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Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method

SIAM Journal on Scientific Computing, 2001
Summary: 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 ...
Andrew V Knyazev
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Preconditioning conjugate gradient method for nonsymmetric systems

International Journal of Computer Mathematics, 1995
It is well known that the preconditioned conjugate gradient algorithms (PCG) work very well (for both symmetric and nonsymmetric problems) if the preconditioned is “good enough”. But, in many cases, “good enough” means that for solving (during the application of (PCG)) the systems in which the preconditioning matrix appears too much computational work ...
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Conjugate gradient method with preconditioning by projector

International Journal of Computer Mathematics, 1988
Preconditioning of the conjugate gradient method by a conjugate projector has been suggested. We describe an algorithm and prove its correctness. An estimate of the preconditioning effect in terms of the gap between the invariant subspace of smooth eigenvectors of a matrix of original system and the complement of the range of the preconditioning ...
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Preconditioned conjugate gradient algorithms for nonconvex problems

2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601), 2004
The paper describes a new conjugate gradient algorithm for large scale nonconvex problems. In order to speed up the convergence the algorithm employs a scaling matrix which transforms the space of original variables into the space in which Hessian matrices of functionals describing the problems have more clustered eigenvalues.
R. Pytlak, T. Tarnawski
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Preconditioned conjugate gradient algorithms with column scaling

2008 47th IEEE Conference on Decision and Control, 2008
The paper describes new conjugate gradient algorithms which use preconditioning. The algorithms are intended for general nonlinear unconstrained problems. In order to speed up the convergence the algorithms employ scaling matrices which transform the space of original variables into the space in which Hessian matrices of functionals describing the ...
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SAOR Preconditioned Conjugate Gradient Method

Workshop on Intelligent Information Technology Application (IITA 2007), 2007
Jianguo Wang, Guoyan Meng
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Preconditioning of the Nonlinear Conjugate Gradient Algorithms

2020
Preconditioning is a technique to accelerate the conjugate gradient algorithms. In Chapter 2, the preconditioning of the linear conjugate gradient algorithm has been presented. For linear systems \( Ax = b, \) preconditioning modifies the system of equations in order to improve the eigenvalue distribution of A.
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GPU implementation of deflated preconditioned conjugate gradient

2010
Electrical Engineering, Mathematics and Computer ...
Gupta, R. (author), Vuik, C. (author), Lemmens, C.W.J. (author)   +2 more
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Preconditioned Conjugate Gradients for Givens Plane Rotations

1991
Assume, as in the previous four chapters, that A ∈ R mxn and b ∈ R mx1 are given and that m ≥ n as well as rank(A)=n are satisfied. Assume also that an approximation to x = A↑b = (ATA)−1ATb is to be calculated. In this chapter it will be shown that this problem can be transformed into an equivalent problem, which is a system of linear algebraic ...
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Preconditioned Conjugate Gradients on the PUMA Architecture

1992
This paper examines using the Preconditioned Conjugate Gradient Method (PCGM) to solve sparse systems of linear equations. A timing model for a parallel implementation of the PCGM is developed for the PUMA architecture. The PUMA architecture is a message passing multi-processor with fast point to point communication implemented in hardware.
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