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. Ferrari, Federico; Sigmund, Ole: Towards solving large-scale topology optimization problems with buckling constraints at the cost of linear analyses (2020)
  2. Nakatsukasa, Yuji: Sharp error bounds for Ritz vectors and approximate singular vectors (2020)
  3. Altmann, R.; Peterseim, D.: Localized computation of eigenstates of random Schrödinger operators (2019)
  4. Elman, Howard C.; Su, Tengfei: Low-rank solution methods for stochastic eigenvalue problems (2019)
  5. Goldenberg, Steven; Stathopoulos, Andreas; Romero, Eloy: A Golub-Kahan Davidson method for accurately computing a few singular triplets of large sparse matrices (2019)
  6. Heinlein, Alexander; Klawonn, Axel; Knepper, Jascha; Rheinbach, Oliver: Adaptive GDSW coarse spaces for overlapping Schwarz methods in three dimensions (2019)
  7. Hu, Jiang; Jiang, Bo; Lin, Lin; Wen, Zaiwen; Yuan, Ya-Xiang: Structured quasi-Newton methods for optimization with orthogonality constraints (2019)
  8. Imakura, Akira; Yamamoto, Yusaku: Efficient implementations of the modified Gram-Schmidt orthogonalization with a non-standard inner product (2019)
  9. Lin, Lin; Lu, Jianfeng; Ying, Lexing: Numerical methods for Kohn-Sham density functional theory (2019)
  10. Lin, Lin; Zepeda-Nunez, Leonardo: Projection-based embedding theory for solving Kohn-Sham density functional theory (2019)
  11. Li, Ruipeng; Xi, Yuanzhe; Erlandson, Lucas; Saad, Yousef: The eigenvalues slicing library (EVSL): algorithms, implementation, and software (2019)
  12. Li, Yingzhou; Lin, Lin: Globally constructed adaptive local basis set for spectral projectors of second order differential operators (2019)
  13. Li, Yingzhou; Lu, Jianfeng; Wang, Zhe: Coordinatewise descent methods for leading eigenvalue problem (2019)
  14. Miao, Cun-Qiang; Liu, Hao: Rayleigh quotient minimization method for symmetric eigenvalue problems (2019)
  15. Shen, Yedan; Kuang, Yang; Hu, Guanghui: An asymptotics-based adaptive finite element method for Kohn-Sham equation (2019)
  16. Sousedík, Bedřich: On adaptive BDDC for the flow in heterogeneous porous media. (2019)
  17. Wu, Lingfei; Xue, Fei; Stathopoulos, Andreas: TRPL+K: thick-restart preconditioned Lanczos+K method for large symmetric eigenvalue problems (2019)
  18. Xie, Hehu; Zhang, Lei; Owhadi, Houman: Fast eigenpairs computation with operator adapted wavelets and hierarchical subspace correction (2019)
  19. Yin, Jianyuan; Zhang, Lei; Zhang, Pingwen: High-index optimization-based shrinking dimer method for finding high-index saddle points (2019)
  20. Zhou, Ming; Neymeyr, Klaus: Cluster robust estimates for block gradient-type eigensolvers (2019)

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