sphere_exactness

sphere_exactness, a MATLAB code which investigates the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3D.

The polynomial exactness of a quadrature rule is defined as the highest total degree D such that the quadrature rule is guaranteed to integrate exactly all polynomials of total degree DEGREE_MAX or less, ignoring roundoff. The total degree of a polynomial is the maximum of the degrees of all its monomial terms. For the unit sphere in 3D, the degree of a monomial term is the sum of the exponents of x, y and z. Thus, for instance, the DEGREE of

x²yz⁵

To be thorough, the program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates every possible monomial term, applies the quadrature rule to it, and determines the quadrature error.

The quadrature rule is read by the program, and there are a number of options for how that information is stored.

Usage:

sphere_exactness ( 'files', 'prefix', degree_max )

• 'files' indicates how the quadrature rule is stored (see below);
• 'prefix' is the common prefix for the files;
• degree_max is the maximum total monomial degree to check. A value of 5 or 10 might be reasonable, but a value of 50 or 100 is probably never a good input!

The input value of files indicates how the quadrature rule information is stored. It might be in one or two files; the abscissas might be stored as (X,Y,Z) or as angles. The angles might be in radians or degrees. The weights might be assumed to be equal, in which case they need not be input.

• XYZW for file prefix.xyzw containing (X,Y,Z,Weight);
• RTPW for file prefix.rtpw containing (Theta, Phi, Weight) (radians);
• DTPW for file prefix.dtpw containing (Theta, Phi, Weight) (degrees);
• XYZ+W for file prefix.xyz containing (X,Y,Z) and file prefix.w containing Weight;
• RTP+W for file prefix.rtp containing (Theta, Phi ) in radians, and file prefix.w containing Weight;
• DTP+W for file prefix.dtp containing (Theta, Phi ) in degrees, and file prefix.w containing Weight;
• XYZ1 for file prefix.xyz containing (X,Y,Z), and equal weights, which do not need to be read in.
• RTP1 for file prefix.rtp containing (Theta, Phi ) in radians, and equal weights, which do not need to be read in.
• DTP1 for file prefix.dtp containing (Theta, Phi ) in degrees, and equal weights, which do not need to be read in.

Licensing:

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

Languages:

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

Related Data and Programs:

cube_exactness, a MATLAB code which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3d.

hypercube_exactness, a MATLAB code which measures the monomial exactness of an m-dimensional quadrature rule over the interior of the unit hypercube in m dimensions.

pyramid_exactness, a MATLAB code which investigates the polynomial exactness of a quadrature rule over the interior of the unit pyramid in 3d.

sphere_design_rule, a dataset directory which contains files defining point sets on the surface of the unit sphere, known as "designs", which can be useful for estimating integrals on the surface, among other uses.

sphere_exactness_test

sphere_lebedev_rule, a dataset directory which contains sets of lebedev points on a sphere, which can be used for quadrature rules of a known precision;

sphere_lebedev_rule, a MATLAB code which computes lebedev quadrature rules over the surface of the unit sphere in 3d.

sphere_quad, a MATLAB code which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

square_exactness, a MATLAB code which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a square (rectangle/quadrilateral) in 2d.

tetrahedron_exactness, a MATLAB code which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3d.

triangle_exactness, a MATLAB code which investigates the monomial exactness quadrature rule over the interior of a triangle in 2d.

wedge_exactness, a MATLAB code which investigates the monomial exactness of a quadrature rule over the interior of the unit wedge in 3d.

Reference:

1. Gerald Folland,
   How to Integrate a Polynomial Over a Sphere,
   American Mathematical Monthly,
   Volume 108, Number 5, May 2001, pages 446-448.
2. Ronald Hardin, Neil Sloane,
   McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions,
   Discrete and Computational Geometry,
   Volume 15, 1996, pages 429-441.
3. Vyacheslav Lebedev, Dmitri Laikov,
   A quadrature formula for the sphere of the 131st algebraic order of accuracy,
   Russian Academy of Sciences Doklady Mathematics,
   Volume 59, Number 3, 1999, pages 477-481.
4. AD McLaren,
   Optimal Numerical Integration on a Sphere,
   Mathematics of Computation,
   Volume 17, Number 84, October 1963, pages 361-383.

Source Code:

• sphere_exactness.m, the source code.