Zdeněk Strakoš
Univerzita Karlova v Praze, Matematicko-fyzikální fakulta, Katedra numerické matematiky

On numerical stability in large scale linear algebraic computations
(Plenary lecture at the GAMM Annual Meeting, Dresden, March 2004)
Numerical solving of real-world problems typically consists of several stages. After describing the problem in a mathematical language, its proper reformulation and discretisation, the resulting linear algebraic problem has to be solved. We focus on this last stage, and specifically consider numerical stability of iterative methods in matrix computations.

In iterative methods, rounding errors have two main effects: They can delay convergence and they can limit the attainable accuracy. It is, however, important to realize that numerical stability analysis is not about derivation of error bounds or estimates. Rather the goal is to find algorithms and their parts that are safe (``numerically stable''), and to identify algorithms and their parts that are not. This classical idea guides our work and also our presentation.

We first recall the concept of backward stability and discuss its use in the context of iterative methods. Then we analyse the question to which extent the accuracy of a computed result relies on the accuracy of the intermediate quantities. Finally, we present some examples of rounding error analysis that are fundamental to justify numerically computed results. We illustrate our points on the Lanczos method, the conjugate gradient method and the generalised minimal residual method (GMRES).