Milan Hladík. Optimal preconditioning for the interval parametric Gauss-Seidel method. In M. Nehmeier et al., editor, Scientific Computing, Computer Arithmetic, and Validated Numerics: 16th International Symposium, SCAN 2014, Würzburg, Germany, September 21-26, LNCS, pp. 116–125, Springer, 2016.
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We deal with an interval parametric system of linear equations, and focus on the problem how to find an optimal preconditioning matrix for the interval parametric Gauss-Seidel method. The optimality criteria considered are to minimize the width of the resulting enclosure, to minimize its upper end-point or to maximize its lower end-point. We show that such optimal preconditioners can be computed by solving suitable linear programming problems. We also show by examples that, in some cases, such optimal preconditioners are able to significantly decrease an overestimation of the results of common methods.
@inCollection{Hla2016b,
author = "Milan Hlad\'{\i}k",
title = "Optimal preconditioning for the interval parametric {Gauss}--{Seidel} method",
webtitle = "Optimal preconditioning for the interval parametric {Gauss}-{Seidel} method",
editor = "Nehmeier et al., M.",
feditor = "Nehmeier, Marco and {Wolff von Gudenberg}, J{\"u}rgen and Tucker, Warwick",
booktitle = "Scientific Computing, Computer Arithmetic, and Validated Numerics: 16th International Symposium, {SCAN} 2014, W{\"u}rzburg, Germany, September 21-26",
publisher = "Springer",
volume = "9553",
series = "LNCS",
fseries = "Lecture Notes in Computer Science",
pages = "116-125",
year = "2016",
doi = "10.1007/978-3-319-31769-4_10",
issn = "0302-9743",
isbn = "978-3-319-31769-4",
bib2html_dl_html = "http://dx.doi.org/10.1007/978-3-319-31769-4_10",
abstract = "We deal with an interval parametric system of linear equations, and focus on the problem how to find an optimal preconditioning matrix for the interval parametric Gauss-Seidel method. The optimality criteria considered are to minimize the width of the resulting enclosure, to minimize its upper end-point or to maximize its lower end-point. We show that such optimal preconditioners can be computed by solving suitable linear programming problems. We also show by examples that, in some cases, such optimal preconditioners are able to significantly decrease an overestimation of the results of common methods.",
keywords = "Interval computation; Interval parametric system; Preconditioner; Linear programming",
}
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