LEGENDRE_POLYNOMIAL
Legendre Polynomials

LEGENDRE_POLYNOMIAL is a MATLAB library 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 GNU LGPL license.

Languages:

LEGENDRE_POLYNOMIAL is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CHEBYSHEV_POLYNOMIAL, a MATLAB library which evaluates the Chebyshev polynomial and associated functions.

HERMITE_POLYNOMIAL, a MATLAB library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

INT_EXACTNESS_LEGENDRE, a MATLAB program which tests the polynomial exactness of Gauss-Legendre quadrature rules.

JACOBI_POLYNOMIAL, a MATLAB library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a MATLAB library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_RULE, a MATLAB program which computes a 1D Gauss-Legendre quadrature rule.

PCE_LEGENDRE, a MATLAB program which assembles the system matrix of a 2D stochastic PDE, using a polynomal chaos expansion in terms of Legendre polynomials;

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

TEST_VALUES, a MATLAB 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. 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;
• p_exponential_product.m, exponential products for P(n,x).
• p_integral.m, evaluates a monomial integral associated with P(n,x).
• p_polynomial.m, evaluates the Legendre polynomials 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_values.m, returns 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.m, evaluates the Legendre polynomials Pm(n,m,x).
• pm_polynomial_values.m, returns values of Legendre polynomials Pm(n,m,x).
• pn_polynomial.m, evaluates the normalized Legendre polynomials Pn(n,x).
• pn_pair_product.m, pair products for normalized Legendre polynomial Pn(n,x).
• r8_factorial.m, computes the factorial function;
• 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_print.m, prints an R8VEC;
• r8vec2_print.m, prints a pair of R8VEC's;
• timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• legendre_polynomial_test.m, calls all the tests.
• legendre_polynomial_test_output.txt, the output file.
• legendre_polynomial_test01.m, calls tests P_POLYNOMIAL.
• legendre_polynomial_test02.m, tests P_POLYNOMIAL_COEFFICIENTS.
• legendre_polynomial_test03.m, tests P_POLYNOMIAL_ZEROS.
• legendre_polynomial_test04.m, tests P_QUADRATURE_RULE.
• legendre_polynomial_test05.m, tests P_EXPONENTIAL_PRODUCT.
• legendre_polynomial_test06.m, tests P_POWER_PRODUCT.
• legendre_polynomial_test07.m, tests PM_POLYNOMIAL.
• legendre_polynomial_test08.m, tests PMN_POLYNOMIAL.
• legendre_polynomial_test09.m, tests PMNS_POLYNOMIAL.
• legendre_polynomial_test10.m, tests PN_PAIR_PRODUCT.
• legendre_polynomial_plot01.m, tests p_polynomial_plot;
• p_polynomial_plot.png, a plot of Legendre functions 0 through 5;

You can go up one level to the MATLAB source codes.