chebyshev_polynomial

chebyshev_polynomial, a MATLAB 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)

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

chebyshev_polynomial is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

chebyshev_polynomial_test

chebyshev, a MATLAB code which computes the Chebyshev interpolant/approximant to a given function over an interval.

chebyshev_series, a MATLAB 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 MATLAB code which computes and prints a gauss-chebyshev type 1 quadrature rule.

chebyshev2_rule, a MATLAB code which compute and print a gauss-chebyshev type 2 quadrature rule.

companion_matrix, a MATLAB code which computes the companion matrix for a polynomial. The polynomial may be represented in the standard monomial basis, or as a sum of Chebyshev, Gegenbauer, Hermite, Laguerre, or Lagrange basis polynomials. All the roots of the polynomial can be determined as the eigenvalues of the corresponding companion matrix.

matlab_polynomial, a MATLAB code which analyzes a variety of polynomial families, returning the polynomial values, coefficients, derivatives, integrals, roots, or other information.

polpak, a MATLAB code which evaluates a variety of mathematical functions.

polynomial_conversion, a MATLAB code which converts representations of a polynomial between monomial, Bernstein, Chebyshev, Hermite, Lagrange, Laguerre and other forms.

test_values, a MATLAB code 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.

Source Code:

Last revised on 12 January 2021.