Proving endpoint dependence in solving interval parametric linear systems

Abstract

Linear algebraic systems involving linear dependencies between interval valued parameters and the so-called united parametric solution set of such systems are considered. The focus is on systems, such that the vertices of their interval hull solution are attained at particular endpoints of some or all parameter intervals. An essential part of finding this endpoint dependence is an initial determination of the parameters which influence the components of the solution set, and of the corresponding kind of monotonicity. In this work, we review a variety of interval approaches for the initial monotonicity proof and compare them with respect to both computational complexity and monotonicity proving efficiency. Some quantitative measures are proposed for the latter. We present a novel methodology for the initial monotonicity proof, which is highly efficient from a computational point of view, and which is also very efficient to prove the monotonicity, for a wide class of interval linear systems involving parameters with rank 1 dependency structure. The newly proposed method is illustrated on some numerical examples and compared with other approaches.

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Notes

  1. 1.

    An interval a = [a,a+] is degenerate if a = a+.

  2. 2.

    True rank 1 parameters are defined in [13].

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Acknowledgments

The author thanks the anonymous reviewers for their thorough reviews and the numerous remarks which helped improving the readability of the paper.

Funding

This work is supported by the Grant No BG05M2OP001-1.001-0003, financed by the Bulgarian Operational Programme “Science and Education for Smart Growth” (2014-2020) and co-financed by the European Union through the European structural and investment funds.

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Correspondence to Evgenija D. Popova.

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Popova, E.D. Proving endpoint dependence in solving interval parametric linear systems. Numer Algor 86, 1339–1358 (2021). https://doi.org/10.1007/s11075-020-00936-3

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Keywords

  • Interval linear systems
  • Data dependencies
  • Interval hull solution
  • Monotonicity
  • Parameterized solution