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"Special Functions"
Chebyshev Polynomial Method

    Chebyshev polynomial using "ChebyshevT method"

    Chebyshev polynomial using "ChebyshevU method"




Order of Chebyshev Polynomial:
Entries:



[ Initial Order of Chebyshev Polynomial: {15) ]
[ Initial entries number: {11} ]

IMPLEMENTATION
Chebyshev Polynomial Method

Chebyshev polynomials are important in numerical analysis. There are two types of Chebyshev polynomials, both of which are solutions to the Chebyshev difference equations:

(1 - x2)y'' - xy' + n2y = 0     (first kind)
(1 - x2)y'' - 3xy' + n(n + 2)y = 0     (second kind)

The Chebyshev polynomials of the first kind are defined by the recurrence relation:

T0(x) = 1
T1(x) = x
T2(x) = 2x2 - 1
Tn+1(x) = 2xTn(x) - Tn-1(x)

The Chebyshev polynomials of the second kind are defined by:

U0(x) = 1
U1(x) = 2x
U2(x) = 4x2 - 1
Un+1(x) = 2xUn(x) - Un-1(x)


Testing the Chebyshev Polynomial Method

To test the Chebyshev Polynomial method, new static methods ChevyshevT - (first kind) and ChevyshevU - (second kind) has been added. The implementation of these type of polynomials are based on their recurrence relation. The TestChebyshevPolynomial() method has been written, added and executed:

           static void TestChebyshev Polynomial();
              {
                 for (int i = 0; i < t2; i++)
                 {
                     double x = 0.25 * (i - 5.0);
                     ListBox1.Items.Add(" x = " + x + ".00, " + "T" + t1 + "(x) = " + SpecialFunctions.ChebyshevT(x, t1).ToString());
                     ListBox2.Items.Add(" x = " + x + ".00, " + "U" + t1 + "(x) = " + SpecialFunctions.ChebyshevU(x, t1).ToString());
                 }
              }



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