[32] Xiangde Guo, Zhezhao Zeng, The Neural-Network Approaches to Solve

Nonlinear Equation, Journal Of Computers, 5/3, March 2010

Secant method [1, 2], which uses divided differences instead of the first derivative of the nonlinear operator, is one of the most famous iterative methods for solving the

nonlinear equation. Secant method reads as follows:

The analytical surface derivatives resulting from the interpolation process are replaced into the MA equation and the boundary condition at the reflector's edge leads to a system of

nonlinear equations. To solve the system, an iterative numerical technique is employed.

Caption: Figure 1: Schematic diagram of a two-dimensional curve C which is defined by a parameterized scalar

nonlinear equation E (i.e., C = {E : (U, [lambda]) [member of] [R.sup.2] [right arrow] R | E(U, [lambda]) = 0}) which presents only one critical point ([U.sub.cr], [[lambda].sub.cr]) in four different situations ((a), (b), (c), and (d)).

In this article, optimization problem like (8) is used to deal with one

nonlinear equation system.

Kwun, A new second- order iteration method for solving

nonlinear equations", Abstr.

The construction of the nonlinear TDFEM formulation will mainly be based on the most widely used numerical method for solving

nonlinear equations, the Newton-Raphson method [30], due to its quadratic local convergence.

Many researchers developed various models [1-10] to investigate this problem, among which the two-layer model is one of the analytical research models and is formulated with highly

nonlinear equations set.

In this paper, we explore numerical methods for solving

nonlinear equations using MATLAB.

Let one consider x* as a root of

nonlinear equation (1) in the domain D and assume that f(x) is sufficiently differentiable in the neighbourhood of the root.

Furthermore, in order to examine the CPU time changes with respect to the size of the network, we made synthetic networks with 10 and 20 nodes by making the

nonlinear equation with corresponding number of variables.