# LEGENDRE_POLYNOMIAL Legendre Polynomials

LEGENDRE_POLYNOMIAL is a Python 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)
```
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.
```

### Languages:

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

### Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a Python library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a Python 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.

GEGENBAUER_POLYNOMIAL, a Python library which evaluates the Gegenbauer polynomial and associated functions.

LEGENDRE_PRODUCT_POLYNOMIAL, a Python library which defines Legendre product polynomials, creating a multivariate polynomial as the product of univariate Legendre polynomials.

LEGENDRE_SHIFTED_POLYNOMIAL, a Python library which evaluates the shifted Legendre polynomial, with domain [0,1].

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

TEST_VALUES, a Python library which supplies test values of various mathematical functions.

### Examples and Tests:

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

Last revised on 17 March 2016.