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<html>
<head>
<title>
STROUD - Numerical Integration in M Dimensions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
STROUD <br> Numerical Integration<br> in M Dimensions
</h1>
<hr>
<p>
<b>STROUD</b>
is a FORTRAN90 library which
implements multidimensional quadrature rules of Arthur Stroud.
</p>
<p>
A few other rules have been collected as well,
particularly for quadrature over the interior of a triangle, which is
useful in finite element calculations.
</p>
<p>
Arthur Stroud published his vast collection of quadrature formulas
for multidimensional regions in 1971. In a few cases, he printed
sample FORTRAN77 programs to compute these integrals. Integration
regions included:
<ul>
<li>
<b>C2</b>, the interior of the square;
</li>
<li>
<b>C3</b>, the interior of the cube;
</li>
<li>
<b>CN</b>, the interior of the N-dimensional hypercube;
</li>
<li>
<b>CN:C2</b>, a 3-dimensional pyramid;
</li>
<li>
<b>CN:S2</b>, a 3-dimensional cone;
</li>
<li>
<b>CN_SHELL</b>, the region contained between two concentric
N-dimensional hypercubes;
</li>
<li>
<b>ELP</b>, the interior of the 2-dimensional ellipse with
weight function 1/sqrt((x-c)^2+y^2)/(sqrt((x+c)^2+y^2);
</li>
<li>
<b>EN_R</b>, all of N-dimensional space, with the weight function:<br>
w(x) = exp ( - sqrt ( sum ( 1 <= i < n ) x(i)^2 ) );
</li>
<li>
<b>EN_R2</b>, all of N-dimensional space, with the Hermite weight function:
w(x) = product ( 1 <= i <= n ) exp ( - x(i)^2 );
</li>
<li>
<b>GN</b>, the interior of the N-dimensional octahedron;
</li>
<li>
<b>H2</b>, the interior of the 2-dimensional hexagon;
</li>
<li>
<b>PAR</b>, the first parabolic region;
</li>
<li>
<b>PAR2</b>, the second parabolic region;
</li>
<li>
<b>PAR3</b>, the third parabolic region;
</li>
<li>
<b>S2</b>, the interior of the circle;
</li>
<li>
<b>S3</b>, the interior of the sphere;
</li>
<li>
<b>SN</b>, the interior of the N-dimensional hypersphere;
</li>
<li>
<b>SN_SHELL</b>, the region contained between two concentric N-dimensional hyperspheres;
</li>
<li>
<b>T2</b>, the interior of the triangle;
</li>
<li>
<b>T3</b>, the interior of the tetrahedron;
</li>
<li>
<b>TN</b>, the interior of the N-dimensional simplex;
</li>
<li>
<b>TOR3:S2</b>, the interior of a 3-dimensional torus with circular cross-section;
</li>
<li>
<b>TOR3:C2</b>, the interior of a 3-dimensional torus with square cross-section;
</li>
<li>
<b>U2</b>, the "surface" of the circle;
</li>
<li>
<b>U3</b>, the surface of the sphere;
</li>
<li>
<b>UN</b>, the surface of the N-dimensional sphere;
</li>
</ul>
</p>
<p>
We have added a few new terms for regions:
<ul>
<li>
<b>CN_GEG</b>, the N dimensional hypercube [-1,+1]^N, with the Gegenbauer
weight function:<br>
w(alpha;x) = product ( 1 <= i <= n ) ( 1 - x(i)^2 )^alpha;
</li>
<li>
<b>CN_JAC</b>, the N dimensional hypercube [-1,+1]^N, with the Beta or
Jacobi weight function:<br>
w(alpha,beta;x) = product ( 1 <= i <= n ) ( 1 - x(i) )^alpha * ( 1 + x(i) )^beta;
</li>
<li>
<b>CN_LEG</b>, the N dimensional hypercube [-1,+1]^N, with the Legendre
weight function:<br>
w(x) = 1;
</li>
<li>
<b>EPN_GLG</b>, the positive space [0,+oo)^N, with the generalized
Laguerre weight function:<br>
w(alpha;x) = product ( 1 <= i <= n ) x(i)^alpha exp ( - x(i) );
</li>
<li>
<b>EPN_LAG</b>, the positive space [0,+oo)^N, with the exponential or
Laguerre weight function:<br>
w(x) = product ( 1 <= i <= n ) exp ( - x(i) );
</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>STROUD</b> is available in
<a href = "../../c_src/stroud/stroud.html">a C version</a> and
<a href = "../../cpp_src/stroud/stroud.html">a C++ version</a> and
<a href = "../../f77_src/stroud/stroud.html">a FORTRAN77 version</a> and
<a href = "../../f_src/stroud/stroud.html">a FORTRAN90 version</a> and
<a href = "../../m_src/stroud/stroud.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/dunavant/dunavant.html">
DUNAVANT</a>,
a FORTRAN90 library which
defines Dunavant rules for quadrature on a triangle.
