Milan Hladík and Jiří Rohn. Radii of solvability and unsolvability of linear systems. Linear Algebra Appl., 503:120–134, 2016.
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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.
@article{HlaRoh2016a,
author = "Hlad\'{\i}k, Milan and Rohn, Ji\v{r}\'{\i}",
title = "Radii of solvability and unsolvability of linear systems",
journal = "Linear Algebra Appl.",
fjournal = "Linear Algebra and its Applications",
volume = "503",
pages = "120-134",
year = "2016",
doi = "10.1016/j.laa.2016.03.028",
issn = "0024-3795",
bib2html_dl_html = "http://dx.doi.org/10.1016/j.laa.2016.03.028",
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",
}
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