stroud_rule

stroud_rule, a Fortran90 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 Fortran90 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).

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

simplex_gm_rule, a Fortran90 code which defines Grundmann-Moeller quadrature rules over the interior of a triangle in 2D, a tetrahedron in 3D, or over the interior of the simplex in M dimensions.

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 Fortran90 code which computes Lebedev quadrature rules on the surface of the unit sphere in 3D.

tetrahedron_arbq_rule, a Fortran90 code which returns quadrature rules, with exactness up to total degree 15, over the interior of a tetrahedron in 3D, by Hong Xiao and Zydrunas Gimbutas.

tetrahedron_keast_rule, a Fortran90 code which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

tetrahedron_ncc_rule, a Fortran90 code which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a tetrahedron in 3D.

tetrahedron_nco_rule, a Fortran90 code which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a tetrahedron in 3D.

triangle_dunavant_rule, a Fortran90 code which defines Dunavant rules for quadrature over the interior of a triangle in 2D.

tetrahedron_keast_rule, a Fortran90 code which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

triangle_dunavant_rule, a Fortran90 code which defines Dunavant rules for quadrature over the interior of a triangle in 2D.

triangle_fekete, a Fortran90 code which defines Fekete rules for quadrature or interpolation over the interior of a triangle in 2D.

triangle_lyness_rule, a Fortran90 code which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2D.

triangle_ncc_rule, a Fortran90 code which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

triangle_nco_rule, a Fortran90 code which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

triangle_wandzura_rule, a Fortran90 code which returns quadrature rules of exactness 5, 10, 15, 20, 25 and 30 over the interior of the triangle in 2D.

Reference:

1. Arthur Stroud,
   Approximate Calculation of Multiple Integrals,
   Prentice Hall, 1971,
   ISBN: 0130438936,
   LC: QA311.S85.
2. Arthur Stroud, Don Secrest,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.

Source Code:

• stroud_rule.f90, the source code;
• stroud_rule.sh, compiles the source code;