Finding all real roots of 3  3 nonlinear algebraic systems using neural networks Konstantinos Goulianas a , Athanasios Margaris b,⇑ , Miltiades Adamopoulos c a TEI of Thessaloniki, Department of Informatics, Thessaloniki, Greece b TEI of Larissa, Department of Computer Science and Telecommunications, Greece c University of Macedonia, Thessaloniki, Greece article info Keywords: Nonlinear algebraic systems Neural networks Numerical analysis Computational methods abstract The objective of this research is the description of a feed-forward neural network capable of solving nonlinear algebraic systems with polynomials equations. The basic features of the proposed structure, include among other things, product units trained by the back- propagation algorithm and a fixed input unit with a constant input of unity. The presented theory is demonstrated by solving complete 3  3 nonlinear algebraic system paradigms, and the accuracy of the method is tested by comparing the experimental results produced by the network, with the theoretical values of the systems roots. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction A typical nonlinear algebraic system is defined as Fð ~ zÞ¼ 0 with the mapping function F : R n ! R n ðn > 1Þ to be described as an n-dimensional vector F ¼½f 1 ; f 2 ; ... ; f n  T ; ð1Þ where f i : R n ! R (i ¼ 1; 2; ... ; n). Generally speaking, there are no good methods for solving such systems: even in the simple case of only two equations in the form f 1 ðz 1 ; z 2 Þ¼ 0 and f 2 ðz 1 ; z 2 Þ¼ 0, the estimation of the system roots is reduced to the identification of the common points of the zero contours of the functions f 1 ðz 1 ; z 2 Þ and f 2 ðz 1 ; z 2 Þ. But this is a very difficult task, since in general, these two functions have no relation to each other at all. In the general case of N nonlinear equations, solving the system requires the identification of points that are mutually common to N unrelated zero-contour hyper- surfaces each of dimension N  1 [28]. 2. Nonlinear algebraic systems According to the basic principles of the nonlinear algebra [26], a complete nonlinear algebraic system of n polynomial equations with n unknowns ~ z ¼ðz 1 ; z 2 ; ... ; z n Þ is identified completely by the number of equations n, and their degrees ðs 1 ; s 2 ; ... ; s n Þ, it is expressed mathematically as F i ð ~ zÞ¼ X n j 1 ;j 2 ;...;j s i A j 1 j 2 ...j s i i z j 1 z j 2 ... z j s i ¼ 0 ð2Þ 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.10.049 ⇑ Corresponding author. E-mail addresses: gouliana@it.teithe.gr (K. Goulianas), amarg@teilar.gr, amarg@uom.gr (A. Margaris), miltos@uom.gr (M. Adamopoulos). Applied Mathematics and Computation 219 (2013) 4444–4464 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc