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ANN Based Solution of Uncertain Linear Systems of Equations

  • S. K. Jeswal
  • S. ChakravertyEmail author
Article
  • 46 Downloads

Abstract

Linear systems of equations have many applications in the area of engineering sciences, mathematics, operations research and statistics. It is worth mentioning that the coefficient matrix of the linear systems of equations may not be always crisp due to various uncertainties. These uncertainties may be in the form of interval. Likewise, the solution set and right hand side vector may also be in interval. In this respect, a fully interval linear system of equations \( (\tilde{P}\tilde{z} = \tilde{q}) \) is one where the coefficient matrix, the unknown vector and the right hand side vector all are in the form of interval. Although various authors proposed different methods to handle the fully linear systems of equations but those are sometimes problem specific etc. As such, in this paper \( n \times n \) fully interval linear systems of equations has been solved based on artificial neural network (ANN) model. In this regard, step by step algorithm has been included. Further, a convergence theorem has also been discussed for choosing suitable learning parameter. Few numerical examples and an application problem related to electrical circuit have been solved using the proposed method. Detail procedure has been discussed with numerical results to show the efficacy and powerfulness of the method.

Keywords

Artificial neural network (ANN) Interval arithmetic Linear systems of equations (LSEs) Fully interval linear systems of equations 

