Abstract
In this paper, we introduce an algorithm for presentation of an inner estimation of the solution set of a complex interval linear system, where the coefficient matrix is a crisp complex-valued matrix and the right-hand-side vector is an interval complex-valued vector. Also, we show that under some certain conditions, the obtained inner estimation is, in fact, an algebraic solution.
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References
Alefeld G (1968) Intervallrechnung uber den komplexen Zahlen und einige Anwendungen. PhD thesis, Universität Karlsruhe, Karlsruhe
Alefeld G, Mayer G (1995) On the symmetric and unsymmetric solution set of interval systems. SIAM J Matrix Anal Appl 16:1223–1240
Alefeld G, Mayer G (2000) Interval analysis: theory and applications. J Comput Appl Math 121:421–464
Barmish BR (1994) New tools for robustness of linear systems. MacMillan, New York
Boche R (1966) Complex interval arithmetic with some applications. Technical report LMSC4-22-66-1, Lockheed Missiles & Space Company, Sunnyvale
Candau Y, Raissi T, Ramdani N, Ibos L (2006) Complex interval arithmetic using polar form. Reliab Comput 12:1–20
Djanybekov BS (2006) Interval householder method for complex linear systems. Reliab Comput 12:35–43
Dreyer A (2005) Interval analysis of analog circuits with component tolerances. Shaker Verlag, Aachen. Doctoral thesis, TU Kaiserslautern
Friedman M, Ming M, Kandel A (1998) Fuzzy linear systems. Fuzzy Sets Syst 96:201–209
Garajová E, Mečiar M (2016) Solving and visualizing nonlinear set inversion problems. Reliab Comput 22:104–115
Gargantini I, Henrici P (1971) Circular arithmetic and the determination of polynomial zeros. Numer Math 18(4):305–320
Garloff J (2009) Interval Gaussian elimination with pivot tightening. SIAM J Matrix Anal Appl 30:1761–1772
Ghanbari M, Allahviranloo T, Haghi E (2012) Estimation of algebraic solution by limiting the solution set of an interval linear system. Soft Comput 16:2135–2142
Henrici P (1971) Circular arithmetic and the determination of polynomial zeros. In: Conference on applications of numerical analysis, volume 228 of Lecture Notes in Mathematics, Dundee, pp 86–92
Hladik M (2010) Solution sets of complex linear interval systems of equations. Reliab Comput 14:78–87
Hladik M (2014) Strong solvability of linear interval systems of inequalities with simple dependencies. Int J Fuzzy Comput Modell 1:3–14
Kolev LV (1993) Interval methods for circuit analysis. World Scientific, Singapore
Kolev LV, Vladov SS (1989) Linear circuit tolerance analysis via systems of linear interval equations. In: ISYNT89 6th international symposium on networks, systems and signal processing, June 28 July 1, Zagreb, pp 57–60
Mayer G (2006) A contribution to the feasibility of the interval Gaussian algorithm. Reliab Comput 12:79–98
Popova E, Kolev L, Kramer W (2010) A Solver for complex-valued parametric linear systems. Serdica J Comput 4:123–132
Rump SM (1999) INTLAB-INTerval LABoratory. In: Csendes T (ed) Developments in reliable computing. Kluwer Academic Publishers, Dordrecht, pp 77–104
Sevastjanov P, Dymova L (2009) A new method for solving interval and fuzzy equations: linear case. Inf Sci 179:925–937
Shary SP (2002) A new technique in systems analysis under interval uncertainty and ambiguity. Reliab Comput 8(5):321–418
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Ghanbari, M. An estimation of algebraic solution for a complex interval linear system. Soft Comput 22, 2881–2890 (2018). https://doi.org/10.1007/s00500-017-2538-2
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DOI: https://doi.org/10.1007/s00500-017-2538-2