Tridiagonal matrix algorithm

Abstract: We analyze the residuals of GMRES [Y Saad and M H Schultz, SIAM J Sci Statist Comput, 7 (1986), pp 856--859], when the method is applied to tridiagonal Toeplitz matrices We first derive formulas for the residuals as well as their norms when GMRES is applied to scaled Jordan blocks This problem has been studied previously by Ipsen [BIT, 40 (2000), pp 524--535] and Eiermann and Ernst [Private communication, 2002], but we formulate and prove our results in a different way We then extend the (lower) bidiagonal Jordan blocks to tridiagonal Toeplitz matrices and study extensions of our bidiagonal analysis to the tridiagonal case Intuitively, when a scaled Jordan block is extended to a tridiagonal Toeplitz matrix by a superdiagonal of small modulus (compared to the modulus of the subdiagonal), the GMRES residual norms for both matrices and the same initial residual should be close to each other We confirm and quantify this intuitive statement We also demonstrate principal difficulties of any GMRES convergence analysis which is based on eigenvector expansion of the initial residual when the eigenvector matrix is ill-conditioned Such analyses are complicated by a cancellation of possibly huge components due to close eigenvectors, which can prevent achieving well-justified conclusions

Abstract: If some of the smallest or some of the largest eigenvalues of a symmetric (tridiagonal) matrix are wanted, it suggests itself to use monotonic Newton corput rections in combination with Q R steps. If an initial shift has rendered the matrix positive or negative definite, then this property is preserved throughout the iteration. Thus, the Q R step may be achieved by two successive Cholesky L R steps or equivalently, since the matrix is tridiagonal, by two Q D steps which are numerically stable [4] and avoid square roots. The rational Q R step used here needs slightly fewer additions than the Ortega-Kaiser step [3].