# jacobi_rule

jacobi_rule, a MATLAB code which generates a specific Gauss-Jacobi quadrature rule, based on user input.

The rule is written to three files for easy use as input to other programs.

The Gauss-Jacobi quadrature rule is used as follows:

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

a b

### Usage:

jacobi_rule ( order, alpha, beta, a, b, 'filename' )
where
• order is the number of points in the quadrature rule.
• alpha is the exponent of (B-x), which must be greater than -1.
• beta is the exponent of (x-A), which must be greater than -1.
• a is the left endpoint;
• b is the right endpoint.
• 'filename' specifies how the rule is to be reported: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

### Languages:

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

### Related Data and Programs:

alpert_rule, a MATLAB code which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

ccn_rule, a MATLAB code which defines a nested Clenshaw Curtis quadrature rule.

chebyshev1_rule, a MATLAB code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

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

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

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

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

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

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

jacobi_polynomial, a MATLAB code which evaluates the Jacobi polynomial and associated functions.

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

legendre_rule, a MATLAB code which computes a Gauss-Legendre quadrature rule.

line_felippa_rule, a MATLAB code which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

patterson_rule, a MATLAB 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_jacobi, a dataset directory which contains triples of files defining Gauss-Jacobi quadrature rules.

truncated_normal_rule, a MATLAB 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,