A divide-and-conquer method for the tridiagonal generalized eigenvalue problem
This paper presents a new method to solve the generalized eigenvalue problem \[ Ax= \lambda Bx,\quad \tag{1} \] where \(A\) and \(B\) are real symmetric tridiagonal matrices, with \(B\) positive definite. In the first phase, by means of a convenient rank one modification, the problem is reduced to solving two similar problems whose dimensions can be arbitrarily chosen so that their sum is the dimension of problem (1). With the solutions of these smaller problems, (1) is transformed into a different generalized eigenvalue problem. The core of the paper is devoted to show that this modified problem can be nicely solved. As pointed out by the authors, the method thus proposed is an extension of the one analysed by \textit{J. J. Cuppen} [Numer. Math. 36, 177-195 (1981; Zbl 0431.65022)] to solve (1) when \(B=I\). Also some aspects of the method are discussed in relation to the one proposed by \textit{C. F. Borges} and \textit{W. B. Gragg} [Proc. Conf. Numer. Linear Algebra Sci. Comput. 1992, 11-29 (1993; Zbl 0799.65043)] for the same problem, which show the interest of its further examination.
scientific article; zbMATH DE number 1115929