</p>
<p>
<a href = "../../f_src/fekete/fekete.html">
FEKETE</a>,
a FORTRAN90 library which
defines a Fekete rule for quadrature or interpolation over a triangle.
</p>
<p>
<a href = "../../f_src/felippa/felippa.html">
FELIPPA</a>,
a FORTRAN90 library which
defines quadrature rules for lines, triangles, quadrilaterals,
pyramids, wedges, tetrahedrons and hexahedrons.
</p>
<p>
<a href = "../../f_src/gm_rule/gm_rule.html">
GM_RULE</a>,
a FORTRAN90 library which
defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general
M-dimensional simplex.
</p>
<p>
<a href = "../../f_src/keast/keast.html">
KEAST</a>,
a FORTRAN90 library which
defines a number of quadrature rules for a tetrahedron.
</p>
<p>
<a href = "../../f_src/ncc_tetrahedron/ncc_tetrahedron.html">
NCC_TETRAHEDRON</a>,
a FORTRAN90 library which
defines Newton-Cotes closed quadrature rules on a tetrahedron.
</p>
<p>
<a href = "../../f_src/ncc_triangle/ncc_triangle.html">
NCC_TRIANGLE</a>,
a FORTRAN90 library which
defines Newton-Cotes closed quadrature rules on a triangle.
</p>
<p>
<a href = "../../f_src/nco_tetrahedron/nco_tetrahedron.html">
NCO_TETRAHEDRON</a>,
a FORTRAN90 library which
defines Newton-Cotes open quadrature rules on a tetrahedron.
</p>
<p>
<a href = "../../f_src/nco_triangle/nco_triangle.html">
NCO_TRIANGLE</a>,
a FORTRAN90 library which
defines Newton-Cotes open quadrature rules on a triangle.
</p>
<p>
<a href = "../../f_src/product_rule/product_rule.html">
PRODUCT_RULE</a>,
a FORTRAN90 program which
constructs a product rule
from 1D factor rules.
</p>
<p>
<a href = "../../f_src/pyramid_rule/pyramid_rule.html">
PYRAMID_RULE</a>,
a FORTRAN90 program which
computes a quadrature rule for a pyramid.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
contains sets of files that define quadrature
rules over various 1D intervals or multidimensional hypercubes.
</p>
<p>
<a href = "../../datasets/quadrature_rules_tet/quadrature_rules_tet.html">
QUADRATURE_RULES_TET</a>,
a dataset directory which
contains triples of files defining various quadrature
rules on tetrahedrons.
</p>
<p>
<a href = "../../datasets/quadrature_rules_tri/quadrature_rules_tri.html">
QUADRATURE_RULES_TRI</a>,
a collection of quadrature rules which
can be applied to triangular
regions.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules on a
variety of intervals with different weight functions.
</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.
</p>
<p>
<a href = "../../f_src/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a FORTRAN90 library which
computes Lebedev quadrature rules for the unit sphere;
</p>
<p>
<a href = "../../f_src/sphere_triangle_quad/sphere_triangle_quad.html">
SPHERE_TRIANGLE_QUAD</a>,
a FORTRAN90 library which
estimates the integral of a function over a spherical triangle.
</p>
<p>
<a href = "../../f_src/test_nint/test_nint.html">
TEST_NINT</a>,
a FORTRAN90 library which
tests N-dimensional
quadrature routines.
</p>
<p>
<a href = "../../f_src/test_tri_int/test_tri_int.html">
TEST_TRI_INT</a>,
a FORTRAN90 library which
tests algorithms
for quadrature over a triangle.
</p>
<p>
<a href = "../../f_src/testpack/testpack.html">
TESTPACK</a>,
a FORTRAN90 library which
tests multidimensional quadrature.