Notes

References

  1. 1.
    Abolmasoumi S, Alavi M (2014) A method for calculating interval linear system. J Math Comput Sci 8:193–204CrossRefGoogle Scholar
  2. 2.
    Allahdadi M, Khorram Z (2015) Solving interval linear equations with modified interval arithmetic. Br J Math Comput Sci 10(2):1–8CrossRefGoogle Scholar
  3. 3.
    Das S, Chakraverty S (2012) Numerical solution of interval and fuzzy system of linear equations. Appl Appl Math Int J 7:334–356MathSciNetzbMATHGoogle Scholar
  4. 4.
    Karunakar P, Chakraverty S (2018) Solving fully interval linear systems of equations using tolerable solution criteria. Soft Comput 22(14):4811–4818zbMATHCrossRefGoogle Scholar
  5. 5.
    Krämer W (2007) Computing and visualizing solution sets of interval linear systems. Serdica J Comput 1(4):455–468MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kolev LV (2004) A method for outer interval solution of linear parametric systems. Reliable Comput 10(3):227–239MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Beaumont O (1998) Solving interval linear systems with linear programming techniques. Linear Algebra Appl 281(1–3):293–309MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Skalna I (2003) Methods for solving systems of linear equations of structure mechanics with interval parameters. Comput Assist Mech Eng Sci 10(3):281–293zbMATHGoogle Scholar
  9. 9.
    Hansen ER (1992) Bounding the solution of interval linear equations. SIAM J Numer Anal 29(5):1493–1503MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Zhen J, Den Hertog D (2017) Centered solutions for uncertain linear equations. CMS 14(4):585–610MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Siahlooei E, Shahzadeh Fazeli SA (2018) Two iterative methods for solving linear interval systems. Appl Comput Intell Soft Comput 2018:1–14zbMATHCrossRefGoogle Scholar
  12. 12.
    Siahlooei E, Shahzadeh Fazeli SA (2018) An application of interval arithmetic for solving fully fuzzy linear systems with trapezoidal fuzzy numbers. Adv Fuzzy Syst 2018:1–11zbMATHCrossRefGoogle Scholar
  13. 13.
    Nayak S, Chakraverty S (2013) A new approach to solve fuzzy system of linear equations. J Math Comput Sci 7(3):205–212CrossRefGoogle Scholar
  14. 14.
    Dehghan M, Hashemi B (2006) Solution of the fully fuzzy linear systems using the decomposition procedure. Appl Math Comput 182(2):1568–1580MathSciNetzbMATHGoogle Scholar
  15. 15.
    Alefeld G, Kreinovich V, Mayer G (2003) On the solution sets of particular classes of linear interval systems. J Comput Appl Math 152(1):1–15MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. Society for Industrial and Applied Mathematics, PhiladelphiazbMATHCrossRefGoogle Scholar
  17. 17.
    Alefeld G, Herzberger J (1983) Introduction to interval computations. Academic Press, New YorkzbMATHGoogle Scholar
  18. 18.
    Neumaier A (1990) Interval methods for systems of equations. Cambridge University Press, CambridgezbMATHGoogle Scholar
  19. 19.
    Cichocki A, Unbehauen R (1992) Neural networks for solving systems of linear equations and related problems. IEEE Trans Circuits Syst I Fundam Theory Appl 39(2):124–138zbMATHCrossRefGoogle Scholar
  20. 20.
    Zhou Z, Chen L, Wan L (2009) Neural network algorithm for solving system of linear equations. In: Computational intelligence and natural computing. CINC’09. International conference on IEEE, vol 2, pp 7–10Google Scholar
  21. 21.
    Viet NH, Kleiber M (2006) AI methods in solving systems of interval linear equations. In: International conference on artificial intelligence and soft computing, pp 150–159. Springer, BerlinCrossRefGoogle Scholar
  22. 22.
    Abbasbandy S, Otadi M, Mosleh M (2008) Numerical solution of a system of fuzzy polynomials by fuzzy neural network. Inf Sci 178(8):1948–1960zbMATHCrossRefGoogle Scholar
  23. 23.
    Buckley JJ, Eslami E, Hayashi Y (1997) Solving fuzzy equations using neural nets. Fuzzy Sets Syst 86(3):271–278zbMATHCrossRefGoogle Scholar
  24. 24.
    Otadi M, Mosleh M (2011) Simulation and evaluation of dual fully fuzzy linear systems by fuzzy neural network. Appl Math Model 35(10):5026–5039MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Otadi M, Mosleh M, Abbasbandy S (2011) Numerical solution of fully fuzzy linear systems by fuzzy neural network. Soft Comput 15(8):1513–1522zbMATHCrossRefGoogle Scholar
  26. 26.
    Rajchakit G, Pratap A, Raja R, Cao J, Alzabut J, Huang C (2019) Hybrid control scheme for projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks with mixed delays. Mathematics 7(8):759CrossRefGoogle Scholar
  27. 27.
    Huang C, Liu B (2019) New studies on dynamic analysis of inertial neural networks involving non-reduced order method. Neurocomputing 325:283–287CrossRefGoogle Scholar
  28. 28.
    Tang R, Yang X, Wan X (2019) Finite-time cluster synchronization for a class of fuzzy cellular neural networks via non-chattering quantized controllers. Neural Netw 113:79–90CrossRefGoogle Scholar
  29. 29.
    Yang X, Yang Z (2014) Synchronization of TS fuzzy complex dynamical networks with time-varying impulsive delays and stochastic effects. Fuzzy Sets Syst 235:25–43MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Wang LX, Mendel JM (1992) Back-propagation fuzzy system as nonlinear dynamic system identifiers. In: IEEE international conference on fuzzy systems, pp 1409–1418Google Scholar
  31. 31.
    Yang X, Zhu Q, Huang C (2011) Generalized lag-synchronization of chaotic mix-delayed systems with uncertain parameters and unknown perturbations. Nonlinear Anal Real World Appl 12(1):93–105MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Huang C, Su R, Cao J, Xiao S (2019) Asymptotically stable of high-order neutral cellular neural networks with proportional delays and D operators. Math Comput Simul.  https://doi.org/10.1016/j.matcom.2019.06.001 CrossRefGoogle Scholar
  33. 33.
    Huang C, Liu B, Tian X, Yang L, Zhang X (2019) Global convergence on asymptotically almost periodic SICNNs with nonlinear decay functions. Neural Process Lett 49(2):625–641CrossRefGoogle Scholar
  34. 34.
    Chakraverty S, Hladík M, Mahato NR (2017) A sign function approach to solve algebraically interval system of linear equations for nonnegative solutions. Fund Inf 152(1):13–31MathSciNetzbMATHGoogle Scholar
  35. 35.
    Behera D, Chakraverty S (2015) New approach to solve fully fuzzy system of linear equations using single and double parametric form of fuzzy numbers. Sadhana 40(1):35–49MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Ning S, Kearfott RB (1997) A comparison of some methods for solving linear interval equations. SIAM J Numer Anal 34(4):1289–1305MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Chakraverty S, Tapaswini S, Behera D (2016) Fuzzy differential equations and applications for engineers and scientists. CRC Press, Boca RatonzbMATHCrossRefGoogle Scholar
  38. 38.
    Rahgooy T, Yazdi HS, Monsefi R (2009) Fuzzy complex system of linear equations applied to circuit analysis. Int J Comput Electr Eng 1(5):535–541CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia

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