CHEBYSHEV_POLYNOMIAL
Chebyshev Polynomials


CHEBYSHEV_POLYNOMIAL is a Python library 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)
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

CHEBYSHEV_POLYNOMIAL is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a Python library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CLAUSEN, a Python library which evaluates a Chebyshev interpolant to the Clausen function Cl2(x).

GEGENBAUER_POLYNOMIAL, a Python library which evaluates the Gegenbauer polynomial and associated functions.

LEGENDRE_POLYNOMIAL, a Python library which evaluates the Legendre polynomial and associated functions.

LEGENDRE_SHIFTED_POLYNOMIAL, a Python library which evaluates the shifted Legendre polynomial, with domain [0,1].

POLPAK, a Python library which evaluates a variety of mathematical functions.

TEST_VALUES, a Python library which supplies test values of various mathematical functions.

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.

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Last revised on 21 July 2015.