A Back Propagation-Type Neural Network Architecture for Solving the Complete n × n Nonlinear Algebraic System of Equations — Oak Academic Publishing
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A Back Propagation-Type Neural Network Architecture for Solving the Complete n × n Nonlinear Algebraic System of Equations
TEI of Thessaloniki, Department of Informatics, Thessaloniki, Greece
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TEI of Larissa, Department of Computer Science and Engineering, Larissa, Greece
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Department of Applied Informatics, University of Macedonia, Thessaloniki, Greece
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TEI of Thessaloniki, Department of Informatics, Thessaloniki, Greece
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Department of Economics, Democritus University of Thrace, Komotini, Greece
1 TEI of Thessaloniki, Department of Informatics, Thessaloniki, Greece
2 TEI of Larissa, Department of Computer Science and Engineering, Larissa, Greece
3 Department of Applied Informatics, University of Macedonia, Thessaloniki, Greece
4 TEI of Thessaloniki, Department of Informatics, Thessaloniki, Greece
5 Department of Economics, Democritus University of Thrace, Komotini, Greece
The objective of this research is the presentation of a neural network capable of solving complete nonlinear algebraic systems of n equations with n unknowns. The proposed neural solver uses the classical back propagation algorithm with the identity function as the output function, and supports the feature of the adaptive learning rate for the neurons of the second hidden layer. The paper presents the fundamental theory associated with this approach as well as a set of experimental results that evaluate the performance and accuracy of the proposed method against other methods found in the literature.
Zhang, G. and Bai, L. (2009) Existence of Solutions for a Nonlinear Algebraic System. Discrete Dynamics in Nature and Society, 2009, Article ID: 785068.
Kowalski, K. and Tankowski, K. (1998) Towards Complete Solutions to Systems of Nonlinear Equations of Many-Electron Theories. Physical Review Letters, 81, 1195-1998. http://dx.doi.org/10.1103/PhysRevLett.81.1195
Holstad, A. (1999) Numerical Solution of Nonlinear Equations in Chemical Speciation Calculations. Computational Geosciences, 3, 229-257. http://dx.doi.org/10.1023/A:1011595429513
Argyros, K. (1993) On the Solution of Underdetermined Systems of Nonlinear Equations in Euclidean Spaces. Pure Mathematics and Applications, 4, 199-209.
Morgan, A.P. (1987) Solving Polynomial Systems Using Continuation for Scientific and Engineering Problems. Prentice-Hall, Eaglewood Cliffs.
Dolotin, V. and Morozov, A. (2007) Introduction to Non-Linear Algebra. World Scientific Publishing Company, Singapore. http://dx.doi.org/10.1142/6508
Broyden, C.G. (1965) A Class of Methods for Solving Nonlinear Simultaneous Equations. Mathematical Computations, 19, 577-593. http://dx.doi.org/10.1090/S0025-5718-1965-0198670-6
Broyden, C.G., Dennis, J.E. and More, J.J. (1973) On the Local and Superlinear Convergence of Quasi-Newton Methods. IMA Journal of Applied Mathematics, 12, 223-245. http://dx.doi.org/10.1093/imamat/12.3.223
Dennis, J.E. and Wolkowicz, H. (1993) Least Change Secant Methods, Sizing, and Shifting. SIAM Journal on Numerical Analysis, 30, 1291-1314.
Hentenryck, P., McAllester, D. and Kapur, D. (1997) Solving Polynomial Systems Using a Branch and Prune Approach. SIAM Journal on Numerical Analysis, 34, 797-827. http://dx.doi.org/10.1137/S0036142995281504
Abaffy, J. and Spedicato, E. (1989) ABS Projection Algorithms: Mathematical Techniques for Linear and Nonlinear Equations. Ellis Horwood, Hemel Hemstead.
Abaffy, J. and Galantai, A. (1987) Conjugate Direction Methods for Linear and Nonlinear Systems of Algebraic Equations. Numerical Methods, 50, 481-502.
Abaffy, J., Galantai, A. and Spedicato, E. (1987) The Local Convergence of ABS Methods for Nonlinear Algebraic Equations. Numerische Mathematik, 51, 429-439. http://dx.doi.org/10.1007/BF01397545
Galantai, A. and Jeney, A. (1996) Quasi-Newton ABS Methods for Solving Nonlinear Algebraic Systems of Equations. Journal of Optimization Theory and Applications, 89, 561-573. http://dx.doi.org/10.1007/BF02275349
Ren, H., Wu, L., Bi, W.H. and Argyros, I.K. (2013) Solving Nonlinear Equations System via an Efficient Genetic Algorithm, with Symmetric and Harmonious Individuals. Applied Mathematics and Computation, 219, 10967-10973.
El-Emary, I.M.M. and El-Kareem, M.M.A. (2008) Towards Using Genetic Algorithms for Solving Nonlinear Equation Systems. World Applied Sciences Journal, 5, 282-289.
