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- LOBPCG abstract "Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG) is a matrix-free method, proposed in , for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric positive definite generalized eigenvalue problem.for a given pair of complex Hermitian or real symmetric matrices, wherethe matrix is also assumed positive-definite.The method performs an iterative maximization (or minimization) of the generalized Rayleigh quotientwhich results in finding largest (or smallest) eigenpairs of The direction of the steepest ascent, which is the gradient, of the generalized Rayleigh quotient is positively proportional to the vectorcalled the eigenvector residual. If a preconditioner is available, it is applied to the residual giving vectorcalled the preconditioned residual. Without preconditioning, we set and so . An iterative methodor, in short,is known as preconditioned steepest ascent (or descent), where the scalar is called the step size. The optimal step size can be determined by maximizing the Rayleigh quotient, i.e.,(or in case of minimizing), in which case the method is called locally optimal. To further accelerate the convergence of the locally optimal preconditioned steepest ascent (or descent), one can add one extra vector to the two-term recurrence relation to make it three-term:(use in case of minimizing). The maximization/minimization of the Rayleigh quotient in a 3-dimensional subspace can be performed numerically by the Rayleigh–Ritz method.This is a single-vector version of the LOBPCG method. It is one of possible generalization of the preconditioned conjugate gradient linear solvers to the case of symmetric eigenvalue problems. Even in the trivial case and the resulting approximation with will be different from that obtained by the Lanczos algorithm, although both approximations will belong to the same Krylov subspace.Iterating several approximate eigenvectors together in a block in a similar locally optimal fashion, gives the full block version of the LOBPCG. It allows robust computation of eigenvectors corresponding to nearly-multiple eigenvalues.An implementation of LOBPCG is available in the public software package BLOPEX, maintained by Andrew Knyazev. The LOBPCG algorithm is also implemented in many other libraries, e.g.,: ABINIT, Octopus (software), PESCAN, Anasazi (Trilinos), SciPy, NGSolve, and PYFEMax.LOBPCG has been successfully used for multi-billion size matrices in the papers and both Gordon Bell Prize finalists, on the Earth Simulator supercomputer in Japan.".
- LOBPCG wikiPageExternalLink scipy.sparse.linalg.lobpcg.html.
- LOBPCG wikiPageExternalLink lobpcg.html.
- LOBPCG wikiPageExternalLink 48-lobpcg-m.
- LOBPCG wikiPageExternalLink blopex.
- LOBPCG wikiPageExternalLink sparse-eigensolvers-java.
- LOBPCG wikiPageID "26951185".
- LOBPCG wikiPageRevisionID "599628393".
- LOBPCG hasPhotoCollection LOBPCG.
- LOBPCG subject Category:Numerical_linear_algebra.
- LOBPCG subject Category:Scientific_simulation_software.
- LOBPCG comment "Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG) is a matrix-free method, proposed in , for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric positive definite generalized eigenvalue problem.for a given pair of complex Hermitian or real symmetric matrices, wherethe matrix is also assumed positive-definite.The method performs an iterative maximization (or minimization) of the generalized Rayleigh quotientwhich results in finding largest (or smallest) eigenpairs of The direction of the steepest ascent, which is the gradient, of the generalized Rayleigh quotient is positively proportional to the vectorcalled the eigenvector residual. ".
- LOBPCG label "LOBPCG".
- LOBPCG sameAs m.01187jg9.
- LOBPCG sameAs Q6459462.
- LOBPCG sameAs Q6459462.
- LOBPCG wasDerivedFrom LOBPCG?oldid=599628393.
- LOBPCG isPrimaryTopicOf LOBPCG.