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<html>
<head>
<title>
SPHERE_EXACTNESS - Exactness of Quadrature on the Unit Sphere
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPHERE_EXACTNESS <br>
Exactness of Quadrature on the Unit Sphere
</h1>
<hr>
<p>
<b>SPHERE_EXACTNESS</b>
is a C++ program which
investigates the polynomial exactness of a quadrature rule
over the surface of the unit sphere in 3D.
</p>
<p>
The polynomial exactness of a quadrature rule is defined as the
highest total degree <b>D</b> such that the quadrature rule is
guaranteed to integrate exactly all polynomials of total degree
<b>DEGREE_MAX</b> 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 <b>x</b>, <b>y</b> and <b>z</b>.
Thus, for instance, the <b>DEGREE</b> of
<blockquote><b>
x<sup>2</sup>yz<sup>5</sup>
</b></blockquote>
is 2+1+5=8.
</p>
<p>
To be thorough, the program starts at <b>DEGREE</b> = 0, and then
proceeds to <b>DEGREE</b> = 1, 2, and so on up to a maximum degree
<b>DEGREE_MAX</b> specified by the user. At each value of <b>DEGREE</b>,
the program generates every possible monomial term, applies the
quadrature rule to it, and determines the quadrature error.
</p>
<p>
The quadrature rule is read by the program, and there are a number of
options for how that information is stored.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>sphere_exactness</b> <i>files</i> <i>prefix</i> <i>degree_max</i>
</blockquote>
where
<ul>
<li>
<i>files</i> indicates how the quadrature rule is stored (see below);
</li>
<li>
<i>prefix</i> is the common prefix for the files;
</li>
<li>
<i>degree_max</i> 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!
</li>
</ul>
</p>
<p>
The input value of <b>files</b> 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.
<ul>
<li>
<b>XYZW</b> for file <i>prefix</i>.<b>xyzw</b> containing (X,Y,Z,Weight);
</li>
<li>
<b>RTPW</b> for file <i>prefix</i>.<b>rtpw</b> containing (Theta, Phi, Weight) (radians);
</li>
<li>
<b>DTPW</b> for file <i>prefix</i>.<b>dtpw</b> containing (Theta, Phi, Weight) (degrees);
</li>
<li>
<b>XYZ+W</b> for file <i>prefix</i>.<b>xyz</b> containing (X,Y,Z)
and file <i>prefix</i>.<b>w</b> containing Weight;
</li>
<li>
<b>RTP+W</b> for file <i>prefix</i>.<b>rtp</b> containing (Theta, Phi ) in radians,
and file <i>prefix</i>.<b>w</b> containing Weight;
</li>
<li>
<b>DTP+W</b> for file <i>prefix</i>.<b>dtp</b> containing (Theta, Phi ) in degrees,
and file <i>prefix</i>.<b>w</b> containing Weight;
</li>
<li>
<b>XYZ1</b> for file <i>prefix</i>.<b>xyz</b> containing (X,Y,Z),
and equal weights, which do not need to be read in.
</li>
<li>
<b>RTP1</b> for file <i>prefix</i>.<b>rtp</b> containing (Theta, Phi ) in radians,
and equal weights, which do not need to be read in.
</li>
<li>
<b>DTP1</b> for file <i>prefix</i>.<b>dtp</b> containing (Theta, Phi ) in degrees,
and equal weights, which do not need to be read in.
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SPHERE_EXACTNESS</b> is available in
<a href = "../../c_src/sphere_exactness/sphere_exactness.html">a C version</a> and
<a href = "../../cpp_src/sphere_exactness/sphere_exactness.html">a C++ version</a> and
<a href = "../../f77_src/sphere_exactness/sphere_exactness.html">a FORTRAN77 version</a> and
<a href = "../../f_src/sphere_exactness/sphere_exactness.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sphere_exactness/sphere_exactness.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/cube_exactness/cube_exactness.html">
CUBE_EXACTNESS</a>,
a C++ library which
investigates the polynomial exactness of quadrature rules
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/hypercube_exactness/hypercube_exactness.html">
HYPERCUBE_EXACTNESS</a>,
a C++ program which
measures the monomial exactness of an M-dimensional quadrature rule
over the interior of the unit hypercube in M dimensions.
</p>
<p>
<a href = "../../cpp_src/pyramid_exactness/pyramid_exactness.html">
PYRAMID_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of the unit pyramid in 3D.
