legendre_polynomial, a FORTRAN77 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.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
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.
bernstein_polynomial, a FORTRAN77 library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;
CHEBYSHEV_POLYNOMIAL, a FORTRAN77 library 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.
HERMITE_POLYNOMIAL, a FORTRAN77 library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.
JACOBI_POLYNOMIAL, a FORTRAN77 library which evaluates the Jacobi polynomial and associated functions.
LAGUERRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.
LEGENDRE_EXACTNESS, a FORTRAN77 program 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_PRODUCT_POLYNOMIAL, a FORTRAN77 library which defines Legendre product polynomials, creating a multivariate polynomial as the product of univariate Legendre polynomials.
LEGENDRE_RULE, a FORTRAN77 program which computes a 1D Gauss-Legendre quadrature rule.
POLPAK, a FORTRAN77 library which evaluates a variety of mathematical functions.
TEST_VALUES, a FORTRAN77 library which supplies test values of various mathematical functions.