</p>
<p>
<a href = "../../f_src/wandzura/wandzura.html">
WANDZURA</a>,
a FORTRAN90 library which
defines Wandzura rules for quadrature on a triangle.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
Jarle Berntsen, Terje Espelid,<br>
Algorithm 706:
DCUTRI: an algorithm for adaptive cubature
over a collection of triangles,<br>
ACM Transactions on Mathematical Software,<br>
Volume 18, Number 3, September 1992, pages 329-342.
</li>
<li>
SF Bockman,<br>
Generalizing the Formula for Areas of Polygons to Moments,<br>
American Mathematical Society Monthly,<br>
Volume 96, Number 2, February 1989, pages 131-132.
</li>
<li>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Second Edition,<br>
Springer, 1987,<br>
ISBN: 0387964673,<br>
LC: QA76.9.C65.B73.
</li>
<li>
William Cody, Kenneth Hillstrom,<br>
Chebyshev Approximations for the Natural Logarithm of the
Gamma Function,
Mathematics of Computation,<br>
Volume 21, Number 98, April 1967, pages 198-203.
</li>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
<li>
Elise deDoncker, Ian Robinson,<br>
Algorithm 612:
Integration over a Triangle Using Nonlinear Extrapolation,<br>
ACM Transactions on Mathematical Software,<br>
Volume 10, Number 1, March 1984, pages 17-22.
</li>
<li>
Hermann Engels,<br>
Numerical Quadrature and Cubature,<br>
Academic Press, 1980,<br>
ISBN: 012238850X,<br>
LC: QA299.3E5.
</li>
<li>
Thomas Ericson, Victor Zinoviev,<br>
Codes on Euclidean Spheres,<br>
Elsevier, 2001,<br>
ISBN: 0444503293,<br>
LC: QA166.7E75
</li>
<li>
Carlos Felippa,<br>
A compendium of FEM integration formulas for symbolic work,<br>
Engineering Computation,<br>
Volume 21, Number 8, 2004, pages 867-890.
</li>
<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>
Bennett Fox,<br>
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,<br>
ACM Transactions on Mathematical Software,<br>
Volume 12, Number 4, December 1986, pages 362-376.
</li>
<li>
Axel Grundmann, Michael Moeller,<br>
Invariant Integration Formulas for the N-Simplex
by Combinatorial Methods,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 15, Number 2, April 1978, pages 282-290.
</li>
<li>
John Harris, Horst Stocker,<br>
Handbook of Mathematics and Computational Science,<br>
Springer, 1998,<br>
ISBN: 0-387-94746-9,<br>
LC: QA40.S76.
</li>
<li>
Patrick Keast,<br>
Moderate Degree Tetrahedral Quadrature Formulas,<br>
Computer Methods in Applied Mechanics and Engineering,<br>
Volume 55, Number 3, May 1986, pages 339-348.
</li>
<li>
Vladimir Krylov,<br>
Approximate Calculation of Integrals,<br>
Dover, 2006,<br>
ISBN: 0486445798,<br>
LC: QA311.K713.
</li>
<li>
Dirk Laurie,<br>
Algorithm 584:
CUBTRI, Automatic Cubature Over a Triangle,<br>
ACM Transactions on Mathematical Software,<br>
Volume 8, Number 2, 1982, pages 210-218.
</li>
<li>
Frank Lether,<br>
A Generalized Product Rule for the Circle,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 8, Number 2, June 1971, pages 249-253.
</li>
<li>
James Lyness, Dennis Jespersen,<br>
Moderate Degree Symmetric Quadrature Rules for the Triangle,<br>
Journal of the Institute of Mathematics and its Applications,<br>
Volume 15, Number 1, February 1975, pages 19-32.
</li>
<li>
James Lyness, BJJ McHugh,<br>
Integration Over Multidimensional Hypercubes,
A Progressive Procedure,<br>
The Computer Journal,<br>
Volume 6, 1963, pages 264-270.
</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>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
<li>
William Peirce,<br>
Numerical Integration Over the Planar Annulus,<br>
Journal of the Society for Industrial and Applied Mathematics,<br>
Volume 5, Number 2, June 1957, pages 66-73.
</li>
<li>
Hans Rudolf Schwarz,<br>
Finite Element Methods,<br>
Academic Press, 1988,<br>
ISBN: 0126330107,<br>
LC: TA347.F5.S3313.
