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)
The computer code and data files described and made available on this web page are distributed under the MIT license
chebyshev_polynomial is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
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.
CLAUSEN, a C code which evaluates a Chebyshev interpolant to the Clausen function Cl2(x).
GEGENBAUER_POLYNOMIAL, a C code which evaluates the Gegenbauer polynomial and associated functions.
HERMITE_POLYNOMIAL, a C code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.
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.
POLPAK, a C code which evaluates a variety of mathematical functions.
TEST_VALUES, a C code which supplies test values of various mathematical functions.