Numerical Behavior of the DQR Method for Rank-Structured Matrices

In this paper we consider fast numerical algorithms for solving certain modified matrix eigenvalue problems associated with algebraic equations. The matrices under consideration have the form A = T + uv , where u, v ∈ Rn×n and T = (ti,j) ∈ Rn×n is a tridiagonal matrix such that tj+1,j = ±tj,j+1, 1 ≤ j ≤ n−1. We show that the DQR approach proposed in [Uhlig F., Numer. Math. 76 (1997), no. 4, 515–553] can compute the eigenvalues of such matrices efficiently. Our algorithm employs a fast Hessenberg reduction scheme for tridiagonal plus rank-one matrices combined with an efficient eigenvalue method for the resulting Hessenberg matrices. Exploiting the rank structure of the input matrix enables both steps to be carried out using quadratic time and linear storage. Numerical implementation of these techniques is presented and discussed for a number of test problems. AMS classification: 65F15