On solving the definite tridiagonal symmetric generalized eigenvalue problem

In this manuscript we will present a new fast technique for solving the generalized eigenvalue problem T x = λSx, in which both matrices T and S are symmetric tridiagonal matrices and the matrix S is assumed to be positive definite.1 A method for computing the eigenvalues is translating it to a standard eigenvalue problem of the following form: L−1T L−T (LT x) = λ(LT x), where S = LLT is the Cholesky decomposition of the matrix S. We will prove in this manuscript, that the matrix L−1T L−T , with S = LLT is of structured rank form. More precisely the matrix is of quasiseparable form, meaning that all submatrices taken out of the strictly lower triangular part of the matrix, have rank at most 1. These matrices admit an order O(n) representation, instead of O(n2) in case the matrix was dense. It will be shown that the computation of the generators of this quasiseparable matrix is only linear in time. Exploiting the properties of structured rank matrices one can use different techniques for computing all of the eigenvalues. Either one can reduce the quasiseparable matrix L−1T L−T to tridiagonal form in O(n2) operations, and hence compute the eigenvalues of the tridiagonal matrix. One can also immediately compute the eigenvalues of this matrix, via the QR-algorithm for quasiseparable matrices. Both approaches lead to O(n2) methods for computing all the eigenvalues of the initial generalized eigenvalue problem, instead of the O(n3) methods in case the structure was left unexploited. Moreover for quasiseparable matrices also bisection and divide and conquer methods exist.