New Methods for Calculations of the Lowest Eigenvalues of the Real Symmetric Generalized Eigenvalue Problem

A new iterative method based on a Newton correction vector for extension of the Krylov subspace, its diagonal, and band versions are proposed for calculation of selected lowest eigenvalues and corresponding eigenvectors of the generalized symmetric eigenvalue problem. Additionally, diagonal and band Jacobi–Davidson methods are introduced. Test calculations show that the new iterative method usually converges faster than quadratic near a solution. The new iterative method along with its band version uses a smaller number of iterative steps to obtain a solution compared to the Jacobi–Davidson, band Jacobi–Davidson method, and generalized Davidson method correspondingly. The diagonal version of the new method preserves an advantage over the diagonal Jacobi–Davidson and the Davidson method.