# legendre_rule

legendre_rule, an Octave code which generates a specific Gauss-Legendre quadrature rule, based on user input.

The Gauss-Legendre quadrature rule is used as follows:

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

### Usage:

legendre_rule ( order, a, b, 'filename' )
where
• order is the number of points in the quadrature rule.
• a is the left endpoint;
• b is the right endpoint.
• 'filename' specifies the output filenames. filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

### Languages:

legendre_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

### Related Data and Programs:

alpert_rule, an Octave code which sets up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

ccn_rule, an Octave code which defines a nested clenshaw curtis quadrature rule.

chebyshev1_rule, an Octave code which can compute and print a gauss-chebyshev type 1 quadrature rule.

chebyshev2_rule, an Octave code which can compute and print a gauss-chebyshev type 2 quadrature rule.

clenshaw_curtis_rule, an Octave code which defines a clenshaw curtis quadrature rule.

gegenbauer_rule, an Octave code which can compute and print a gauss-gegenbauer quadrature rule.

gen_hermite_rule, an Octave code which can compute and print a generalized gauss-hermite quadrature rule.

gen_laguerre_rule, an Octave code which can compute and print a generalized gauss-laguerre quadrature rule.

hermite_rule, an Octave code which can compute and print a gauss-hermite quadrature rule.

jacobi_rule, an Octave code which can compute and print a gauss-jacobi quadrature rule.

laguerre_rule, an Octave code which can compute and print a gauss-laguerre quadrature rule.

legendre_rule_fast, an Octave code which uses a fast (order n) algorithm to compute a gauss-legendre quadrature rule of given order.

line_felippa_rule, an Octave code which returns the points and weights of a felippa quadrature rule over the interior of a line segment in 1d.

patterson_rule, an Octave code which computes a gauss-patterson quadrature rule.

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.

truncated_normal_rule, an Octave code which computes a quadrature rule for a normal probability density function (pdf), also called a gaussian distribution, that has been truncated to [a,+oo), (-oo,b] or [a,b].

### Reference:

1. Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
3. Sylvan Elhay, Jaroslav Kautsky,
Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
ACM Transactions on Mathematical Software,
Volume 13, Number 4, December 1987, pages 399-415.
4. Jaroslav Kautsky, Sylvan Elhay,
Calculation of the Weights of Interpolatory Quadratures,
Numerische Mathematik,
Volume 40, 1982, pages 407-422.
5. Roger Martin, James Wilkinson,
The Implicit QL Algorithm,
Numerische Mathematik,
Volume 12, Number 5, December 1968, pages 377-383.
6. Arthur Stroud, Don Secrest,