stroud_rule

stroud_rule, a Fortran77 code 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.

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.

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:

C2, the interior of the square;
C3, the interior of the cube;
CN, the interior of the N-dimensional hypercube;
CN:C2, a 3-dimensional pyramid;
CN:S2, a 3-dimensional cone;
CN_SHELL, the region contained between two concentric N-dimensional hypercubes;
ELP, the interior of the 2-dimensional ellipse with weight function 1/sqrt((x-c)^2+y^2)/(sqrt((x+c)^2+y^2);
EN_R, all of N-dimensional space, with the weight function:
w(x) = exp ( - sqrt ( sum ( 1 <= i < n ) x(i)^2 ) );
EN_R2, all of N-dimensional space, with the Hermite weight function: w(x) = product ( 1 <= i <= n ) exp ( - x(i)^2 );
GN, the interior of the N-dimensional octahedron;
H2, the interior of the 2-dimensional hexagon;
PAR, the first parabolic region;
PAR2, the second parabolic region;
PAR3, the third parabolic region;
S2, the interior of the circle;
S3, the interior of the sphere;
SN, the interior of the N-dimensional hypersphere;
SN_SHELL, the region contained between two concentric N-dimensional hyperspheres;
T2, the interior of the triangle;
T3, the interior of the tetrahedron;
TN, the interior of the N-dimensional simplex;
TOR3:S2, the interior of a 3-dimensional torus with circular cross-section;
TOR3:C2, the interior of a 3-dimensional torus with square cross-section;
U2, the "surface" of the circle;
U3, the surface of the sphere;
UN, the surface of the N-dimensional sphere;

We have added a few new terms for regions:

CN_GEG, the N dimensional hypercube [-1,+1]^N, with the Gegenbauer weight function:
w(alpha;x) = product ( 1 <= i <= n ) ( 1 - x(i)^2 )^alpha;
CN_JAC, the N dimensional hypercube [-1,+1]^N, with the Beta or Jacobi weight function:
w(alpha,beta;x) = product ( 1 <= i <= n ) ( 1 - x(i) )^alpha * ( 1 + x(i) )^beta;
CN_LEG, the N dimensional hypercube [-1,+1]^N, with the Legendre weight function:
w(x) = 1;
EPN_GLG, the positive space [0,+oo)^N, with the generalized Laguerre weight function:
w(alpha;x) = product ( 1 <= i <= n ) x(i)^alpha exp ( - x(i) );
EPN_LAG, the positive space [0,+oo)^N, with the exponential or Laguerre weight function:
w(x) = product ( 1 <= i <= n ) exp ( - x(i) );

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

stroud_rule is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

stroud_rule_test

disk_rule, a Fortran77 code which computes quadrature rules for the unit disk in 2D, that is, the interior of the circle of radius 1 and center (0,0).

dunavant, a Fortran90 code which defines Dunavant rules for quadrature on a triangle.

fekete, a Fortran90 code which defines a Fekete rule for quadrature or interpolation over a triangle.

felippa, a Fortran90 code which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

gm_rule, a Fortran90 code which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

keast, a Fortran90 code which defines a number of quadrature rules for a tetrahedron.

ncc_tetrahedron, a Fortran90 code which defines Newton-Cotes closed quadrature rules on a tetrahedron.

ncc_triangle, a Fortran90 code which defines Newton-Cotes closed quadrature rules on a triangle.

nco_tetrahedron, a Fortran90 code which defines Newton-Cotes open quadrature rules on a tetrahedron.

nco_triangle, a Fortran90 code which defines Newton-Cotes open quadrature rules on a triangle.

product_rule, a Fortran90 code which constructs a product rule from 1D factor rules.

pyramid_rule, a Fortran90 code which computes a quadrature rule for a pyramid.

quad_rule, a Fortran77 code which defines quadrature rules on a variety of intervals with different weight functions.

sphere_design_rule, a Fortran90 code which returns point sets on the surface of the unit sphere, known as "designs", which can be useful for estimating integrals on the surface.

sphere_lebedev_rule, a Fortran77 code which computes Lebedev quadrature rules for the unit sphere;

sphere_triangle_quad, a Fortran90 code which estimates the integral of a function over a spherical triangle.

test_nint, a Fortran90 code which tests N-dimensional quadrature routines.

test_tri_int, a Fortran90 code which tests algorithms for quadrature over a triangle.

testpack, a Fortran77 code which tests multidimensional quadrature.

wandzura, a Fortran90 code which defines Wandzura rules for quadrature on a triangle.

Reference:

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Handbook of Mathematical Functions,
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Algorithm 706: DCUTRI: an algorithm for adaptive cubature over a collection of triangles,
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Generalizing the Formula for Areas of Polygons to Moments,
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A Guide to Simulation,
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Algorithm 612: Integration over a Triangle Using Nonlinear Extrapolation,
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Numerical Quadrature and Cubature,
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Codes on Euclidean Spheres,
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How to Integrate a Polynomial Over a Sphere,
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Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
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Invariant Integration Formulas for the N-Simplex by Combinatorial Methods,
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Handbook of Mathematics and Computational Science,
Springer, 1998,
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Moderate Degree Tetrahedral Quadrature Formulas,
Computer Methods in Applied Mechanics and Engineering,
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Approximate Calculation of Integrals,
Dover, 2006,
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Algorithm 584: CUBTRI, Automatic Cubature Over a Triangle,
ACM Transactions on Mathematical Software,
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Frank Lether,
A Generalized Product Rule for the Circle,
SIAM Journal on Numerical Analysis,
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James Lyness, Dennis Jespersen,
Moderate Degree Symmetric Quadrature Rules for the Triangle,
Journal of the Institute of Mathematics and its Applications,
Volume 15, Number 1, February 1975, pages 19-32.
James Lyness, BJJ McHugh,
Integration Over Multidimensional Hypercubes, A Progressive Procedure,
The Computer Journal,
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AD McLaren,
Optimal Numerical Integration on a Sphere,
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Source Code:

stroud.f, the source code;
stroud.sh, commands to compile the source code.

Last revised on 15 December 2023.