On condition numbers of polynomial eigenvalue problems with nonsingular leading coefficients

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented.Comment: 4 figure

Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm

An efficient way of solving 2D stability problems in fluid mechanics is to use, after discretization of the equations that cast the problem in the form of a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This method preserves the banded structure sparsity of matrices of the algebraic eigenvalue problem and thus decreases memory use and CPU-time consumption. The errors that affect computed eigenvalues and eigenvectors are due to the truncation in the discretization and to finite precision in the computation of the discretized problem. In this paper we analyze those two errors and the interplay between them. We use as a test case the two-dimensional eigenvalue problem yielded by the computation of inertial modes in a spherical shell. This problem contains many difficulties that make it a very good test case. It turns out that that single modes (especially most-damped modes i.e. with high spatial frequency) can be very sensitive to round-off errors, even when apparently good spectral convergence is achieved. The influence of round-off errors is analyzed using the spectral portrait technique and by comparison of double precision and extended precision computations. Through the analysis we give practical recipes to control the truncation and round-off errors on eigenvalues and eigenvectors.Comment: 15 pages, 9 figure

Wave Packet Pseudomodes of Variable Coefficient Differential Operators

The pseudospectra of nonselfadjoint linear ordinary differential operators with variable coefficients are considered. It is shown that when a certain winding number or twist condition is satisfied, closely related to Hörmander's commutator condition for partial differential equations, \varepsilon-pseudoeigenfunctions of such operators for exponentially small values of \varepsilon exist in the form of localized wave packets. In contrast to related results of Davies and of Dencker, Sjöstrand, and Zworski, the symbol need not be smooth

Stable high-order finite-difference methods based on non-uniform grid point distributions

It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundstr¨om theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resulting finite-difference operator. Several heuristic explanations exist for the goodness of the clustering, and attempts have been made to link it to the Runge phenomenon present in polynomial interpolations of high degree. By following the philosophy behind the Chebyshev polynomials, a non-uniform grid for piecewise polynomial interpolations of degree q_N is introduced in this paper, where N + 1 is the total number of grid nodes. It is shown that when q = N, this polynomial interpolation coincides with the Chebyshev interpolation, and the resulting finite-difference schemes are equivalent to Chebyshev collocation methods. Finally, test cases are run showing how stability and correct transient behaviours are achieved for any degree q<N through the use of the proposed non-uniform grids. Discussions are complemented by spectra and pseudospectra of the finite-difference operators