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.

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  1. Jolivet, Pierre; Roman, Jose E.; Zampini, Stefano: KSPHPDDM and PCHPDDM: extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners (2021)
  2. Krumnow, Christian; Pfeffer, Max; Uschmajew, André: Computing eigenspaces with low rank constraints (2021)
  3. Ezvan, Olivier; Zeng, Xiaoshu; Ghanem, Roger; Gencturk, Bora: Multiscale modal analysis of fully-loaded spent nuclear fuel canisters (2020)
  4. Ferrari, Federico; Sigmund, Ole: Towards solving large-scale topology optimization problems with buckling constraints at the cost of linear analyses (2020)
  5. Kelley, C. T.; Bernholc, J.; Briggs, E. L.; Hamilton, Steven; Lin, Lin; Yang, Chao: Mesh independence of the generalized Davidson algorithm (2020)
  6. Lu, Ding: Nonlinear eigenvector methods for convex minimization over the numerical range (2020)
  7. Miao, Cun-Qiang: On Chebyshev-Davidson method for symmetric generalized eigenvalue problems (2020)
  8. Nakatsukasa, Yuji: Sharp error bounds for Ritz vectors and approximate singular vectors (2020)
  9. Xu, Fei; Xie, Hehu; Zhang, Ning: A parallel augmented subspace method for eigenvalue problems (2020)
  10. Zhang, Lei-Hong; Yang, Wei Hong; Shen, Chungen; Ying, Jiaqi: An eigenvalue-based method for the unbalanced Procrustes problem (2020)
  11. Altmann, R.; Peterseim, D.: Localized computation of eigenstates of random Schrödinger operators (2019)
  12. Dax, Achiya: Computing the smallest singular triplets of a large matrix (2019)
  13. Elman, Howard C.; Su, Tengfei: Low-rank solution methods for stochastic eigenvalue problems (2019)
  14. Goldenberg, Steven; Stathopoulos, Andreas; Romero, Eloy: A Golub-Kahan Davidson method for accurately computing a few singular triplets of large sparse matrices (2019)
  15. Heinlein, Alexander; Klawonn, Axel; Knepper, Jascha; Rheinbach, Oliver: Adaptive GDSW coarse spaces for overlapping Schwarz methods in three dimensions (2019)
  16. Hu, Jiang; Jiang, Bo; Lin, Lin; Wen, Zaiwen; Yuan, Ya-Xiang: Structured quasi-Newton methods for optimization with orthogonality constraints (2019)
  17. Imakura, Akira; Yamamoto, Yusaku: Efficient implementations of the modified Gram-Schmidt orthogonalization with a non-standard inner product (2019)
  18. Kong, Yuan; Fang, Yong: Behavior of the correction equations in the Jacobi-Davidson method (2019)
  19. Lin, Lin; Lu, Jianfeng; Ying, Lexing: Numerical methods for Kohn-Sham density functional theory (2019)
  20. Lin, Lin; Zepeda-Nunez, Leonardo: Projection-based embedding theory for solving Kohn-Sham density functional theory (2019)

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