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Arthur Stroud,<br>
Approximate Calculation of Multiple Integrals,<br>
Prentice Hall, 1971,<br>
ISBN: 0130438936,<br>
LC: QA311.S85.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Stephen Wandzura, Hong Xiao,<br>
Symmetric Quadrature Rules on a Triangle,<br>
Computers and Mathematics with Applications,<br>
Volume 45, 2003, pages 1829-1840.
</li>
<li>
Stephen Wolfram,<br>
The Mathematica Book,<br>
Fourth Edition,<br>
Cambridge University Press, 1999,<br>
ISBN: 0-521-64314-7,<br>
LC: QA76.95.W65.
</li>
<li>
Dongbin Xiu,<br>
Numerical integration formulas of degree two,<br>
Applied Numerical Mathematics,<br>
Volume 58, 2008, pages 1515-1520.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200,<br>
LC: TA640.2.Z54
</li>
<li>
Daniel Zwillinger, editor,<br>
CRC Standard Mathematical Tables and Formulae,<br>
30th Edition,<br>
CRC Press, 1996,<br>
ISBN: 0-8493-2479-3,<br>
LC: QA47.M315.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "stroud.f90">stroud.f90</a>, the source code;
</li>
<li>
<a href = "stroud.sh">stroud.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "stroud_prb.f90">stroud_prb.f90</a>, the calling program;
</li>
<li>
<a href = "stroud_prb.sh">stroud_prb.sh</a>,
commands to compile, link and run the calling program;
</li>
<li>
<a href = "stroud_prb_output.txt">stroud_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ARC_SINE</b> computes the arc sine function, with argument truncation.
</li>
<li>
<b>BALL_F1_ND</b> approximates an integral inside a ball in ND.
</li>
<li>
<b>BALL_F3_ND</b> approximates an integral inside a ball in ND.
</li>
<li>
<b>BALL_MONOMIAL_ND</b> integrates a monomial on a ball in ND.
</li>
<li>
<b>BALL_UNIT_07_3D</b> approximates an integral inside the unit ball in 3D.
</li>
<li>
<b>BALL_UNIT_14_3D</b> approximates an integral inside the unit ball in 3D.
</li>
<li>
<b>BALL_UNIT_15_3D</b> approximates an integral inside the unit ball in 3D.
</li>
<li>
<b>BALL_UNIT_F1_ND</b> approximates an integral inside the unit ball in ND.
</li>
<li>
<b>BALL_UNIT_F3_ND</b> approximates an integral inside the unit ball in ND.
</li>
<li>
<b>BALL_UNIT_VOLUME_3D</b> computes the volume of the unit ball in 3D.
</li>
<li>
<b>BALL_UNIT_VOLUME_ND</b> computes the volume of the unit ball in ND.
</li>
<li>
<b>BALL_VOLUME_3D</b> computes the volume of a ball in 3D.
</li>
<li>
<b>BALL_VOLUME_ND</b> computes the volume of a ball in ND.
</li>
<li>
<b>C1_GEG_MONOMIAL_INTEGRAL:</b> integral of monomial with Gegenbauer weight on C1.
</li>
<li>
<b>C1_JAC_MONOMIAL_INTEGRAL:</b> integral of a monomial with Jacobi weight over C1.
</li>
<li>
<b>C1_LEG_MONOMIAL_INTEGRAL:</b> integral of monomial with Legendre weight on C1.
</li>
<li>
<b>CIRCLE_ANNULUS</b> approximates an integral in an annulus.
</li>
<li>
<b>CIRCLE_ANNULUS_AREA_2D</b> returns the area of a circular annulus in 2D.
</li>
<li>
<b>CIRCLE_ANNULUS_SECTOR</b> approximates an integral in a circular annulus sector.
</li>
<li>
<b>CIRCLE_ANNULUS_SECTOR_AREA_2D:</b> area of a circular annulus sector in 2D.
</li>
<li>
<b>CIRCLE_AREA_2D</b> returns the area of a circle in 2D.
</li>
<li>
<b>CIRCLE_CAP_AREA_2D</b> computes the area of a circle cap in 2D.
</li>
<li>
<b>CIRCLE_CUM</b> approximates an integral on the circumference of a circle in 2D.
</li>
<li>
<b>CIRCLE_RT_SET</b> sets an R, THETA product quadrature rule in the unit circle.
