Andrew Knyazev, Software & Simulation
Preconditioned Conjugate Gradient Methods for Eigenproblems

This project is in progress. Please visit my Preconditioned Eigensolvers Web page for more info.

The LOBPCG (Locally Optimal Block Preconditioned Conjugate Gradient) code is now publicly available as as a part of the BLOPEX package. MATLAB sources are also available directly from MathWorks.

Main features:
a matrix-free iterative method for computing several extreme eigenpairs of symmetric positive generalized eigenproblems;
a user-defined symmetric positive preconditioner (a good preconditioner for a stiffness matrix works well for the corresponding eigenvalue problem, too!);
robustness with respect to random initial approximations, variable preconditioners, and ill-conditioning (up to 10^16) of the stiffness matrix;
apparently optimal convergence speed.

Numerical comparisons suggest that LOBPCG is a genuine analog for eigenproblems of the standard preconditioned conjugate gradient method for symmetric linear systems.

The LOBPCG algorithm is presented in the paper Andrew Knyazev, Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method. SIAM Journal on Scientific Computing 23 (2001), no. 2, pp. 517-541.

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