Special issue on

Linear Algebra in Self-Validating Methods

These methods have a connection to linear algebra since problems are frequently transformed into linearized problems with uncertain data. Then the linearization and discretization errors are estimated, possibly together with an infinite dimensional part of the problem.

It has turned out that computation of an inclusion of the solution complex of even a linear system of equations with uncertain data is NP-hard. This has given rise to interesting connections between self-validating methods and complexity theory. Despite this, in many cases a reasonably sharp inclusion can be calculated. The class of problems being solvable in this sense has been extended in recent years.

The possibility to estimate the range of a function is a main ingredient of self-validating methods. Beside the naive way to get error bounds by replacing every operation by the corresponding interval operation, much more elaborate methods have come up using gradients, slopes, lp- and qp-approaches and more.

In the past few decades the area of self-validating methods has been evolving, with rapidly growing number of researchers. We want to take this opportunity to publish a special issue on self-validating methods. A preliminary list of topics would include:

- systems of linear equations and inequalities
- range of functions
- complexity theory for problems with uncertain data
- componentwise distance to singularity and/or stability
- sparse systems of equations
- algebraic eigenvalue problems
- iterative methods
- matrix methods in validation methods for differential equations
- use of M-matrices and H-matrices in validation methods
- analysis of zeros and connection to controllability
- combination of computer algebra with floating point methods.

This is a sample, but not an exclusive list of topics. If there is doubt about suitability of a particular paper, contact one of the editors of the special issue.

Please submit three (3) hard copies to one of the special issue editors
listed below. The deadline for submission is *September 30, 1999.*

Jiri Rohn

Faculty of Mathematics and Physics

Charles University

Malostranske nam. 25

118 00 Prague

Czech Republic

e-mail rohn@uivt.cas.cz

Siegfried M. Rump

Inst. f. Computer Science III

Technical University Hamburg-Harburg

Eissendorfer Str. 38

21071 Hamburg, Germany

e-mail rump@tu-harburg.de

Tetsuro Yamamoto

Department of Mathematics

Faculty of Science

Ehime University

Matsuyama 790, Japan

e-mail yamamoto@dpc.ehime-u.ac.jp