Abstract
In this paper, fuzzy polynomial systems in dual form are considered and an algebraic approach for finding their solutions is presented. A dual fuzzy polynomial system in the form \(AX+B=CX+D \), where \(A, B, C,\) and \(D\) are fuzzy matrices, is converted to a system with real coefficients and variables first. Then, Wu’s algorithm is used as a solution procedure for solving this system. This algorithm leads to solving characteristic sets that are amenable to easy solution. Finally, the accuracy of the presented algorithm is shown via some examples.
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Abbasbandy, S., Asady, B.: Newton’s method for solving fuzzy nonlinear equations. Appl. Math. Comput. 159(2), 349–356 (2004)
Abbasbandy, S., Jafarian, A.: Steepest descent method for solving fuzzy nonlinear equations. Appl. Math. Comput. 174(1), 669–675 (2006)
Abbasbandy, S., Otadi, M., Mosleh, M.: Minimal solution of general dual fuzzy linear systems. Chaos Solut. Fract. 37, 1113–1124 (2008)
Abbasbandy, S., Otadi, M., Mosleh, M.: Numerical solution of a system of fuzzy polynomials by fuzzy neural network. Inf. Sci. 178(8), 1948–1960 (2008)
Buckley, J., Feuring, T., Hayashi, Y.: Solving fuzzy equations using evolutionary algorithms and neural nets. Soft Comput. 6(2), 116–123 (2002)
Cechlarova, K.: A note on unsolvable systems of max–min (fuzzy) equations. Linear Algebra Appl. 310, 123–128 (2000)
Chou, S., Gao, X., McPhee, N.: A Combination of Ritt-Wu’s Method and Collins’ Method. Technical report, Austin, Texax, USA (1989)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms : An Introduction To computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer, New York (2007)
Gao, X., Hou, X., Tang, J., Cheng, H.: Complete solution classification for the perspective-three-point problem. IEEE Trans. Pattern Anal. Mach. Intell. 25(8), 930–943 (2003)
Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets Syst. 18(1), 31–43 (1986)
Gong, Z., Guo, X., Liu, K.: Approximate solution of dual fuzzy matrix equations. Inf. Sci. 266, 112–133 (2014)
Jafarian, A., Measoomy, S.: Nia: on the solution of nonlinear fuzzy equations system by fuzzy neural networks method. Int. J. Fuzzy Syst. 15(3), 376–380 (2013)
Jin, M., Li, X., Wang, D.: A new algorithmic scheme for computing characteristic sets. J. Symb. Comput. 50, 431–449 (2013)
Tavassoli Kajani, M., Asady, B., Hadi-Vencheh, A.: An iterative method for solving dual fuzzy nonlinear equations. Appl. Math. Comput. 167(1), 316–323 (2005)
Lee, K.H.: First Course Fuzzy Theory Applications. Advances in intelligent and soft computing. Springer, Berlin (2005)
Liang, L.R., Lu, S., Wang, X., Lu, Y., Mandal, V., Patacsil, D., Kumar, D.: Fm-test: a fuzzy-set-theory-based approach to differential gene expression data analysis. BMC Bioinform 7(S–4), S7 (2006)
Ming, M., Friedman, M., Kandel, A.: Duality in fuzzy linear systems. Fuzzy Sets Syst 109, 55–58 (2000)
Abbasi Molai, A., Basiri, A., Rahmany, S.: Resolution of a system of fuzzy polynomial equations using the gröbner basis. Inf. Sci. 220, 541–558 (2013)
Mosleh, M., Otadi, M., Abbasbandy, S.: Solution of fully fuzzy linear systems by st method. J Appl Math 8, 23–31 (2011)
Otadi, M., Mosleh, M.: Solution of fuzzy polynomial equations by modified adomain decomposition method. Soft Comput 15, 187–192 (2011)
Salahshour, S., Homayoun Nejad, M.: Approximating solution of fully fuzzy linear systems in dual form. Int. J. Ind. Math. 5(1), 19–23 (2013)
Wen-tsun, W.: Basic principles of mechanical theorem proving in geometrics. J. Sys. Sci. Math. 4(3), 207–235 (1984)
Wen-Tsun, W.: On the decision problem and the mechanization of theorem-proving in elementary geometry. Contemp. Math. 29, 213–234 (1984)
Wen-tsun, W.: Mathematics Mechanization: Mechanical Geometry Theorem-Proving, Mechanical Geometry Problem-Solving and Polynomial Equations-Solving. Mathematics and Its Applications. BeijingbScience Press, London (2001). (This is an updated revised translation of the original Chinese work of the same title, Science Press, c2000)
Wen-Tsun, W., Gao, X.: Mathematics mechanization and applications after thirty years. Front. Comput. Sci. China 1(1), 1–8 (2007)
Wen-Tsun, W., Gao, X.S.: Automated reasoning and equation solving with the characteristic set method. J. Comput. Sci. Technol. 21(5), 756–764 (2006)
Yeh, Ch-T: Reduction of fuzzy linear systems of dual equations. Int. J. Fuzzy Syst. 9(3), 173–178 (2007)
Zadeh, L.A.: Fuzzy sets. Inf Control 8, 338–353 (1965)
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The authors would like to thank Ali Abbasi Molai, Xiao-Shan Gao, and Donming Wang for their helpful discussions.
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Boroujeni, M., Basiri, A., Rahmany, S. et al. Solving Fuzzy Systems in Dual Form Using Wu’s Method. Int. J. Fuzzy Syst. 17, 170–180 (2015). https://doi.org/10.1007/s40815-015-0033-4
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DOI: https://doi.org/10.1007/s40815-015-0033-4