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Laguerre Polynomial Method

Order of Laguerre Polynomial:

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

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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