A block Chebyshev-Davidson method for linear response eigenvalue problems

We present a Chebyshev-Davidson method to compute a few smallest positive eigenvalues and corresponding eigenvectors of the linear response eigenvalue problem. The method is actually applicable to the slightly more general linear response eigenvalue problem where purely imaginary eigenvalues may occur. For the Chebyshev filter, a tight upper bound is obtained by a computable bound estimator constructed under a reasonable condition. When the condition fails, we give an adaptive strategy for updating the upper bounds to guarantee the effectiveness of the ChebyshevDavidson method. We also obtain an estimate of the rate of convergence for the Rize values computed in our algorithm. Finally, we present numerical results to demonstrate the performance of the proposed Chebyshev-Davidson method. 2000 Mathematics Subject Classification. 65F15, 15A18. key words and phrases. eigenvalue, eigenvector, Chebyshev polynomials, Davidson type method, convergence rate, upper bound estimator.