It is also possible to use the Newton's root seeking method to find the minimum, maximum, or saddle point of a function,
because the derivative of the targeted function is zero at these points.
In this method, the minimum is not bracketed and
only one 1 initial guess value of the solution is needed to get the iterative process started to find the minimum of a
linear function.
Algorithm Creation
The Newton method uses the following iteration relation:
x_{n+1} = x_{n}  f'(x_{n}) / f''(x_{n})

where f'(x) and f''(x) are the first and second derivatives of a function f(x).
Testing the Newton Method
We use the Newton method to find the minimum of a nonlinear function. To test it out, we find the minimum of the same function
used to test the Bisection method. Supporting code and methods are not shown.
static void TestNewton();
{
ListBox1.Items.Clear();
ListBox2.Items.Clear();
double result = Optimization.Newton(f, t1, 1.0e5);
ListBox1.Items.Add("x = " + result.ToString());
ListBox2.Items.Add("f(x) = " + f(result).ToString());
}
