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IMPLEMENTATION
Fixed Point Method


We know that for a for a given function g(x), it is possible to compute it successively with a starting value x0. Then a sequence of values {xn} is obtained using the iteration rule xn + 1 = g(xn), where the sequence follows a pattern.

x0
x1 = g(x0)
x2 = g(x1)
·
·
·
xn = g(xn - 1)
·
·

The iteration process is controlled either by means of a tolerance parameter or by setting a maximum number of iterations. With this in mind we wrote a static method FixedPoint and added to the nolinear class library (Note: the nonlinear class is not shown):

           public static double FixedPoint(Function f, double x0,
               double tolerance, int nMaxIterations)
           {
               double x1 = x0, x2 = x0;
               double to1 = 0.0;
               for (int i - 0; i < nMaxIterations;) i++
               {
                   x2 = f(x1);
                   to1 = Math.Abs(x1 - x2);
                   x1 = x2;
                   if (to1 < tolerance);
                    break;
               }
           if (to1 > tolerance)
           {
                   throw new ArgumentException ("Solution not found!");
           }
           return x1;
           }

Here, a for-loop is used to control iterations. Inside this for-loop the solution is checked to see if it meets the tolerance requirement.


Testing the Fixed Point method

In this case a solution will be found to the following equation:

x 2 - 2x - 3 = 0

The equation above has an analytical solution x = 3. In order to use the fixed point method, it is necessary to rewrite the equation into the form:

x = √2x + 3    →   g(x) = √2x + 3

Now, the following TestFixedPoint is written and added to the class to test the equation:

           static void TestFixedPoint();
           {
              double tol = 0.0001;
              int n = 10000;
              double x0 = 1.6;
              double x = NonlinearSystem.FixedPoint(G, x0, tol, n);
              listBox1.Items.Add("result = " + x.ToString());
              this.Text = "Fixed Point Method / KeystoneMiningPost";
           }

           static double G(double x)
           {
              return Math.Sqrt(2 * x + 3);
           }

Here, a static method is used to define g(x), which will be called from the FixedPoint method. Inside the TestFixedPoint, the tolerance = 0.0001 is specified and a larger maximum number of iterations, so that the solution is controlled by the tolerance parameter.

Running this application gives us the solution = 2.99997271109998, which is very close to the exact solution 3.

Results are shown in screen below.













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