stroud


stroud, 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:

We have added a few new terms for regions:

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL 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:

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.

stroud_test

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:


Last revised on 31 August 2020.