Error bounds in the isometric Arnoldi process

Error bounds for the eigenvalues computed in the isometric Arnoldi method are derived. The Arnoldi method applied to a unitary matrix U successively computes a sequence of unitary upper Hessenberg matrices Hk, k = 1,2,… The eigenvalues of the Hkʹs are increasingly better approximations to eigenvalues of U. An upper bound for the distance of the spectrum of Hk from the spectrum of U, and an upper bound for the distance between each individual eigenvalue of Hk and one of U are given. Between two eigenvalues of Hk on the unit circle, there is guaranteed to lie an eigenvalue of U. The results are applied to a problem in signal processing.