# chebyshev_polynomial

chebyshev_polynomial, a C++ code which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

The Chebyshev polynomial T(n,x), or Chebyshev polynomial of the first kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:

```        cos ( t ) = x
T(n,x) = cos ( n * t )
```
For any value of x, T(n,x) may be evaluated by a three term recurrence:
```        T(0,x) = 1
T(1,x) = x
T(n+1,x) = 2x T(n,x) - T(n-1,x)
```

The Chebyshev polynomial U(n,x), or Chebyshev polynomial of the second kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:

```        cos ( t ) = x
U(n,x) = sin ( ( n + 1 ) t ) / sin ( t )
```
For any value of x, U(n,x) may be evaluated by a three term recurrence:
```        U(0,x) = 1
U(1,x) = 2x
U(n+1,x) = 2x U(n,x) - U(n-1,x)
```

The Chebyshev polynomial V(n,x), or Chebyshev polynomial of the third kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:

```        cos ( t ) = x
V(n,x) = cos ( (2n+1)*t/2) / cos ( t/2)
```
For any value of x, V(n,x) may be evaluated by a three term recurrence:
```        V(0,x) = 1
V(1,x) = 2x-1
V(n+1,x) = 2x V(n,x) - V(n-1,x)
```

The Chebyshev polynomial W(n,x), or Chebyshev polynomial of the fourth kind, may be defined, for 0 <= n, and -1 <= x <= +1 by:

```        cos ( t ) = x
W(n,x) = sin((2*n+1)*t/2)/sin(t/2)
```
For any value of x, W(n,x) may be evaluated by a three term recurrence:
```        W(0,x) = 1
W(1,x) = 2x+1
W(n+1,x) = 2x W(n,x) - W(n-1,x)
```

### Languages:

chebyshev_polynomial is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a C++ code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV, a C++ code which computes the Chebyshev interpolant/approximant to a given function over an interval.

CHEBYSHEV_SERIES, a C++ code which can evaluate a Chebyshev series approximating a function f(x), while efficiently computing one, two or three derivatives of the series, which approximate f'(x), f''(x), and f'''(x), by Manfred Zimmer.

CHEBYSHEV1_RULE, a C++ code which computes and prints a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a C++ code which compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLAUSEN, a C++ code which evaluates a Chebyshev interpolant to the Clausen function Cl2(x).

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HERMITE_POLYNOMIAL, a C++ code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

INT_EXACTNESS_CHEBYSHEV1, a C++ code which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

INT_EXACTNESS_CHEBYSHEV2, a C++ code which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.

JACOBI_POLYNOMIAL, a C++ code which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a C++ code which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a C++ code which evaluates the Legendre polynomial and associated functions.

LEGENDRE_SHIFTED_POLYNOMIAL, a C++ code which evaluates the shifted Legendre polynomial, with domain [0,1].

JACOBI_POLYNOMIAL, a C++ code which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a C++ code which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a C++ code which evaluates the Legendre polynomial and associated functions.

LEGENDRE_SHIFTED_POLYNOMIAL, a C++ code which evaluates the shifted Legendre polynomial, with domain [0,1].

LOBATTO_POLYNOMIAL, a C++ code which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

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### Reference:

1. Theodore Chihara,
An Introduction to Orthogonal Polynomials,
Gordon and Breach, 1978,
ISBN: 0677041500,
LC: QA404.5 C44.
2. Walter Gautschi,
Orthogonal Polynomials: Computation and Approximation,
Oxford, 2004,
ISBN: 0-19-850672-4,
LC: QA404.5 G3555.
3. John Mason, David Handscomb,
Chebyshev Polynomials,
CRC Press, 2002,
ISBN: 0-8493-035509,
LC: QA404.5.M37.
4. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
NIST Handbook of Mathematical Functions,
Cambridge University Press, 2010,
ISBN: 978-0521192255,
LC: QA331.N57.
5. Gabor Szego,
Orthogonal Polynomials,
American Mathematical Society, 1992,
ISBN: 0821810235,
LC: QA3.A5.v23.

### Source Code:

Last revised on 17 February 2020.