Iwona Skalna and Milan Hladík. Direct and iterative methods for interval parametric algebraic systems producing parametric solutions. Numer. Linear Algebra Appl., 26(3):e2229:1–e2229:24, 2019.
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This paper deals with interval parametric linear systems with general dependencies. Motivated by the so-called parameterized solution introduced by Kolev, we consider the enclosures of the solution set in a revised affine form. This form is advantageous to a classical interval solution because it enables us to obtain both outer and inner bounds for the parametric solution set and, thus, intervals containing the endpoints of the hull solution, among others. We propose two solution methods, a direct method called the generalized expansion method and an iterative method based on interval-affine Krawczyk iterations. For the iterative method, we discuss its convergence and show the respective sufficient criterion. For both methods, we perform theoretical and numerical comparisons with some other approaches. The numerical experiments, including also interval parametric linear systems arising in practical problems of structural and electrical engineering, indicate the great usefulness of the proposed methodology and its superiority over most of the existing approaches to solving interval parametric linear systems.
We mentioned in the introduction that the shape of the solution set is described by quadrics, which is true when embedding it to a higher dimensional space by using additional variables. In the original space, the solution set is characterized by polynomials then.
@article{SkaHla2019a,
author = "Iwona Skalna and Milan Hlad\'{\i}k",
title = "Direct and iterative methods for interval parametric algebraic systems producing parametric solutions",
journal = "Numer. Linear Algebra Appl.",
fjournal = "Numerical Linear Algebra with Applications",
volume = "26",
number = "3",
pages = "e2229:1-e2229:24",
year = "2019",
doi = "10.1002/nla.2229",
issn = "1099-1506",
url = "https://onlinelibrary.wiley.com/doi/abs/10.1002/nla.2229",
bib2html_dl_html = "http://dx.doi.org/10.1002/nla.2229",
bib2html_dl_pdf = "https://onlinelibrary.wiley.com/doi/epdf/10.1002/nla.2229",
bib2html_errata = "We mentioned in the introduction that the shape of the solution set is described by quadrics, which is true when embedding it to a higher dimensional space by using additional variables. In the original space, the solution set is characterized by polynomials then.",
abstract = "This paper deals with interval parametric linear systems with general dependencies. Motivated by the so-called parameterized solution introduced by Kolev, we consider the enclosures of the solution set in a revised affine form. This form is advantageous to a classical interval solution because it enables us to obtain both outer and inner bounds for the parametric solution set and, thus, intervals containing the endpoints of the hull solution, among others. We propose two solution methods, a direct method called the generalized expansion method and an iterative method based on interval-affine Krawczyk iterations. For the iterative method, we discuss its convergence and show the respective sufficient criterion. For both methods, we perform theoretical and numerical comparisons with some other approaches. The numerical experiments, including also interval parametric linear systems arising in practical problems of structural and electrical engineering, indicate the great usefulness of the proposed methodology and its superiority over most of the existing approaches to solving interval parametric linear systems.",
keywords = "Affine arithmetic; Interval computation; Linear equations; Parametric system",
}
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