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HOME › AREAS OF EXPERTISE › Special Functions  › ~ Laguerre Polynomial Method

 "Special Functions" Laguerre Polynomial Method
 x = -1.00, L20(x) = 709.931791843359 x = 0.00, L20(x) = 1 x = 1.00, L20(x) = -0.164258811827793 x = 2.00, L20(x) = 0.50072156756003 x = 3.00, L20(x) = -0.557509325217191 x = 4.00, L20(x) = -0.150791428102111 x = 5.00, L20(x) = 2.02022574447691

 Order of Laguerre Polynomial: Entries: [ Initial Order of Laguerre Polynomial: {20) ] [ Initial entries number: {7} ]

IMPLEMENTATION
Laguerre Polynomial Method

The Laguerre polynomial is the standard solution of the Laguerre equation:

 xy'' + ( 1-x )y' + ny = 0

Which is a second-order linear differential equation. This equation has non-singular solutions only if n is a positive integer.

The polynomials from the standard solutions of the above equation, usually denoted L0,,L1,2,... form a polynomial sequence which can be defined by the Rodrigues formula:

 Ln(x) = ex / n! dn / dxn(e-xxn)

Can easily derive the first few polynomials from the above equation:

 L0(x) = 1 L1(x) = -x + 1 L2(x) = 1/2( x2- 4x + 2)

Testing the Laguerre Polynomial Method

To test the Laguerre Polynomial method, a new static method has been added. The implementation of the type of polynomials are based on their recurrence relation. The TestLaguerrePolynomial() method has been written, added and executed:

static void TestLaGuerrePolynomial();
{
for (int i = 0; i < t2; i++)
{
double x = 1.0 * i - 1.0;
ListBox1.Items.Add(" x = " + x + ".00, " + "L" + t1 + "(x) = " + SpecialFunctions.Laguerre(x, t1).ToString());
}
}

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