## lobpcg.m

 swMATH ID: 24008 Software Authors: Knyazev, Andrew V. Description: 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. Homepage: https://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m Related Software: JDQZ; JDQR; ARPACK; BLOPEX; PRIMME; IRAM; FEAST; EIGIFP; KSSOLV; SparseMatrix; eigs; JDCG; LAPACK; SLEPc; Quantum Espresso; TRLan; Matlab; PETSc; PARSEC; ABINIT Referenced in: 147 Publications
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### Referenced by 256 Authors

 12 Lin, Lin 6 Wen, Zaiwen 6 Yang, Chao 5 Miao, Cun-Qiang 5 Nakatsukasa, Yuji 5 Sousedík, Bedřich 4 Hu, Guanghui 4 Knyazev, Andrew V. 4 Li, Rencang 4 Lu, Jianfeng 4 Vecharynski, Eugene 4 Zhang, Yin 4 Zhou, Yunkai 3 Bergamaschi, Luca 3 Huang, Tsungming 3 Klawonn, Axel 3 Kuang, Yang 3 Liu, Xin 3 Mandel, Jan 3 Neymeyr, Klaus 3 Ovtchinnikov, Evgueni Eduardovich 3 Pini, Giorgio 3 Rheinbach, Oliver 3 Saad, Yousef 3 Shao, Meiyue 3 Stathopoulos, Andreas 3 Ying, Lexing 3 Zhang, Lei-Hong 2 Aishima, Kensuke 2 Dolgov, Sergey V. 2 E, Weinan 2 Hwang, Fengnan 2 Imakura, Akira 2 Khoromskiĭ, Boris Nikolaevich 2 Kressner, Daniel 2 Kühn, Martin Joachim 2 Lehoucq, Richard B. 2 Li, Yingzhou 2 Lu, Zhongjie 2 Martínez, Ángeles 2 Oseledets, Ivan V. 2 Pavarino, Luca Franco 2 Pollock, Sara 2 Román, Jose E. 2 Romero, Eloy 2 Shen, Chungen 2 Šístek, Jakub 2 Spence, Alastair 2 Wang, Lin-Wang 2 Wang, Weichung 2 Wei, Zih-Hao 2 Xie, Hehu 2 Xu, Ze 2 Xue, Fei 2 Ye, Qiang 2 Yin, Jianyuan 2 Zhang, Lei 2 Zhang, Pingwen 2 Zhang, Shao-Liang 1 Abdel-Rehim, Abdou M. 1 Abdelmoula, Radhi 1 Ainsworth, Mark 1 Altmann, Robert 1 An, Dong 1 Antoine, Xavier 1 Arbenz, Peter 1 Bai, Zhaojun 1 Baker, C. G. 1 Banerjee, Amartya S. 1 Bao, Gang 1 Benner, Peter 1 Bernholc, J. 1 Berns-Müller, Jörg 1 Binante, Vincenzo 1 Boffi, Daniele 1 Bollhöfer, Matthias 1 Bozzo, Enrico 1 Bramble, James H. 1 Briggs, E. L. 1 Cai, Jian-Feng 1 Canning, Andrew 1 Carson, Erin Claire 1 Chelikowsky, James R. 1 Cho, Heyrim 1 Chung, Tsz Shun Eric 1 Cichocki, Andrzej 1 da Jornada, Felipe H. 1 Dargush, Gary F. 1 Dax, Achiya 1 Deparis, Simone 1 Deslippe, Jack 1 Dolean, Victorita 1 Dongarra, Jack J. 1 Duersch, Jed A. 1 Dunning, Peter D. 1 Dziekonski, A. 1 Eisenmann, Henrik 1 Elliott, Ryan S. 1 Elman, Howard C. 1 Erlandson, Lucas ...and 156 more Authors
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### Referenced in 52 Serials

 24 SIAM Journal on Scientific Computing 21 Journal of Computational Physics 10 Linear Algebra and its Applications 6 Journal of Computational and Applied Mathematics 6 SIAM Journal on Matrix Analysis and Applications 6 Journal of Scientific Computing 5 Mathematics of Computation 4 Computer Methods in Applied Mechanics and Engineering 4 Numerische Mathematik 4 Numerical Algorithms 4 Multiscale Modeling & Simulation 4 Science China. Mathematics 3 Japan Journal of Industrial and Applied Mathematics 3 Communications in Computational Physics 2 Computers & Mathematics with Applications 2 Computer Physics Communications 2 Applied Mathematics and Computation 2 BIT 2 Numerical Linear Algebra with Applications 1 Computers and Fluids 1 Wave Motion 1 ACM Transactions on Mathematical Software 1 Calcolo 1 Computing 1 International Journal for Numerical Methods in Engineering 1 Journal of Optimization Theory and Applications 1 Mathematics and Computers in Simulation 1 SIAM Journal on Numerical Analysis 1 Journal of Computational Mathematics 1 Physica D 1 Applied Numerical Mathematics 1 Applications of Mathematics 1 Journal of Elasticity 1 Pattern Recognition 1 SIAM Journal on Optimization 1 Communications in Numerical Methods in Engineering 1 Computational and Applied Mathematics 1 ETNA. Electronic Transactions on Numerical Analysis 1 Advances in Computational Mathematics 1 Mathematical Problems in Engineering 1 Mathematical & Computational Applications 1 Computational Geosciences 1 Computational Methods in Applied Mathematics 1 Journal of Applied Mathematics 1 Acta Numerica 1 International Journal of Numerical Analysis and Modeling 1 Numerical Algebra, Control and Optimization 1 S$$\vec{\text{e}}$$MA Journal 1 East Asian Journal on Applied Mathematics 1 Nanoscale Systems: Mathematical Modeling, Theory and Applications 1 Research in the Mathematical Sciences 1 Results in Applied Mathematics
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### Referenced in 22 Fields

 135 Numerical analysis (65-XX) 28 Linear and multilinear algebra; matrix theory (15-XX) 23 Partial differential equations (35-XX) 15 Statistical mechanics, structure of matter (82-XX) 12 Mechanics of deformable solids (74-XX) 11 Operations research, mathematical programming (90-XX) 9 Quantum theory (81-XX) 5 Calculus of variations and optimal control; optimization (49-XX) 4 Ordinary differential equations (34-XX) 4 Fluid mechanics (76-XX) 4 Optics, electromagnetic theory (78-XX) 3 Operator theory (47-XX) 2 Combinatorics (05-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Statistics (62-XX) 2 Computer science (68-XX) 2 Mechanics of particles and systems (70-XX) 1 Special functions (33-XX) 1 Differential geometry (53-XX) 1 Probability theory and stochastic processes (60-XX) 1 Biology and other natural sciences (92-XX) 1 Information and communication theory, circuits (94-XX)