# legendre_fast_rule

legendre_fast_rule, a FORTRAN90 code 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_fast_rule 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_fast_rule 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 FORTRAN90 code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

chebyshev2_rule, a FORTRAN90 code which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

clenshaw_curtis_rule, a FORTRAN90 code which defines a Clenshaw Curtis quadrature rule.

gegenbauer_rule, a FORTRAN90 code which can compute and print a Gauss-Gegenbauer quadrature rule.

gen_hermite_rule, a FORTRAN90 code which can compute and print a generalized Gauss-Hermite quadrature rule.

gen_laguerre_rule, a FORTRAN90 code which can compute and print a generalized Gauss-Laguerre quadrature rule.

hermite_rule, a FORTRAN90 code which can compute and print a Gauss-Hermite quadrature rule.

intlib, a FORTRAN90 code which contains routines for numerical estimation of integrals in 1D.

jacobi_rule, a FORTRAN90 code which can compute and print a Gauss-Jacobi quadrature rule.

laguerre_rule, a FORTRAN90 code which can compute and print a Gauss-Laguerre quadrature rule.

legendre_rule, a FORTRAN90 code which can compute and print a Gauss-Legendre quadrature rule.

patterson_rule, a FORTRAN90 code which computes a Gauss-Patterson quadrature rule.

product_rule, a FORTRAN90 code which constructs a product rule from 1D factor rules.

quadpack, a FORTRAN90 code which contains routines for numerical estimation of integrals in 1D.

quadrature_rules, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

quadrature_rules_legendre, a dataset directory which contains triples of files defining standard Gauss-Legendre quadrature rules.

test_int, a FORTRAN90 code which contains number of functions that may be used as test integrands for quadrature rules in 1D.

### 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 26 July 2020.