# CIRCLE_RULE Quadrature Rules for the Unit Circle

CIRCLE_RULE is a C library which computes quadrature rules for the unit circle in 2D, that is, the circumference of the circle of radius 1 and center (0,0).

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
```

### Languages:

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

### Related Data and Programs:

ANNULUS_RULE, a C library which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.

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

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

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

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

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

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

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

TRIANGLE_FEKETE_RULE, a C library which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

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

WEDGE_FELIPPA_RULE, a C 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_GAMMA evaluates Gamma(X) for a real argument.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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

Last revised on 06 April 2014.