STROUD
Numerical Integration
in M Dimensions

STROUD is a FORTRAN90 library, using double precision arithmetic, which implements multidimensional quadrature rules of Arthur Stroud.

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:

• the surface of the circle, sphere, or M-dimensional sphere;
• the interior of the circle, sphere, or M-dimensional sphere;
• the interior of the triangle, tetrahedron, or M-dimensional simplex;
• the interior of the the square, cube, or the M-dimensional block;
• the interior or surface of the torus.

Licensing:

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

Related Data and Programs:

CLENSHAW_CURTIS is a FORTRAN90 library which defines a Clenshaw Curtis quadrature grid in multiple dimensions.

DUNAVANT is a FORTRAN90 library which defines Dunavant rules for quadrature on a triangle.

FEKETE is a FORTRAN90 library which defines a Fekete rule for quadrature or interpolation over a triangle.

FELIPPA is a MATHEMATICA library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

GM_RULES is a FORTRAN90 library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

INTLIB is a FORTRAN90 library which numerically estimates integrals in one dimension.

KEAST is a FORTRAN90 library which defines a number of quadrature rules for a tetrahedron.

NCC_TETRAHEDRON is a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

NCC_TRIANGLE is a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TETRAHEDRON is a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a tetrahedron.

NCO_TRIANGLE is a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a triangle.

NINT_EXACTNESS is a FORTRAN90 program which demonstrates how to measure the polynomial exactness of a multidimensional quadrature rule.

NINT_EXACTNESS_TRI is an executable FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the triangle.

NINTLIB is a FORTRAN90 library which numerically estimates integrals in multiple dimensions.

PRODUCT_FACTOR is a FORTRAN90 program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is a FORTRAN90 program which constructs a product rule from identical 1D factor rules.

QUADPACK is a FORTRAN90 library which estimates integrals in one dimension.

QUADRATURE_RULES is a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_TET is a dataset directory which contains triples of files defining various quadrature rules on tetrahedrons.

QUADRATURE_RULES_TRI is a collection of quadrature rules which can be applied to triangular regions.

QUADRATURE_TEST an executable 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 is a FORTRAN90 library which defines quadrature rules on a variety of intervals with different weight functions.

SIMPACK is a FORTRAN77 library which approximate the integral of a function or vector of functions over a multidimensional simplex, or a region which is the sum of multidimensional simplexes.

STROUD is also available in a C++ version and a MATLAB version.

STRI_QUAD is a FORTRAN90 library which can approximate the integral of a function over the surface of a sphere.

TEST_NINT is a FORTRAN90 library which tests N-dimensional quadrature routines.

TEST_TRI_INT is a FORTRAN90 library which tests algorithms for quadrature over a triangle.

TESTPACK is a FORTRAN90 library which tests multidimensional quadrature.

TOMS351 is a FORTRAN77 library which estimates an integral using Romberg integration.

TOMS379 is a FORTRAN77 library which estimates an integral.

TOMS418 is a FORTRAN77 library which estimates the integral of a function with a sine or cosine factor.

TOMS424 is a FORTRAN77 library which estimates the integral of a function using Clenshaw-Curtis quadrature.

TOMS468 is a FORTRAN77 library which attempts the "automatic" integration of a function.

TRIANGULATION is a FORTRAN90 library which works with triangulations and includes some quadrature rules on triangles.

WANDZURA is a FORTRAN90 library which defines Wandzura rules for quadrature on a triangle.

Reference:

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   Handbook of Mathematical Functions,
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2. Jarle Berntsen, Terje Espelid,
   Algorithm 706: DCUTRI: an algorithm for adaptive cubature over a collection of triangles,
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   Volume 18, Number 3, September 1992, pages 329-342.
3. SF Bockman,
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   Volume 10, Number 1, March 1984, pages 17-22.
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   LC: QA166.7E75
10. Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
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    Volume 21, Number 8, 2004, pages 867-890.
11. Gerald Folland,
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    Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
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13. Axel Grundmann, Michael Moeller,
    Invariant Integration Formulas for the N-Simplex by Combinatorial Methods,
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14. John Harris, Horst Stocker,
    Handbook of Mathematics and Computational Science,
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    ISBN: 0-387-94746-9,
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15. Patrick Keast,
    Moderate Degree Tetrahedral Quadrature Formulas,
    Computer Methods in Applied Mechanics and Engineering,
    Volume 55, Number 3, May 1986, pages 339-348.
16. Vladimir Krylov,
    Approximate Calculation of Integrals,
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17. Dirk Laurie,
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    Volume 8, Number 2, 1982, pages 210-218.
18. Frank Lether,
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19. 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.
20. James Lyness, BJJ McHugh,
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21. AD McLaren,
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23. William Peirce,
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24. Hans Rudolf Schwarz,
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26. Arthur Stroud,
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    LC: QA47.M315.

