which will be useful in our

backward error analysis. The steps described above are summarized in Algorithm 1.

However, the perturbation analysis and the

backward error analysis for these algorithms have not been studied, which are important to the analysis of the accuracy and stability for computing the largest eigenvalue by these algorithms.

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.

The

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

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.g., [30, 31]), is a widely used technique employed in the study of effects of rounding errors in numerical algorithms.

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 (1.1), where the system data A and f are perturbed.

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

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

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.

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.

BACKWARD ERROR ANALYSIS. In this section we assume that the elements of the monic Jacobi matrix J(B, G) are real floating point numbers.