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 "Special Functions" Chebyshev Polynomial Method

Chebyshev polynomial using "ChebyshevT method"

 x = -1.25.00, T15(x) = -16384.0000152588 x = -1.00, T15(x) = -1 x = -0.75.00, T15(x) = 0.153945922851563 x = -0.5.00, T15(x) = 1 x = -0.25.00, T15(x) = -0.604080200195313 x = 0.00, T15(x) = 0 x = 0.25.00, T15(x) = 0.604080200195313 x = 0.5.00, T15(x) = -1 x = 0.75.00, T15(x) = -0.153945922851563 x = 1.00, T15(x) = 1 x = 1.25.00, T15(x) = 16384.0000152588

Chebyshev polynomial using "ChebyshevU method"

 x = -1.25.00, U15(x) = -43690.6666564941 x = -1.00, U15(x) = -16 x = -0.75.00, U15(x) = 1.27432250976563 x = -0.5.00, U15(x) = 1 x = -0.25.00, U15(x) = -0.809844970703125 x = 0.00, U15(x) = 0 x = 0.25.00, U15(x) = 0.809844970703125 x = 0.5.00, U15(x) = -1 x = 0.75.00, U15(x) = -1.27432250976563 x = 1.00, U15(x) = 16 x = 1.25.00, U15(x) = 43690.6666564941

 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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