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 "Numerical Integration Solutions" Gauss-Chebyshev Method

 N RESULT
 1 2 3 4 5 6 7 8 9 10 11 3.14159265358979 0.785398163397448 1.17809724509617 1.17809724509617 1.17809724509617 1.17809724509617 1.17809724509617 1.17809724509617 1.17809724509617 1.17809724509617 1.17809724509617

 (Actually after 2 is a repetition) Results from the Gauss-Chebyshev method

IMPLEMENTATION
Gauss-Chebyshev Integration

To compute the following Gaussian integral using the Gauss-Chebyshev method:

 I = ∫-11 (1 - x2) 3/2 dx

The delegate function is simply (1 - x2)2, since the weighting function w(x) = 1 / √1 - x2 for the Gauss-Chebyshev integration.

Running this example creates the results shown above. The exact result of this integral is equal to 3π / 8. The result for n = 2 already gives the exact result, which is expected because the function (1 - x2)2 is a polynomial of degree four, meaning the Gauss-Chebyshev integration is exact with three nodes.

Testing the Gauss-Chebyshev Integration Method

In order to test the Gauss-Hermite method as defined above, a new TestGaussHermite() static method has been added and executed. Supporting code and methods are not shown.

 ► static void TestGaussChebyshev();               {                  ListBox1.Items.Clear();                  ListBox2.Items.Clear();                  double result;                  for (int n = 1; n < t1; n++)                  (                    result = Integration.GaussChebyshev(f5, n);                    ListBox1.Items.Add(" " + n + );                    ListBox2.Items.Add(" " + result);                  )               }

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