Marzieh Dehghani-Madiseh and Milan Hladík. Efficient approaches for enclosing the united solution set of the interval generalized Sylvester matrix equations. Appl. Numer. Math., 126:18–33, 2018.
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We investigate the interval generalized Sylvester matrix equation [A]X[B]+[C]X[D]=[F]. We propose a necessary condition for its solutions, and also a sufficient condition for boundedness of the whole solution set. The main effort is performed to develop techniques for computing outer estimations of the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator, reducing significantly computational complexity, compared to the Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of [A], [B], [C] and [D] and in both of them we suppose that the midpoints of [A] and [C] are simultaneously diagonalizable as well as for the midpoints of the matrices [B] and [D]. Numerical experiments are given to illustrate the performance of the proposed methods.
@article{DehHla2018a,
author = "Dehghani-Madiseh, Marzieh and Milan Hlad\'{\i}k",
title = "Efficient approaches for enclosing the united solution set of the interval generalized {Sylvester} matrix equations",
journal = "Appl. Numer. Math.",
fjournal = "Applied Numerical Mathematics",
volume = "126",
pages = "18-33",
year = "2018",
doi = "10.1016/j.apnum.2017.12.003",
issn = "0168-9274",
url = "https://www.sciencedirect.com/science/article/pii/S0168927417302519",
bib2html_dl_html = "https://doi.org/10.1016/j.apnum.2017.12.003",
bib2html_dl_pdf = "https://www.sciencedirect.com/science/article/pii/S0168927417302519",
abstract = "We investigate the interval generalized Sylvester matrix equation [A]X[B]+[C]X[D]=[F]. We propose a necessary condition for its solutions, and also a sufficient condition for boundedness of the whole solution set. The main effort is performed to develop techniques for computing outer estimations of the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator, reducing significantly computational complexity, compared to the Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of [A], [B], [C] and [D] and in both of them we suppose that the midpoints of [A] and [C] are simultaneously diagonalizable as well as for the midpoints of the matrices [B] and [D]. Numerical experiments are given to illustrate the performance of the proposed methods.",
keywords = "Interval arithmetic; Generalized Sylvester matrix equation; Krawczyk operator; Preconditioning",
}
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