Certain modifications of the $AB$-algorithm

One considers various modifications of the $AB$-algorithm for the solution of the complete (partial) eigenvalue problem of a regular pencil $A-\lambda B$ of square matrices. A modification of the $AB$-algorithm is suggested which allows to eliminate in a finite number of steps the zero and the infinite eigenvalues of the pencil $A-\lambda B$ and to lower its dimensions. For regular pencils with real eigenvalues a modification ot the $AB$-algorithm with a shift is presented. For a well-defined choice of the shifts one proves the quadratic convergence of the algorithm, successively to each eigenvalue of the pencil, starting with the smallest one. For a pencil whose eigenvalues can be divided into the groups of “large” and “small” eigenvalues, one considers a modification of the $AB$-algorithm, allowing to obtain approximations to the indicated groups of eigenvalues as solutions of a problem for pencils of lower dimensions.