Skip to main content
SpringerLink
Log in

ANN Based Solution of Uncertain Linear Systems of Equations

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Abolmasoumi S, Alavi M (2014) A method for calculating interval linear system. J Math Comput Sci 8:193–204

    Article  Google Scholar 

  2. Allahdadi M, Khorram Z (2015) Solving interval linear equations with modified interval arithmetic. Br J Math Comput Sci 10(2):1–8

    Article  Google Scholar 

  3. Das S, Chakraverty S (2012) Numerical solution of interval and fuzzy system of linear equations. Appl Appl Math Int J 7:334–356

    MathSciNet  MATH  Google Scholar 

  4. Karunakar P, Chakraverty S (2018) Solving fully interval linear systems of equations using tolerable solution criteria. Soft Comput 22(14):4811–4818

    Article  MATH  Google Scholar 

  5. Krämer W (2007) Computing and visualizing solution sets of interval linear systems. Serdica J Comput 1(4):455–468

    MathSciNet  MATH  Google Scholar 

  6. Kolev LV (2004) A method for outer interval solution of linear parametric systems. Reliable Comput 10(3):227–239

    Article  MathSciNet  MATH  Google Scholar 

  7. Beaumont O (1998) Solving interval linear systems with linear programming techniques. Linear Algebra Appl 281(1–3):293–309

    Article  MathSciNet  MATH  Google Scholar 

  8. Skalna I (2003) Methods for solving systems of linear equations of structure mechanics with interval parameters. Comput Assist Mech Eng Sci 10(3):281–293

    MATH  Google Scholar 

  9. Hansen ER (1992) Bounding the solution of interval linear equations. SIAM J Numer Anal 29(5):1493–1503

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhen J, Den Hertog D (2017) Centered solutions for uncertain linear equations. CMS 14(4):585–610

    Article  MathSciNet  MATH  Google Scholar 

  11. Siahlooei E, Shahzadeh Fazeli SA (2018) Two iterative methods for solving linear interval systems. Appl Comput Intell Soft Comput 2018:1–14

    Article  MATH  Google Scholar 

  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 2018:1–11

    Article  MATH  Google Scholar 

  13. Nayak S, Chakraverty S (2013) A new approach to solve fuzzy system of linear equations. J Math Comput Sci 7(3):205–212

    Article  Google Scholar 

  14. Dehghan M, Hashemi B (2006) Solution of the fully fuzzy linear systems using the decomposition procedure. Appl Math Comput 182(2):1568–1580

    MathSciNet  MATH  Google Scholar 

  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):1–15

    Article  MathSciNet  MATH  Google Scholar 

  16. Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. Society for Industrial and Applied Mathematics, Philadelphia

    Book  MATH  Google Scholar 

  17. Alefeld G, Herzberger J (1983) Introduction to interval computations. Academic Press, New York

    MATH  Google Scholar 

  18. Neumaier A (1990) Interval methods for systems of equations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  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):124–138

    Article  MATH  Google Scholar 

  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 7–10

  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, Berlin

  22. Abbasbandy S, Otadi M, Mosleh M (2008) Numerical solution of a system of fuzzy polynomials by fuzzy neural network. Inf Sci 178(8):1948–1960

    Article  MATH  Google Scholar 

  23. Buckley JJ, Eslami E, Hayashi Y (1997) Solving fuzzy equations using neural nets. Fuzzy Sets Syst 86(3):271–278

    Article  MATH  Google Scholar 

  24. Otadi M, Mosleh M (2011) Simulation and evaluation of dual fully fuzzy linear systems by fuzzy neural network. Appl Math Model 35(10):5026–5039

    Article  MathSciNet  MATH  Google Scholar 

  25. Otadi M, Mosleh M, Abbasbandy S (2011) Numerical solution of fully fuzzy linear systems by fuzzy neural network. Soft Comput 15(8):1513–1522

    Article  MATH  Google Scholar 

  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):759

    Article  Google Scholar 

  27. Huang C, Liu B (2019) New studies on dynamic analysis of inertial neural networks involving non-reduced order method. Neurocomputing 325:283–287

    Article  Google Scholar 

  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 113:79–90

    Article  Google Scholar 

  29. Yang X, Yang Z (2014) Synchronization of TS fuzzy complex dynamical networks with time-varying impulsive delays and stochastic effects. Fuzzy Sets Syst 235:25–43

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang LX, Mendel JM (1992) Back-propagation fuzzy system as nonlinear dynamic system identifiers. In: IEEE international conference on fuzzy systems, pp 1409–1418

  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):93–105

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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):625–641

    Article  Google Scholar 

  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):13–31

    MathSciNet  MATH  Google Scholar 

  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):35–49

    Article  MathSciNet  MATH  Google Scholar 

  36. Ning S, Kearfott RB (1997) A comparison of some methods for solving linear interval equations. SIAM J Numer Anal 34(4):1289–1305

    Article  MathSciNet  MATH  Google Scholar 

  37. Chakraverty S, Tapaswini S, Behera D (2016) Fuzzy differential equations and applications for engineers and scientists. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  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):535–541

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha, 769008, India

    S. K. Jeswal & S. Chakraverty

Authors
  1. S. K. Jeswal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Chakraverty
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Chakraverty.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jeswal, S.K., Chakraverty, S. ANN Based Solution of Uncertain Linear Systems of Equations. Neural Process Lett 51, 1957–1971 (2020). https://doi.org/10.1007/s11063-019-10183-w

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-019-10183-w

Keywords

Search

Navigation