</p>
<p>
<a href = "../../datasets/sphere_design_rule/sphere_design_rule.html">
SPHERE_DESIGN_RULE</a>,
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.
</p>
<p>
<a href = "../../f_src/sphere_design_rule/sphere_design_rule.html">
SPHERE_DESIGN_RULE</a>,
a FORTRAN90 library which
returns 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.
</p>
<p>
<a href = "../../datasets/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a dataset directory which
contains sets of Lebedev points on a unit sphere, which can be used for
quadrature rules of a known precision;
</p>
<p>
<a href = "../../cpp_src/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a C++ library which
computes Lebedev quadrature rules for the unit sphere;
</p>
<p>
<a href = "../../cpp_src/sphere_monte_carlo/sphere_monte_carlo.html">
SPHERE_MONTE_CARLO</a>,
a C++ library which
applies a Monte Carlo method to estimate the integral of a function
over the surface of the unit sphere in 3D;
</p>
<p>
<a href = "../../cpp_src/sphere_quad/sphere_quad.html">
SPHERE_QUAD</a>,
a C++ library which
approximates an integral over the surface of the unit sphere
by applying a triangulation to the surface;
</p>
<p>
<a href = "../../cpp_src/sphere_triangle_quad/sphere_triangle_quad.html">
SPHERE_TRIANGLE_QUAD</a>,
a C++ library which
estimates the integral of a function over a spherical triangle.
</p>
<p>
<a href = "../../cpp_src/square_exactness/square_exactness.html">
SQUARE_EXACTNESS</a>,
a C++ library which
investigates the polynomial exactness of quadrature rules for f(x,y)
over the interior of a rectangle in 2D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_exactness/wedge_exactness.html">
WEDGE_EXACTNESS</a>,
a C++ program which
investigates the monomial exactness of a quadrature rule
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Gerald Folland,<br>
How to Integrate a Polynomial Over a Sphere,<br>
American Mathematical Monthly,<br>
Volume 108, Number 5, May 2001, pages 446-448.
</li>
<li>
Ronald Hardin, Neil Sloane,<br>
McLaren's Improved Snub Cube and Other New Spherical Designs
in Three Dimensions,<br>
Discrete and Computational Geometry,<br>
Volume 15, 1996, pages 429-441.
</li>
<li>
Vyacheslav Lebedev, Dmitri Laikov,<br>
A quadrature formula for the sphere of the 131st
algebraic order of accuracy,<br>
Russian Academy of Sciences Doklady Mathematics,<br>
Volume 59, Number 3, 1999, pages 477-481.
</li>
<li>
AD McLaren,<br>
Optimal Numerical Integration on a Sphere,<br>
Mathematics of Computation,<br>
Volume 17, Number 84, October 1963, pages 361-383.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sphere_exactness.cpp">sphere_exactness.cpp</a>, the source code.
</li>
<li>
<a href = "sphere_exactness.sh">sphere_exactness.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>DESIGN_04</b> is a spherical design of precision 4. Uniform weights
are used.
<ul>
<li>
<a href = "design_04.xyz">design_04.xyz</a> contains the XYZ coordinates.
</li>
<li>
<a href = "design_04_output.txt">design_04_output.txt</a> is the output
from <b>sphere_exactness xyz1 design_04 5</b>.
</li>
</ul>
</p>
<p>
<b>LEBEDEV_13</b> is the 74 Lebedev rule of precision 13. Weights are
specified, and the data is stored in the "DTPW" format, that is, theta
and phi are specified in degrees.
are used.
<ul>
<li>
<a href = "lebedev_013.dtpw">lebedev_013.dtpw</a> contains the
DTPW information.
</li>
<li>
<a href = "lebedev_013_output.txt">lebedev_013_output.txt</a> is the output
from <b>sphere_exactness dtpw lebedev_013 5</b>.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for SPHERE_EXACTNESS.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>MONOMIAL_INT_SPHERE_UNIT</b> integrates a monomial over the unit tetrahedron.
</li>
<li>
<b>MONOMIAL_QUADRATURE_SPHERE</b> applies quadrature to a monomial in a tetrahedron.
</li>
<li>
<b>MONOMIAL_VALUE</b> evaluates a monomial.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>S_EQI</b> is a case insensitive comparison of two strings for equality.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>SPHERE01_MONOMIAL_INT</b> integrates a monomial on the unit sphere.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 10 September 2010.
</i>
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