LEGENDRE_RULE_FAST, a MATLAB program which implements a fast algorithm for the computation of the points and weights of the Gauss-Legendre quadrature rule.

The standard algorithm for computing the N points and weights of such a rule is by Golub and Welsch. It sets up and solves an eigenvalue problem, whose solution requires work of order N*N.

By contrast, the fast algorithm, by Glaser, Liu and Rokhlin, can compute the same information expending work of order N. For quadrature problems requiring high accuracy, where N might be 100 or more, the fast algorithm provides a significant improvement in speed.

The Gauss-Legendre quadrature rule is designed for the interval [-1,+1].

The Gauss-Legendre quadrature assumes that the integrand has the form:

```        Integral ( -1 <= x <= +1 ) f(x) dx
```

The standard Gauss-Legendre quadrature rule is used as follows:

```        Integral ( -1 <= x <= +1 ) f(x) dx
```
is to be approximated by
```        Sum ( 1 <= i <= order ) w(i) * f(x(i))
```

This program allows the user to request that the rule be transformed from the standard interval [-1,+1] to the interval [a,b].

### Usage:

legendre_rule_fast ( n, a, b )
where
• n is the order (number of points);
• a is the left endpoint (often -1.0 or 0.0);
• b is the right endpoint (usually 1.0).

### Languages:

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

### Related Data and Programs:

CHEBYSHEV1_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a MATLAB program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, a MATLAB program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a MATLAB program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a MATLAB program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a MATLAB program which can compute and print a Gauss-Hermite quadrature rule.

JACOBI_RULE, a MATLAB program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a MATLAB program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a MATLAB program which can compute and print a Gauss-Legendre quadrature rule.

PATTERSON_RULE, a MATLAB program which computes a Gauss-Patterson quadrature rule.

PRODUCT_RULE, a MATLAB program which constructs a product rule from 1D factor rules.

### Reference:

1. Andreas Glaser, Xiangtao Liu, Vladimir Rokhlin,
A fast algorithm for the calculation of the roots of special functions,
SIAM Journal on Scientific Computing,
Volume 29, Number 4, pages 1420-1438, 2007.

### Source Code:

Last revised on 14 February 2019.