</li>
<li>
<b>CIRCLE_RT_SIZE</b> sizes an R, THETA product quadrature rule in the unit circle.
</li>
<li>
<b>CIRCLE_RT_SUM</b> applies an R, THETA product quadrature rule inside a circle.
</li>
<li>
<b>CIRCLE_SECTOR</b> approximates an integral in a circular sector.
</li>
<li>
<b>CIRCLE_SECTOR_AREA_2D</b> returns the area of a circular sector in 2D.
</li>
<li>
<b>CIRCLE_TRIANGLE_AREA_2D</b> returns the area of a circle triangle in 2D.
</li>
<li>
<b>CIRCLE_XY_SET</b> sets an XY quadrature rule inside the unit circle in 2D.
</li>
<li>
<b>CIRCLE_XY_SIZE</b> sizes an XY quadrature rule inside the unit circle in 2D.
</li>
<li>
<b>CIRCLE_XY_SUM</b> applies an XY quadrature rule inside a circle in 2D.
</li>
<li>
<b>CN_GEG_00_1</b> implements the midpoint rule for region CN_GEG.
</li>
<li>
<b>CN_GEG_00_1_SIZE</b> sizes the midpoint rule for region CN_GEG.
</li>
<li>
<b>CN_GEG_01_1</b> implements a precision 1 rule for region CN_GEG.
</li>
<li>
<b>CN_GEG_01_1_SIZE</b> sizes a precision 1 rule for region CN_GEG.
</li>
<li>
<b>CN_GEG_02_XIU</b> implements the Xiu rule for region CN_GEG.
</li>
<li>
<b>CN_GEG_02_XIU_SIZE</b> sizes the Xiu rule for region CN_GEG.
</li>
<li>
<b>CN_GEG_03_XIU</b> implements the Xiu precision 3 rule for region CN_GEG.
</li>
<li>
<b>CN_GEG_03_XIU_SIZE</b> sizes the Xiu precision 3 rule for region CN_GEG.
</li>
<li>
<b>CN_GEG_MONOMIAL_INTEGRAL:</b> integral of monomial with Gegenbauer weight on CN.
</li>
<li>
<b>CN_JAC_00_1</b> implements the midpoint rule for region CN_JAC.
</li>
<li>
<b>CN_JAC_00_1_SIZE</b> sizes the midpoint rule for region CN_JAC.
</li>
<li>
<b>CN_JAC_01_1</b> implements a precision 1 rule for region CN_JAC.
</li>
<li>
<b>CN_JAC_01_1_SIZE</b> sizes a precision 1 rule for region CN_JAC.
</li>
<li>
<b>CN_JAC_02_XIU</b> implements the Xiu rule for region CN_JAC.
</li>
<li>
<b>CN_JAC_02_XIU_SIZE</b> sizes the Xiu rule for region CN_JAC.
</li>
<li>
<b>CN_JAC_MONOMIAL_INTEGRAL:</b> integral of a monomial with Jacobi weight over CN.
</li>
<li>
<b>CN_LEG_01_1</b> implements the midpoint rule for region CN_LEG.
</li>
<li>
<b>CN_LEG_01_1_SIZE</b> sizes the midpoint rule for region CN_LEG.
</li>
<li>
<b>CN_LEG_02_XIU</b> implements the Xiu rule for region CN_LEG.
</li>
<li>
<b>CN_LEG_02_XIU_SIZE</b> sizes the Xiu rule for region CN_LEG.
</li>
<li>
<b>CN_LEG_03_1</b> implements the Stroud rule CN:3-1 for region CN_LEG.
</li>
<li>
<b>CN_LEG_03_1_SIZE</b> sizes the Stroud rule CN:3-1 for region CN_LEG.
</li>
<li>
<b>CN_LEG_03_XIU</b> implements the Xiu precision 3 rule for region CN_LEG.
</li>
<li>
<b>CN_LEG_03_XIU_SIZE</b> sizes the Xiu precision 3 rule for region CN_LEG.
</li>
<li>
<b>CN_LEG_05_1</b> implements the Stroud rule CN:5-1 for region CN_LEG.
</li>
<li>
<b>CN_LEG_05_1_SIZE</b> sizes the Stroud rule CN:5-1 for region CN_LEG.
</li>
<li>
<b>CN_LEG_05_2</b> implements the Stroud rule CN:5-2 for region CN_LEG.
