# CIRCLE_RULE Quadrature Rules for the Unit Circle

CIRCLE_RULE is a FORTRAN90 library which computes quadrature rules for approximating integrals over the circumference of the unit circle.

The user specifies the value NT, the number of equally spaced angles. The program returns vectors T(1:NT) and W(1:NT), which define the rule Q(f).

Given NT and the vectors T and W, the integral I(f) of a function f(x,y) is estimated by Q(f) as follows:

```        q = 0.0
for i = 1, nt
x = cos ( t(i) )
y = sin ( t(i) )
q = q + w(j) * f ( x, y )
end
```

### Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

### Languages:

CIRCLE_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:

CIRCLE_ARC_GRID, a FORTRAN90 program which computes points equally spaced along a circular arc;

CIRCLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the circumference of the unit circle in 2D.

CIRCLE_MONTE_CARLO, a FORTRAN90 library which applies a Monte Carlo method to estimate the integral of a function on the circumference of the unit circle in 2D;

CUBE_FELIPPA_RULE, a FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

DISK_RULE, a FORTRAN90 library which computes quadrature rules over the interior of the unit disk in 2D.

PYRAMID_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

PYRAMID_RULE, a FORTRAN90 program which computes a quadrature rule for approximating integrals over the interior of the unit pyramid in 3D.

SPHERE_LEBEDEV_RULE, a FORTRAN90 library which computes Lebedev quadrature rules on the surface of the unit sphere in 3D.

SQUARE_FELIPPA_RULE, a FORTRAN90 library which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

STROUD, a FORTRAN90 library which defines quadrature rules for a variety of M-dimensional regions, including the interior of the square, cube and hypercube, the pyramid, cone and ellipse, the hexagon, the M-dimensional octahedron, the circle, sphere and hypersphere, the triangle, tetrahedron and simplex, and the surface of the circle, sphere and hypersphere.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a FORTRAN90 library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

### Reference:

1. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.

### List of Routines:

• CIRCLE_RULE computes a quadrature rule for the unit circle.
• CIRCLE01_MONOMIAL_INTEGRAL: integral on circumference of unit circle in 2D.
• R8_EPSILON returns the R8 roundoff unit.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8VEC_PRINT prints an R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.

Last revised on 05 April 2014.