Pourjafari, E. and Mojallali, H. (2012) Solving Nonlinear Equation Systems with a New Approach Based on Invasive Weed Optimization Algorithm and Clustering. Swarm and Evolutionary Computation, 4, 33-43. http://dx.doi.org/10.1016/j.swevo.2011.12.001
Mehrabian, A.R. and Lucas, C. (2006) A Novel Numerical Optimization Algorithm Inspired from Weed Colonization. Ecological Informatics, 1, 355-366. http://dx.doi.org/10.1016/j.ecoinf.2006.07.003
Oliveira, H.A. and Petraglia, A. (2013) Solving Nonlinear Systems of Functional Equations with Fuzzy Adaptive Simulated Annealing. Applied Soft Computing, 13, 4349-4357. http://dx.doi.org/10.1016/j.asoc.2013.06.018
Effati, S. and Nazemi, A.R. (2005) A New Method for Solving a System of the Nonlinear Equations. Applied Mathematics and Computations, 168, 877-894. http://dx.doi.org/10.1016/j.amc.2004.09.029
Mathia, K. and Saeks, R. (1995) Solving Nonlinear Equations Using Recurrent Neural Networks. Proceedings of World Congress on Neural Networks (WCNN’95), Washington DC, 17-21 July 1995, 76-80.
Meng, A. and Zeng, Z. (2011) A Neural Computational Method to Solve Nonlinear Equation Systems. Journal of Computational Information Systems, 7, 3462-3469.
Luo, Y.Z., Tang, G.T. and Zhou, L.N. (2008) Hybrid Approach for Solving Systems of Nonlinear Equations Using Chaos Optimization and Quasi-Newton Method. Applied Soft Computing, 8, 1068-1073. http://dx.doi.org/10.1016/j.asoc.2007.05.013
Kuri-Morales, A.F. (2003) Solution of Simultaneous Nonlinear Equations Using Genetic Algorithms. WSEAS Transactions on Systems, 2, 44-51.
Nasira, G.N. and Devi, D.S. (2012) Solving Nonlinear Equations through Jacobian Sparsity Patterns Using Genetic Algorithms. International Journal of Communications and Engineering, 5, 78-82.
Grosan, C. and Abraham, A. (2008) A New Approach for Solving Nonlinear Equation Systems. IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans, 38, 698-714. http://dx.doi.org/10.1109/TSMCA.2008.918599
Liu, H., Zhou, Y. and Li, Y. (2011) A Quasi-Newton Population Migration Algorithm for Solving Systems of Nonlinear Equations. Journal of Computers, 6, 36-42. http://dx.doi.org/10.4304/jcp.6.1.36-42
Zhou, Y.H. and Mao, Z.Y. (2003) A New Search Algorithm for Global Optimization—Population Migration Algorithm. Journal of South China University of Technology, 21, 1-5.
Zhao, Q. and Li, W. (2012) An Improved Iterative Algorithm of Neural Network for Nonlinear Equation Groups. Proceedings of IEEE 2nd International Conference on Business Computing and Global Informatization, Shanghai, 12-14 October 2012, 522-525. http://dx.doi.org/10.1109/bcgin.2012.142
Mishra, D. and Kalra, P.K. (2007) Modified Hopfield Neural Network Approach for Solving Nonlinear Algebraic Equations. Engineering Letters, 14, 135-142.
Li, G. and Zeng, Z. (2008) A Neural-Network Algorithm for Solving Nonlinear Equation Systems. 9th International Conference on Computational Intelligence and Security, 1, 20-23. http://dx.doi.org/10.1109/cis.2008.65
Margaris, A. and Adamopoulos, M. (2007) Solving Nonlinear Algebraic Systems Using Artificial Neural Networks. Proceedings of the 10th International Conference on Engineering Applications of Artificial Neural Networks, Thessaloniki, 29-31 August 2007, 107-120.
Margaris, A. and Goulianas, K. (2012) Finding All Roots of 2 × 2 Nonlinear Algebraic Systems Using Back-Propagation Neural Networks. Neural Computing and Applications, 21, 891-904. http://dx.doi.org/10.1007/s00521-010-0488-z
Goulianas, K., Margaris, A. and Adamopoulos, M. (2013) Finding All Real Roots of 3 × 3 Nonlinear Algebraic Systems Using Neural Networks. Applied Mathematics and Computation, 219, 4444-4464. http://dx.doi.org/10.1016/j.amc.2012.10.049
Tsoulos, I.G. and Stavrakoudis, A. (2010) On Locating All Roots of Systems of Nonlinear Equations inside Bounded Domain Using Global Optimization Methods. Nonlinear Analysis: Real World Applications, 11, 2465-2471. http://dx.doi.org/10.1016/j.nonrwa.2009.08.003
Waziri, M.Y., Leong, W.J. and Mamat, M. (2012) A Two-Step Matrix-Free Secant Method for Solving Large-Scale Systems of Nonlinear Equations. Journal of Applied Mathematics, 2012, Article ID: 348654.
Leong, W.J., Hassan, M.A. and Yusuf, M.W. (2011) A Matrix-Free Quasi-Newton Method for Solving Large-Scale Nonlinear Systems. Computers and Mathematics with Applications, 62, 2354-2363. http://dx.doi.org/10.1016/j.camwa.2011.07.023
Yu, G., Niu, S., Ma, J. and Song, Y. (2013) An Adaptive Prediction-Correction Method for Solving Large-Scale Nonlinear Systems of Monotone Equations with Applications. Abstract and Applied Analysis, 2013, Article ID: 619123.
Mamat, M., Muhammad, K. and Waziri, M.Y. (2014) Trapezoidal Broyden’s Method for Solving Systems of Nonlinear Equations. Applied Mathematical Sciences, 8, 251-260.
Sun, W. and Yuan, Y-X. (2006) Optimization Theory and Methods, Nonlinear Programming. Springer, New York.