</li>
<li>
<b>CN_LEG_05_2_SIZE</b> sizes the Stroud rule CN:5-2 for region CN_LEG.
</li>
<li>
<b>CN_LEG_MONOMIAL_INTEGRAL:</b> integral of monomial with Legendre weight on CN.
</li>
<li>
<b>CONE_UNIT_3D</b> approximates an integral inside the unit cone in 3D.
</li>
<li>
<b>CONE_VOLUME_3D</b> returns the volume of a cone in 3D.
</li>
<li>
<b>CUBE_SHELL_ND</b> approximates an integral inside a cubic shell in N dimensions.
</li>
<li>
<b>CUBE_SHELL_VOLUME_ND</b> computes the volume of a cubic shell in ND.
</li>
<li>
<b>CUBE_UNIT_3D</b> approximates an integral inside the unit cube in 3D.
</li>
<li>
<b>CUBE_UNIT_ND</b> approximates an integral inside the unit cube in ND.
</li>
<li>
<b>CUBE_UNIT_VOLUME_ND</b> returns the volume of the unit cube in ND.
</li>
<li>
<b>ELLIPSE_AREA_2D</b> returns the area of an ellipse in 2D.
</li>
<li>
<b>ELLIPSE_CIRCUMFERENCE_2D</b> returns the circumference of an ellipse in 2D.
</li>
<li>
<b>ELLIPSE_ECCENTRICITY_2D</b> returns the eccentricity of an ellipse in 2D.
</li>
<li>
<b>ELLIPSOID_VOLUME_3D</b> returns the volume of an ellipsoid in 3d.
</li>
<li>
<b>EN_R2_01_1</b> implements the Stroud rule 1.1 for region EN_R2.
</li>
<li>
<b>EN_R2_01_1_SIZE</b> sizes the Stroud rule 1.1 for region EN_R2.
</li>
<li>
<b>EN_R2_02_XIU</b> implements the Xiu rule for region EN_R2.
</li>
<li>
<b>EN_R2_02_XIU_SIZE</b> sizes the Xiu rule for region EN_R2.
</li>
<li>
<b>EN_R2_03_1</b> implements the Stroud rule 3.1 for region EN_R2.
</li>
<li>
<b>EN_R2_03_1_SIZE</b> sizes the Stroud rule 3.1 for region EN_R2.
</li>
<li>
<b>EN_R2_03_2</b> implements the Stroud rule 3.2 for region EN_R2.
</li>
<li>
<b>EN_R2_03_2_SIZE</b> sizes the Stroud rule 3.2 for region EN_R2.
</li>
<li>
<b>EN_R2_03_XIU</b> implements the Xiu precision 3 rule for region EN_R2.
</li>
<li>
<b>EN_R2_03_XIU_SIZE</b> sizes the Xiu precision 3 rule for region EN_R2.
</li>
<li>
<b>EN_R2_05_1</b> implements the Stroud rule 5.1 for region EN_R2.
</li>
<li>
<b>EN_R2_05_1_SIZE</b> sizes the Stroud rule 5.1 for region EN_R2.
</li>
<li>
<b>EN_R2_05_2</b> implements the Stroud rule 5.2 for region EN_R2.
</li>
<li>
<b>EN_R2_05_2_SIZE</b> sizes the Stroud rule 5.2 for region EN_R2.
</li>
<li>
<b>EN_R2_05_3</b> implements the Stroud rule 5.3 for region EN_R2.
</li>
<li>
<b>EN_R2_05_3_SIZE</b> sizes the Stroud rule 5.3 for region EN_R2.
</li>
<li>
<b>EN_R2_05_4</b> implements the Stroud rule 5.4 for region EN_R2.
</li>
<li>
<b>EN_R2_05_4_SIZE</b> sizes the Stroud rule 5.4 for region EN_R2.
</li>
<li>
<b>EN_R2_05_5</b> implements the Stroud rule 5.5 for region EN_R2.
</li>
<li>
<b>EN_R2_05_5_SIZE</b> sizes the Stroud rule 5.5 for region EN_R2.
</li>
<li>
<b>EN_R2_05_6</b> implements the Stroud rule 5.6 for region EN_R2.
</li>
<li>
<b>EN_R2_05_6_SIZE</b> sizes the Stroud rule 5.6 for region EN_R2.
