A Divide-and-Conquer Method for the Tridiagonal Generalized Eigenvalue Problem

A divide-and-conquer method is developed for solving the generalized eigenvalue problem Ax = Bx, where A; B are real symmetric tridiagonal and B is positive definite. It is a generalization of Cuppens method for the standard eigenvalue problem, where B = I , being based on rank one modifications. This method is complementing a method developed by Borges-Gragg using bordering modifications. 1 Introduction In this note we discuss a numerical method for solving the tridiagonal eigenvalue problem Ax = Bx; (1) where A; B are real symmetric tridiagonal matrices and B is in addition positive definite. In [3] Cuppen introduced a divide-and-conquer method for the case B = I. It is based on rank one modifications. Dedicated to W.B. Gragg at the occasion of his 60. birthday Borges and Gragg, using modifications by bordering, developed later on in [1] a method for the general case (1). Independently Gu studied the bordering approach in [5] for the standard eigenvalue problem B = I. Somewh..