The inherent inaccuracy of implicit tridiagonal QR

Abstract

Recently Demmel and Veselic showed that Jacobi's method has a tighter relative error bound for the computed eigenvalues of a symmetric positive de nite matrix than does QR iteration. Here we show the weaker error bound of QR as implemented in LAPACK's SSTEQR or EISPACK's IMTQL is unavoidable. We do this by presenting a particular symmetric positive de nite tridiagonal matrix for whichQRmust fail, given any reasonable shift strategy