legendre_polynomial

legendre_polynomial, a MATLAB 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)
      

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.

Related Data and Programs:

bernstein_polynomial, a MATLAB code which evaluates the bernstein polynomials, useful for uniform approximation of functions;

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.

gegenbauer_polynomial, a MATLAB code which evaluates the Gegenbauer polynomial and associated functions.

hermite_polynomial, a MATLAB code which evaluates the physicist's hermite polynomial, the probabilist's hermite polynomial, the hermite function, and related functions.

jacobi_polynomial, a MATLAB code which evaluates the jacobi polynomial and associated functions.

laguerre_polynomial, a MATLAB code which evaluates the laguerre polynomial, the generalized laguerre polynomial, and the laguerre function.

legendre_polynomial_test

legendre_product_polynomial, a MATLAB code which defines legendre product polynomials, creating a multivariate polynomial as the product of univariate legendre polynomials.

legendre_rule, a MATLAB code which computes a 1d gauss-legendre quadrature rule.

legendre_shifted_polynomial, a MATLAB code which evaluates the shifted legendre polynomial, with domain [0,1].

lobatto_polynomial, a MATLAB code which evaluates lobatto polynomials, similar to legendre polynomials except that they are zero at both endpoints.

pce_legendre, a MATLAB code which assembles the system matrix of a 2d stochastic pde, using a polynomal chaos expansion in terms of legendre polynomials;

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

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. 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:

• imtqlx.m, diagonalizes a symmetric tridiagonal matrix;
• legendre_to_monomial_matrix.m, returns a matrix that transforms a polynomial expansion from Legendre to monomial form.
• monomial_to_legendre_matrix.m, returns a matrix that transforms a polynomial expansion from monomial to Legendre form.
• p_exponential_product.m, exponential products for P(n,x).
• p_integral.m, evaluates a monomial integral associated with P(n,x).
• p_polynomial_coefficients.m, coefficients of Legendre polynomials P(n,x).
• p_polynomial_plot.m, plots one or more Legendre polynomials P(n,x).
• p_polynomial_prime.m, evaluates the derivative of Legendre polynomials P(n,x).
• p_polynomial_prime2.m, evaluates the second derivative of Legendre polynomials P(n,x).
• p_polynomial_value.m, evaluates the Legendre polynomials P(n,x).
• p_polynomial_values.m, selected values of the Legendre polynomials P(n,x).
• p_polynomial_zeros.m, zeros of Legendre function P(n,x).
• p_power_product.m, power products for Legendre polynomial P(n,x).
• p_quadrature_rule.m, quadrature for Legendre function P(n,x).
• pm_polynomial_value.m, evaluates the Legendre polynomials Pm(n,m,x).
• pm_polynomial_values.m, returns values of Legendre polynomials Pm(n,m,x).
• pmn_polynomial_value.m, evaluates the normalized Legendre polynomials Pmn(n,m,x).
• pmns_polynomial_value.m, evaluates the sphere normalized Legendre polynomials Pmns(n,m,x).
• pn_pair_product.m, pair products for normalized Legendre polynomial Pn(n,x).
• pn_polynomial_coefficients.m, coefficients of the normalized Legendre polynomials Pn(n,x).
• pn_polynomial_value.m, evaluates the normalized Legendre polynomials Pn(n,x).
• pn_polynomial_values.m, selected values of the normalized Legendre polynomials Pn(n,x).
• r8_sign.m, returns the sign of an R8.
• r8mat_print.m, prints an R8MAT;
• r8mat_print_some.m, prints some of an R8MAT;
• r8vec_linspace.m, creates a column vector of linearly spaced values.
• r8vec_print.m, prints an R8VEC;
• r8vec2_print.m, prints a pair of R8VEC's;