TheAB-algorithm and its properties

Abstract

One presents a new algorithm, called the

-algorithm, for solving the generalized eigenvalue problem Ax=λBx, where det (A—λB) ≠ 0, relative to A. The algorithm is iterative, it is based on the application of plane rotations and allows us to pass from the initial problem to the solving of a similar problem having simpler matrices, whose eigenvalues can be easily computed and coincide with the eigenvalues of the initial problem. Thus, if all the eigenvalues of the initial problem are distinct, then the application of the

-algorithm leads to the computation of the eigenvalues of a pencil with triangular matrices. In the case of an arbitrary initial pencil A—λB, the problem reduces to solving the eigenvalue problem for a pencil of quasitriangular form. One proves the convergence of the algorithm. One establishes its properties which in many respects are similar with the properties of the known algorithms QR and QZ, the first of which solves the usual eigenvalue problem while the second one solves the generalized problem of the above-mentioned form.