Enclosing the solution set of parametric interval matrix equation A(p)X = B(p)
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Consider the parametric matrix equation A(p)X = B(p), where the elements of the matrices A(p) and B(p) depend linearly on a number of uncertain parameters varying within given intervals. We prove that the united parametric solution sets of the matrix equation and that of the corresponding linear system with multiple right-hand sides, although different as sets, have the same interval hull. A generalization of the parametric Krawczyk iteration with low computational complexity for the matrix equation is presented. Some details improving the implementation and the application of this method are discussed. An interval method, designed by A. Neumaier and A. Pownuk for enclosing the united solution set of parametric linear systems with particular dependency structure, is generalized for arbitrary linear dependencies between the parameters and for systems with multiple right-hand sides. A new, more powerful, sufficient condition for regularity of a parametric interval matrix is proven. An important application of the linear systems with multiple right-hand sides is presented as a key methodology for feasibility in computing the interval hull of a class of united parametric solution sets that appear in practical problems.
KeywordsLinear matrix equations Interval parameters Solution enclosure
Mathematics subject classification (2010)65G40 15A24 65F10
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The author thanks the anonymous reviewers for their comments which helped improving the manuscript.
- 7.Muhanna, R.L.: Benchmarks for interval finite element computations web site http://rec.ce.gatech.edu/resources/Benchmark_2.pdf (2004)
- 11.Popova, E.D.: Strong regularity of parametric interval matrices. In: Dimovski, I.I. et al. (eds.) Mathematics and Education in Mathematics, Proceedings of the 33rd Spring Conference of the Union of Bulgarian Mathematicians, Borovets, Bulgaria, BAS, pp. 446–451 (2004)Google Scholar
- 13.Popova, E.D.: Computer-assisted proofs in solving linear parametric problems, Post-proceedings of 12Th GAMM–IMACS Int. Symp. on Scientific Computing, Computer Arithmetic and Valiyeard Numerics. IEEE Computer Society Press, Duisburg, Germany (2006)Google Scholar
- 19.Rama Rao, M.V., Muhanna, R.L., Mullen, R.L.: Interval finite element analysis of thin plates. In: Freitag, S., Muhanna, R.L., Mullen, R.L. (eds.) Proceedings of the NSF Workshop on Reliable Engineering Computing, Ruhr Univ. Bochum, Germany, pp. 111–130 (2016)Google Scholar
- 21.Rump, S.M.: Verification methods for dense and sparse systems of equations. In: Herzberger, J. (ed.) Topics in Valiyeard Computations, pp. 63–135. Elsevier Science B. V (1994)Google Scholar
- 26.Skalna, I.: Strong regularity of parametric interval matrices. Linear and Multilinear Algebra. doi: 10.1080/03081087.2016.1277687 (2017)