# jacobi_rule

jacobi_rule, a FORTRAN90 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))
```

### 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 and an Octave version.

### Related Data and Programs:

alpert_rule, a FORTRAN90 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 FORTRAN90 code which defines a nested Clenshaw Curtis quadrature rule.

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 a variety of routines for numerical estimation of integrals in 1D.

JACOBI_POLYNOMIAL, a FORTRAN90 code which evaluates the Jacobi polynomial and associated functions.

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

LEGENDRE_RULE, a FORTRAN90 code which computes a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, a FORTRAN90 code which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

LINE_NCC_RULE, a FORTRAN90 code which computes a Newton Cotes Closed (NCC) quadrature rule for the line, that is, for an interval of the form [A,B], using equally spaced points which include the endpoints.

LINE_NCO_RULE, a FORTRAN90 code which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

LOGNORMAL_RULE, a FORTRAN90 code which can compute and print a quadrature rule for functions of a variable whose logarithm is normally distributed.

PATTERSON_RULE, a FORTRAN90 code which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

PATTERSON_RULE_COMPUTE, a FORTRAN90 code which computes the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

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

TRUNCATED_NORMAL_RULE, a FORTRAN90 code which computes a quadrature rule for a normal 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,