stroud
stroud,
a C code which
defines quadrature rules for a variety of Mdimensional regions,
including the interior of the square, cube and hypercube, the pyramid,
cone and ellipse, the hexagon, the Mdimensional 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 Ndimensional hypercube;

CN:C2, a 3dimensional pyramid;

CN:S2, a 3dimensional cone;

CN_SHELL, the region contained between two concentric
Ndimensional hypercubes;

ELP, the interior of the 2dimensional ellipse with
weight function 1/sqrt((xc)^2+y^2)/(sqrt((x+c)^2+y^2);

EN_R, all of Ndimensional space, with the weight function:
w(x) = exp (  sqrt ( sum ( 1 <= i < n ) x(i)^2 ) );

EN_R2, all of Ndimensional space, with the Hermite weight function:
w(x) = product ( 1 <= i <= n ) exp (  x(i)^2 );

GN, the interior of the Ndimensional octahedron;

H2, the interior of the 2dimensional 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 Ndimensional hypersphere;

SN_SHELL, the region contained between two concentric Ndimensional hyperspheres;

T2, the interior of the triangle;

T3, the interior of the tetrahedron;

TN, the interior of the Ndimensional simplex;

TOR3:S2, the interior of a 3dimensional torus with circular crosssection;

TOR3:C2, the interior of a 3dimensional torus with square crosssection;

U2, the "surface" of the circle;

U3, the surface of the sphere;

UN, the surface of the Ndimensional 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 computer code and data files described and made available on this web page
are distributed under
the MIT license
Languages:
stroud is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
QUADRATURE_RULES,
a dataset directory which
contains sets of files that define quadrature
rules over various 1D intervals or multidimensional hypercubes.
QUADRATURE_RULES_TET,
a dataset directory which
defines various quadrature
rules on tetrahedrons.
QUADRATURE_RULES_TRI,
a dataset directory which
defines quadrature rules to be applied to triangular
regions.
QUADRULE,
a C code which
defines quadrature rules on a
variety of intervals with different weight functions.
SPHERE_LEBEDEV_RULE,
a dataset directory which
contains sets of points on a sphere which can be used for
quadrature rules of a known precision;
stroud_test
TETRAHEDRON_NCC_RULE,
a C code which
defines NewtonCotes Closed (NCC) quadrature rules
over the interior of a tetrahedron in 3D.
TETRAHEDRON_NCO_RULE,
a C code which
defines NewtonCotes Open (NCO) quadrature rules
over the interior of a tetrahedron in 3D.
TRIANGLE_NCC_RULE,
a C code which
defines NewtonCotes Closed (NCC) quadrature rules
over the interior of a triangle in 2D.
TRIANGLE_NCO_RULE,
a C code which
defines NewtonCotes Open (NCO) quadrature rules
over the interior of a triangle in 2D.
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Handbook of Mathematical Functions,
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Algorithm 706:
DCUTRI: an algorithm for adaptive cubature
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ACM Transactions on Mathematical Software,
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SF Bockman,
Generalizing the Formula for Areas of Polygons to Moments,
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A Guide to Simulation,
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Chebyshev Approximations for the Natural Logarithm of the
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Algorithm 612:
Integration over a Triangle Using Nonlinear Extrapolation,
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Algorithm 647:
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Invariant Integration Formulas for the NSimplex
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Approximate Calculation of Integrals,
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Algorithm 584:
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A Generalized Product Rule for the Circle,
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Moderate Degree Symmetric Quadrature Rules for the Triangle,
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Integration Over Multidimensional Hypercubes,
A Progressive Procedure,
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Optimal Numerical Integration on a Sphere,
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Approximate Calculation of Multiple Integrals,
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Numerical integration formulas of degree two,
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Source Code:
Last revised on 11 August 2019.