Abstract. In this paper we propose a new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is based on an old and almost unknown method of Jacobi. Jacobi's approach, combined with Davidson's method, leads to a new method that has improved convergence properties and that may be used for general matrices. We also propose a variant of the new method that may be useful for the computation of nonextremal eigenvalues as well. Jacobi [12] proposed a method for eigenvalue approximation that essentially was a combination of (1) Jacobi rotations, (2) Gauss-Jacobi iterations, and (3) an almost forgotten method that we will refer to as Jacobi's orthogonal component correction (JOCC). Reinvestigation of Jacobi's ideas leads to another view of Davidson's method, and this not only helps us explain the behavior of the method, but it also leads to a new and robust method with superior convergence properties for nondiagonally dominant (unsymmetric) matrices as well. Special variants of this method are already known; see [19] , [28] , and our discussion in section 4.1.
The paper by REF introduces an iterative method for computing extremal eigenvalues of symmetric matrices.
A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems
SIAM J. Matrix Anal. Appl
Mathematics Computer Science