Moslem Zamani and Milan Hladík. Error bounds and a condition number for the absolute value equations. Math. Program., 198(1):85–113, March 2023.
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Due to their relation to the linear complementarity problem, absolute value equations have been intensively studied recently. In this paper, we present error bound conditions for absolute value equations. Along with the error bounds, we introduce a condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discuss basic properties of the condition number, including its computational complexity. We present various bounds on the condition number, and we give exact formulae for special classes of matrices. Moreover, we consider matrices that appear based on the transformation from the linear complementarity problem. Finally, we apply the error bound to convergence analysis of two methods for solving absolute value equations.
@article{ZamHla2023a,
author = "Moslem Zamani and Milan Hlad\'{\i}k",
title = "Error bounds and a condition number for the absolute value equations",
journal = "Math. Program.",
fjournal = "Mathematical Programming",
volume = "198",
number = "1",
month = "March",
pages = "85-113",
year = "2023",
doi = "10.1007/s10107-021-01756-6",
issn = "1862-4480",
url = "https://link.springer.com/article/10.1007/s10107-021-01756-6",
bib2html_dl_html = "https://doi.org/10.1007/s10107-021-01756-6",
bib2html_dl_pdf = "https://rdcu.be/c6COj",
abstract = "Due to their relation to the linear complementarity problem, absolute value equations have been intensively studied recently. In this paper, we present error bound conditions for absolute value equations. Along with the error bounds, we introduce a condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discuss basic properties of the condition number, including its computational complexity. We present various bounds on the condition number, and we give exact formulae for special classes of matrices. Moreover, we consider matrices that appear based on the transformation from the linear complementarity problem. Finally, we apply the error bound to convergence analysis of two methods for solving absolute value equations.",
keywords = "Absolute value equation; Error bounds; Condition number; Linear complementarity problem; Interval matrix; Convergence rate",
}
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