# legendre_polynomial

legendre_polynomial, an Octave 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.
```

### Licensing:

The computer code and data files described and made available on this web page are 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_rule, an Octave code which computes a 1d gauss-legendre quadrature rule.

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

pce_legendre, an Octave code which assembles the system matrix of a 2d stochastic pde, using a polynomal chaos expansion in terms of legendre polynomials;

polpak, an Octave code which evaluates a variety of mathematical functions.

test_values, an Octave 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:

Last revised on 12 January 2021.