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On preconditioning and solving an extended class of interval parametric linear systems

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Abstract

We deal with interval parametric systems of linear equations and the goal is to solve such systems, which basically comes down to finding an enclosure for a parametric solution set. Obviously, we want this enclosure to be tight and cheap to compute; unfortunately, these two objectives are conflicting. The review of the available literature shows that in order to make a system more tractable, most of the solution methods use left preconditioning of the system by the midpoint inverse. Surprisingly, and in contrast to standard interval linear systems, our investigations have shown that double preconditioning can be more efficient than a single one, both in terms of checking the regularity of the system matrix and enclosing the solution set, which is demonstrated by numerical examples. Consequently, right (which was hitherto mentioned in the context of checking regularity of interval parametric matrices) and double preconditioning together with the p-solution concept enable us to solve a larger class of interval parametric linear systems than most existing methods. The applicability of the proposed approach to solving interval parametric linear systems is illustrated by several numerical examples.

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Notes

  1. They usually have closed-form expressions.

  2. The results on the complexity of various problems related to interval matrices and interval linear systems can be found, e.g., in [7,8,9]; since an interval matrix (interval linear system) can be treated as a special case of an interval parametric matrix (interval parametric linear system) with each parameter occurring only once, all these results are valid for interval parametric matrices and interval parametric linear systems, too.

  3. To our best knowledge, both right and double preconditioning in this general form were not yet considered in the context of solving interval parametric linear systems.

  4. In case of any questions regarding the source code, please contact skalna@agh.edu.pl.

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Funding

M. Hladík was supported by the Czech Science Foundation Grant P403-18-04735S.

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Authors and Affiliations

  1. AGH University of Science and Technology, Kraków, Poland

    Iwona Skalna

  2. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Prague, Czech Republic

    Milan Hladík

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  1. Iwona Skalna
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  2. Milan Hladík
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Correspondence to Iwona Skalna.

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Skalna, I., Hladík, M. On preconditioning and solving an extended class of interval parametric linear systems. Numer Algor 87, 1535–1562 (2021). https://doi.org/10.1007/s11075-020-01018-0

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