Numerical Integration
in M Dimensions

STROUD is a C library 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:


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


STROUD is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

GM_RULE, a C library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

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.

QUADRATURE_TEST, a MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.

QUADRULE, a C library 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;

TETRAHEDRON_NCC_RULE, a C library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCO_RULE, a C library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_NCC_RULE, a C library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a C library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.


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Source Code:

Examples and Tests:

List of Routines:

You can go up one level to the C source codes.

Last revised on 08 September 2010.