A new iterative scheme for obtaining eigenvectors of large, real-symmetric matrices

A rapidly convergent block-iterative scheme for the computation of a few of the lowest eigenvalues and eigenvectors of a large dense real symmetric matrix is proposed. The method is especially applicable to matrix eigenvalue problems that arise from the discretization of self-adjoint partial differential equations. One such application to certain symmetric matrices that arise in solid-state band structure calculations is considered in detail. The most timeconsuming parts of the present algorithm are a matrix multiplication and a Gauss-Siedel relaxation step which are performed on each iteration. These two parts can, however, be very efficiently implemented on a vector or parallel processing computer.