</li>
<li>
<b>EN_R2_07_1</b> implements the Stroud rule 7.1 for region EN_R2.
</li>
<li>
<b>EN_R2_07_1_SIZE</b> sizes the Stroud rule 7.1 for region EN_R2.
</li>
<li>
<b>EN_R2_07_2</b> implements the Stroud rule 7.2 for region EN_R2.
</li>
<li>
<b>EN_R2_07_2_SIZE</b> sizes the Stroud rule 7.2 for region EN_R2.
</li>
<li>
<b>EN_R2_07_3</b> implements the Stroud rule 7.3 for region EN_R2.
</li>
<li>
<b>EN_R2_07_3_SIZE</b> sizes the Stroud rule 7.3 for region EN_R2.
</li>
<li>
<b>EN_R2_09_1</b> implements the Stroud rule 9.1 for region EN_R2.
</li>
<li>
<b>EN_R2_09_1_SIZE</b> sizes the Stroud rule 9.1 for region EN_R2.
</li>
<li>
<b>EN_R2_11_1</b> implements the Stroud rule 11.1 for region EN_R2.
</li>
<li>
<b>EN_R2_11_1_SIZE</b> sizes the Stroud rule 11.1 for region EN_R2.
</li>
<li>
<b>EN_R2_MONOMIAL_INTEGRAL</b> evaluates monomial integrals in EN_R2.
</li>
<li>
<b>EP1_GLG_MONOMIAL_INTEGRAL:</b> integral of monomial with GLG weight on EP1.
</li>
<li>
<b>EP1_LAG_MONOMIAL_INTEGRAL:</b> integral of monomial with Laguerre weight on EP1.
</li>
<li>
<b>EPN_GLG_00_1</b> implements the "midpoint rule" for region EPN_GLG.
</li>
<li>
<b>EPN_GLG_00_1_SIZE</b> sizes the midpoint rule for region EPN_GLG.
</li>
<li>
<b>EPN_GLG_01_1</b> implements a precision 1 rule for region EPN_GLG.
</li>
<li>
<b>EPN_GLG_01_1_SIZE</b> sizes a precision 1 rule for region EPN_GLG.
</li>
<li>
<b>EPN_GLG_02_XIU</b> implements the Xiu rule for region EPN_GLG.
</li>
<li>
<b>EPN_GLG_02_XIU_SIZE</b> sizes the Xiu rule for region EPN_GLG.
</li>
<li>
<b>EPN_GLG_MONOMIAL_INTEGRAL:</b> integral of monomial with GLG weight on EPN.
</li>
<li>
<b>EPN_LAG_00_1</b> implements the "midpoint rule" for region EPN_LAG.
</li>
<li>
<b>EPN_LAG_00_1_SIZE</b> sizes the midpoint rule for region EPN_LAG.
</li>
<li>
<b>EPN_LAG_01_1</b> implements a precision 1 rule for region EPN_LAG.
</li>
<li>
<b>EPN_LAG_01_1_SIZE</b> sizes a precision 1 rule for region EPN_LAG.
</li>
<li>
<b>EPN_LAG_02_XIU</b> implements the Xiu rule for region EPN_LAG.
</li>
<li>
<b>EPN_LAG_02_XIU_SIZE</b> sizes the Xiu rule for region EPN_LAG.
</li>
<li>
<b>EPN_LAG_MONOMIAL_INTEGRAL:</b> integral of monomial with Laguerre weight on EPN.
</li>
<li>
<b>GW_02_XIU</b> implements the Golub-Welsch version of the Xiu rule.
</li>
<li>
<b>GW_02_XIU_SIZE</b> sizes the Golub Welsch version of the Xiu rule.
</li>
<li>
<b>HEXAGON_AREA_2D</b> returns the area of a regular hexagon in 2D.
</li>
<li>
<b>HEXAGON_SUM</b> applies a quadrature rule inside a hexagon in 2D.
</li>
<li>
<b>HEXAGON_UNIT_AREA_2D</b> returns the area of the unit regular hexagon in 2D.
</li>
<li>
<b>HEXAGON_UNIT_SET</b> sets a quadrature rule inside the unit hexagon in 2D.
</li>
<li>
<b>HEXAGON_UNIT_SIZE</b> sizes a quadrature rule inside the unit hexagon in 2D.
</li>
<li>
<b>I4_FACTORIAL</b> computes N! (for small values of N).
</li>