Rayleigh-Quotient-Minimierung mit Vorkonditionierung

The smallest eigenvalues and corresponding eigenvectors of Ax = λBx, where A and B are symmetric, positive definite and sparse matrices of high order, can be computed by minimizing the Rayleigh quotient by means of the method of conjugate gradients. An appropriate preconditioning of the eigenvalue problem results in an essential improvement of the convergence. It is interesting that an implicit realization of the preconditioning does not increase the required computational effort per iteration step. The algorithm is highly vectorizable and is indeed a very efficient method with respect to computational work and to storage requirements.