The following are the steps we followed in order to apply the Newton-Raphson method to find the root of a nonlinear equation f (x):

We wrote a static method and added to the our library NonLinear System class (code not shown). The following is the code listing of this method:

           public static double NewtonRaphson(Function f, Function f1,
               double x0, double tolerance)
           {
             double f0 = f(x0);
             double x = x0;
             while (Math.Abs(f(x)) > tolerance)
             {
                 x -= f0 / f1(x);
                 f0 = f(x);
             }
             return x;
           }

This method requires an analytical expression for the first derivative of the function f(x) as an input delegate function fl. Instead of using the analytic first dervative of the function as input, it is also possible to implement a routine that computes the first derivative of the function numerically.

Many scientific and engineering phenomena are characterized by nonlinear behavior and very useful when analytical solutions cannot be found. We often apply solutions using nonlinear behavior like the following:

• Fixed Point method finding solutions usually require rearranging the original function for execution.

• Incremental method one of the most basic numerical methods we use for its simplicity.

• Bisection method a very reliable method for solving nonlinear equations.

• Birge-Vieta method a reliable method when dealing with finding real roots of nth degree polynomial equations.