Finding multiple real roots by neural networks based on complete discrimination system of polynomial

A new method of solving the multiple real roots of polynomial by neural networks is proposed in this paper. This method combines the symbolic method with the numerical method. Based on the complete discrimination system of polynomial, the number and multiplicities of the distinct real roots of polynomials can be explicit determined. According to the number of the distinct real roots, a neural networks model for finding the multiple real roots of polynomial is established. From the description of the new model, it is not difficult to find that the existent neural networks for finding real roots of polynomial is the special case of the new one, where all of the real roots are treated as different values. Through training the new model by the gradient descent method, the approximate real roots of polynomial can be obtained. From the simulation results, it is shown that, comparing to the existent neural networks of finding real roots, the new method is not only more effective, but also can avoid the inequality between two or more equal real roots after finishing to solve the polynomial.