# Gaussian elimination for flexible systems of linear inclusions

## TL;DR

This paper extends Gaussian elimination to flexible systems of linear inclusions involving external numbers with small errors, enabling solutions that account for indeterminacy and robustness in nonstandard analysis contexts.

## Contribution

It introduces a method for solving flexible systems using Gaussian elimination, accommodating external numbers and analyzing robustness and indeterminacy.

## Key findings

- Flexible systems can be transformed into row-echelon form with error terms.
- Solutions can be obtained considering indeterminacy in linear spaces and modules.
- Maximal robustness for flexible systems is characterized.

## Abstract

Flexible systems are linear systems of inclusions in which the elements of the coefficient matrix are external numbers in the sense of nonstandard analysis. External numbers represent real numbers with small, individual error terms. Using Gaussian elimination, a flexible system can be put into a row-echelon form with increasing error terms at the right-hand side. Then parameters are assigned to the error terms and the resulting system is solved by common methods of linear algebra. The solution set may have indeterminacy not only in terms of linear spaces, but also of modules. We determine maximal robustness for flexible systems.