Radii of solvability and unsolvability of linear systems

Abstract

We consider a problem of determining the component-wise distance (called the radius) of a linear system of equations or inequalities to a system that is either solvable or unsolvable. We propose explicit characterization of these radii and show relations between them.

Then the radii are classified in the polynomial vs. NP-hard manner.

We also present a generalization to an arbitrary linear system consisting from both equations and inequalities with both free and nonnegative variables. Eventually, we extend the concept of the component-wise distance to a non-uniform one.

Keywords

• Interval matrix
• Linear equations
• Linear inequalities
• Matrix norm