  6. Miao, Cun-Qiang: Computing eigenpairs in augmented Krylov subspace produced by Jacobi-Davidson correction equation (2018)
  7. Teng, Zhongming; Wang, Xuansheng: Heavy ball restarted CMRH methods for linear systems (2018)
  8. Xue, Fei: A block preconditioned harmonic projection method for large-scale nonlinear eigenvalue problems (2018)
  9. Aishima, Kensuke: On convergence of iterative projection methods for symmetric eigenvalue problems (2017)
  10. Antoine, Xavier; Levitt, Antoine; Tang, Qinglin: Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by preconditioned nonlinear conjugate gradient methods (2017)
  11. Benner, Peter; Dolgov, Sergey; Khoromskaia, Venera; Khoromskij, Boris N.: Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation (2017)
  12. Fox, Alyson; Manteuffel, Thomas; Sanders, Geoffrey: Numerical methods for Gremban’s expansion of signed graphs (2017)
  13. Kim, Hyea Hyun; Chung, Eric; Wang, Junxian: BDDC and FETI-DP preconditioners with adaptive coarse spaces for three-dimensional elliptic problems with oscillatory and high contrast coefficients (2017)
  14. Lin, Lin: Localized spectrum slicing (2017)
  15. Lin, Lin; Xu, Ze; Ying, Lexing: Adaptively compressed polarizability operator for accelerating large scale ab initio phonon calculations (2017)
  16. Lu, Jianfeng; Yang, Haizhao: Preconditioning orbital minimization method for planewave discretization (2017)
  17. Miao, Cun-Qiang: A filtered-Davidson method for large symmetric eigenvalue problems (2017)
  18. Pérez-Jordá, José M.: Fast solution of Schrödinger’s equation using linear combinations of plane waves (2017)
  19. Wen, Zaiwen; Zhang, Yin: Accelerating convergence by augmented Rayleigh-Ritz projections for large-scale eigenpair computation (2017)
  20. Zhang, Junyu; Wen, Zaiwen; Zhang, Yin: Subspace methods with local refinements for eigenvalue computation using low-rank tensor-train format (2017)

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