No abstract

A natural definition of the condition of a matrix is the ratio of its least upper and its greatest lower bound with respect to a pair of norms.Sometimes, the rows (or columns) of a matrix are only essential up to factors; in any case such a scaling is computationally trivial. Therefore, the problem of minimizing the condition by optimal scaling has several applications. We want to determine the infinum or at least to derive lower and upper bounds for it. We want to know the scaling leading to the minimum or at least coming close to it, and we want conditions showing that a matrix is optimally scaled.We show that the problem can be completely solved for the condition subordinate to a pair of maximum norms; and if each of the matrices A and A -1 has some bloekwise checkerboard sign distribution, more generally for a pair of H61der norms. Without this condition, lower and upper bounds, the latter together with appropriate scaling factors, are derived. For deriving the upper bounds, a mutually dual pair of vectors has to be determined; preparatory study of this problem has been made by STOER and WITZGALL [11~.

No abstract

No abstract

No abstract

No abstract

Using graphs for representing computational processes, relative error propagation is described. It is shown how this relates to the condition of a problem and to the property of a process to be benign, i.e., to have only harmless effects of rounding errors. In particular, composition of processes is studied under these aspects. Several examples illustrate the theory.

No abstract