LEGENDRE_RULE is a FORTRAN77 program which generates a Gauss-Legendre quadrature rule, based on user input.

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

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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

CCN_RULE, a FORTRAN77 program which defines one of a set of nested Clenshaw Curtis quadrature rules of any order.

CLENSHAW_CURTIS_RULE, a FORTRAN77 program which defines a Clenshaw Curtis quadrature rule.

HERMITE_RULE, a FORTRAN77 program which can compute and print a Gauss-Hermite quadrature rule.

LAGUERRE_RULE, a FORTRAN77 program which can compute and print a Gauss-Laguerre quadrature rule for estimating the integral of a function with density exp(-x) over the interval [0,+oo).

LEGENDRE_EXACTNESS, a FORTRAN77 program which tests the monomial exactness of quadrature rules for the Legendre problem of integrating a function with density 1 over the interval [-1,+1].

LEGENDRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Legendre polynomial and associated functions.

LEGENDRE_RULE_FAST, a FORTRAN77 program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

LINE_FELIPPA_RULE, a FORTRAN77 library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

LINE_NCC_RULE, a FORTRAN77 library 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 FORTRAN77 library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

PATTERSON_RULE, a FORTRAN77 program 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.

TRUNCATED_NORMAL_RULE, a FORTRAN77 program 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,
Prentice Hall, 1966,
LC: QA299.4G3S7.

### Examples and Tests:

• leg_o4_r.txt, the region file created by the command
```
legendre_rule 4 -1.0 +1.0 leg_o4
```
• leg_o4_w.txt, the weight file created by the command
```
legendre_rule 4 -1.0 +1.0 leg_o4
```
• leg_o4_x.txt, the abscissa file created by the command
```
legendre_rule 4 -1.0 +1.0 leg_o4
```

### List of Routines:

• MAIN is the main program for LEGENDRE_RULE.
• CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
• CGQF computes knots and weights of a Gauss quadrature formula.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• GET_UNIT returns a free FORTRAN unit number.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• PARCHK checks parameters ALPHA and BETA for classical weight functions.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8MAT_WRITE writes an R8MAT file.
• RULE_WRITE writes a quadrature rule to a file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 value from a string.
• SCQF scales a quadrature formula to a nonstandard interval.
• SGQF computes knots and weights of a Gauss Quadrature formula.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN77 source codes.

Last revised on 28 September 2013.