legendre_polynomial


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

The Legendre polynomial P(n,x) can be defined by:

        P(0,x) = 1
        P(1,x) = x
        P(n,x) = (2*n-1)/n * x * P(n-1,x) - (n-1)/n * P(n-2,x)
      
where n is a nonnegative integer.

The N zeroes of P(n,x) are the abscissas used for Gauss-Legendre quadrature of the integral of a function F(X) with weight function 1 over the interval [-1,1].

The Legendre polynomials are orthogonal under the inner product defined as integration from -1 to 1:

        Integral ( -1 <= x <= 1 ) P(i,x) * P(j,x) dx 
          = 0 if i =/= j
          = 2 / ( 2*i+1 ) if i = j.
      

Licensing:

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

Languages:

legendre_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_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.

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_EXACTNESS, a C code which tests the monomial exactness of quadrature rules for the Legendre problem of integrating a function with density 1 over the interval [-1,+1].

legendre_polynomial_test

LEGENDRE_PRODUCT_POLYNOMIAL, a C code which defines Legendre product polynomials, creating a multivariate polynomial as the product of univariate Legendre polynomials.

LEGENDRE_RULE, a C code which computes a 1D Gauss-Legendre quadrature rule.

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.

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. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
    NIST Handbook of Mathematical Functions,
    Cambridge University Press, 2010,
    ISBN: 978-0521192255,
    LC: QA331.N57.
  4. Gabor Szego,
    Orthogonal Polynomials,
    American Mathematical Society, 1992,
    ISBN: 0821810235,
    LC: QA3.A5.v23.

Source Code:


Last revised on 10 July 2019.