In Section 3 and in Section 4 we present the structured backward error analysis of an approximate eigenpair for complex symmetric, complex skew-symmetric, T-even, and T-odd matrix polynomials and compare these results with the corresponding unstructured backward errors.

In particular, to present the backward error analysis for a given approximation to an eigenvalue/eigenvector pair of a matrix polynomial L, we will construct an appropriately structured minimal (in the Frobenius and the spectral norm) perturbation polynomial [increment of L] such that the given eigenvalue/eigenvector pair is exact for L + [increment of L].

Therefore, we will frequently use the matrix description of the Darboux process in the statements of the theorems appearing in the backward error analysis and in the perturbation theory of Darboux transformation as if the input parameters were B and G.

Therefore, to explain these errors it is necessary to find the componentwise condition number with respect to the kind of perturbations suggested by the backward error analysis.

The

backward error analysis of these operations will be a problem when [beta] is much smaller than [absolute value of f].

Organized into sections starting with the Schrodinger equation, moving onto numerical schemes and examples, finite dimensional

backward error analysis, infinite dimensional and semi-discrete Hamiltonian flow, convergence results, modified energy in the linear and semi-linear case and culminates with an introduction to long-time analysis.

Backward error analysis in numerical linear algebra, pioneered by von Neumann and Goldstein [28], Turing [26], Givens [10] and further developed and popularised by Wilkinson (see, e.

Backward error analysis provides an elegant way how to study numerical stability of algorithms, that is, their sensitivity with respect to rounding errors.

In the backward error analysis, the vector U is interpreted as the solution of a problem (1.

A backward error analysis provides a simple and reliable way to recognize and deal with this issue.

The idea of backward error analysis is to view the numerical approximation as the exact solution of a problem close to the given problem.

We have no need of a backward error analysis in the regular case, but it is very advantageous in the irregular case.