FOM-inverse vector iteration method for computing a few smallest (largest) eigenvalues of pair (A,B) (Q2371475)

This article presents a new inner-outer iteration like method for computing a few smallest (largest) eigenvalues of the symmetric positive definite problem: \[ Ax=\lambda Bx. \] The first part is a brief introduction concerning the above problem and the appropriate numerical methods for solving it. The second and the third part are devoted to an explanation and an enumeration of the definitions and theorems used in the paper. The fourth part presents the algorithm of the inverse vector iteration method and some computational remarks. The fifth part details the full orthogonalization method (FOM) for solving the linear system \(Ax=b\) [cf. \textit{Y. Saad}, Math. Comput. 37, 105--126 (1981; Zbl 0474.65019)]. The sixth part focuses on the FOM-inverse vector iteration method. One obtains thus the main algorithm of the paper. Numerical tests performed for the case where \(A\) and \(B\) are \(1000\times 1000\) real matrices, by the implementation of the algorithm, are presented in the seventh and the eighth part and show that the algorithm converges fast and works with high accuracy. The eighth part presents the deflation technique which is used by the main procedure, in order to obtain the next smallest eigenvalue, once the smallest eigenvalue is found. The last section contains some comments and conclusions.