CONJUGATE GRADIENT METHODS FOR SOLVING THE SMALLEST EIGENPAIR OF LARGE SYMMETRIC EIGENVALUE PROBLEMS

Abstract

In this paper, a detailed description of CG for evaluating eigenvalue problems by minimizing the Rayleigh quotient is presented from both theoretical and computational viewpoints. Three variants of CG together with their asymptotic behaviours and restarted schemes are discussed. In addition, it is shown that with a generally selected preconditioning matrix the actual performance of the PCG scheme may not be superior to an accelerated inverse power method. Finally, some test problems in the finite element simulation of 2-D and 3-D large scale structural models with up to 20200 unknowns are performed to examine and demonstrate the performances.