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"Optimization Solution"
Newton Optimization Method
Minimum =
      Value =

f(x) = 1.6 x3 + 3x2 - 2x
Initial Value =
[ Tolerance: 1.0e-5]

Newton Method Optimization

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:

xn+1 = xn - f'(xn) / f''(xn)

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();
                 double result = Optimization.Newton(f, t1, 1.0e-5);
                 ListBox1.Items.Add("x = " + result.ToString());
                 ListBox2.Items.Add("f(x) = " + f(result).ToString());

To test the Newton method, we find the minimum of the same nonlinear function f(x)=1.6x3+3x2-2x used in the Bisection method (see Bisection Method). The user can manipulate initial values as desired.

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