legendre_polynomial, a Python 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 MIT license
legendre_polynomial is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
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