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 information on this web page is 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 an Octave version and a Python version.

Related Data and Programs:

legendre_polynomial_test

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.

legendre_rule, a MATLAB code which computes a 1d Gauss-Legendre quadrature rule.

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

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.

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. 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;