# disk01_rule

disk01_rule, a Python code which computes a quadrature rule over the interior of the unit disk in 2D.

The user specifies values NT and NR, where NT is the number of equally spaced angles, and NR controls the number of radial points. The program returns vectors T(1:NT), R(1:NR) and W(1:NR), which define the rule Q(f).

To use a rule that is equally powerful in R and T, typically, set NT = 2 * NR.

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

```        s = 0.0
for j in range ( 0, nr ):
for i in range ( 0, nt ):
x = r[j] * np.cos ( t[i] )
y = r[j] * np.sin ( t[i] )
s = s + w[j] * f ( x, y )
area = pi
q = area * s
```

To approximate an integral over a circle with center (XC,YC) and radius RC:

```        s = 0.0
for j in range ( 0, nr ):
for i in range ( 0, nt ):
x = xc + rc * r[j] * np.cos ( t[i] )
y = yc + rc * r[j] * np.sin ( t[i] )
s = s + w[j] * f ( x, y )
area = rc * rc * pi
q = area * s
```

### Licensing:

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

### Languages:

disk01_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

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disk_rule, a Python code which computes quadrature rules over the interior of the general disk in 2D, with radius RC and center (XC,YC).

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### Reference:

1. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
2. 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.

### Source Code:

Last revised on 22 January 2020.