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 "Numerical Integration Solutions" Gauss-Hermite Method

 N RESULT
 1 2 3 4 5 6 7 8 9 10 11 0 0.886226925452759 0.886226925452759 0.886226925452758 0.88622692545276 0.88622692545276 0.88622692545276 0.88622692545276 0.886226925452762 0.886226925452758 0.886226925452759

 EXACT RESULT = 0.88622692542758

 (Actually after 4 is a repetition) Results from the Gauss-Hermite method

IMPLEMENTATION
Gauss-Hermite Integration

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

 I = ∫∞∞ e-x2 x2dx

The delegate function is simply x2, since the weighting function w(x) = e-x2 for the Gauss-Hermite integration.

Running this example creates the results shown above. The exact result of this integral is equal to √π / 2. The result for n = 3 is already very close to the exact result.

Testing the Gauss-Hermite 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 TestGaussHermite();               {                  ListBox1.Items.Clear();                  ListBox2.Items.Clear();                  double result;                  for (int n = 1; n < 9; n++)                  (                    result = Integration.GaussHermite(f4, n);                    ListBox1.Items.Add(" " + n + );                    ListBox2.Items.Add(" " + result);                  )               }

 Other Implementations...

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EIGENVALUE
SOLUTIONS...

> Rayleigh-Quotient Method

> Cubic Spline Method

 Applied Mathematical Algorithms
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