Source Code:

• stroud.f90, the source code;
• stroud.csh, commands to compile the source code.

Examples and Tests:

• stroud_prb.f90, the calling program;
• stroud_prb.csh, commands to compile, link and run the calling program;
• stroud_prb_output.txt, the output file.

List of Routines:

• ARC_SINE computes the arc sine function, with argument truncation.
• BALL_F1_ND approximates an integral inside a ball in ND.
• BALL_F3_ND approximates an integral inside a ball in ND.
• BALL_MONOMIAL_ND integrates a monomial on a ball in ND.
• BALL_UNIT_07_3D approximates an integral inside the unit ball in 3D.
• BALL_UNIT_14_3D approximates an integral inside the unit ball in 3D.
• BALL_UNIT_15_3D approximates an integral inside the unit ball in 3D.
• BALL_UNIT_F1_ND approximates an integral inside the unit ball in ND.
• BALL_UNIT_F3_ND approximates an integral inside the unit ball in ND.
• BALL_UNIT_VOLUME_3D computes the volume of the unit ball in 3D.
• BALL_UNIT_VOLUME_ND computes the volume of the unit ball in ND.
• BALL_VOLUME_3D computes the volume of a ball in 3D.
• BALL_VOLUME_ND computes the volume of a ball in ND.
• CIRCLE_ANNULUS approximates an integral in an annulus.
• CIRCLE_ANNULUS_AREA_2D returns the area of a circular annulus in 2D.
• CIRCLE_ANNULUS_SECTOR approximates an integral in a circular annulus sector.
• CIRCLE_ANNULUS_SECTOR_AREA_2D returns the area of a circular annulus sector in 2D.
• CIRCLE_AREA_2D returns the area of a circle in 2D.
• CIRCLE_CAP_AREA_2D computes the area of a circle cap in 2D.
• CIRCLE_CUM approximates an integral on the circumference of a circle in 2D.
• CIRCLE_RT_SET sets an R, THETA product quadrature rule in the unit circle.
• CIRCLE_RT_SIZE sizes an R, THETA product quadrature rule in the unit circle.
• CIRCLE_RT_SUM applies an R, THETA product quadrature rule inside a circle.
• CIRCLE_SECTOR approximates an integral in a circular sector.
• CIRCLE_SECTOR_AREA_2D returns the area of a circular sector in 2D.
• CIRCLE_TRIANGLE_AREA_2D returns the area of a circle triangle in 2D.
• CIRCLE_XY_SET sets an XY quadrature rule inside the unit circle in 2D.
• CIRCLE_XY_SIZE sizes an XY quadrature rule inside the unit circle in 2D.
• CIRCLE_XY_SUM applies an XY quadrature rule inside a circle in 2D.
• CONE_UNIT_3D approximates an integral inside the unit cone in 3D.
• CONE_VOLUME_3D returns the volume of a cone in 3D.
• CUBE_SHELL_ND approximates an integral inside a cubic shell in N dimensions.
• CUBE_SHELL_VOLUME_ND computes the volume of a cubic shell in ND.
• CUBE_UNIT_3D approximates an integral inside the unit cube in 3D.
• CUBE_UNIT_ND approximates an integral inside the unit cube in ND.
• CUBE_UNIT_VOLUME_ND returns the volume of the unit cube in ND.
• ELLIPSE_AREA_2D returns the area of an ellipse in 2D.
• ELLIPSE_CIRCUMFERENCE_2D returns the circumference of an ellipse in 2D.
• ELLIPSE_ECCENTRICITY_2D returns the eccentricity of an ellipse in 2D.
• ELLIPSOID_VOLUME_3D returns the volume of an ellipsoid in 3d.
• HEXAGON_AREA_2D returns the area of a regular hexagon in 2D.
• HEXAGON_SUM applies a quadrature rule inside a hexagon in 2D.
• HEXAGON_UNIT_AREA_2D returns the area of the unit regular hexagon in 2D.
• HEXAGON_UNIT_SET sets a quadrature rule inside the unit hexagon in 2D.
• HEXAGON_UNIT_SIZE sizes a quadrature rule inside the unit hexagon in 2D.
• I4_FACTORIAL computes N! (for small values of N).
• I4_FACTORIAL2 computes the factorial N!!
• KSUB_NEXT2 computes the next K subset of an N set.
• LEGENDRE_SET sets abscissas and weights for Gauss-Legendre quadrature.
• LEGENDRE_SET_X1 sets a Gauss-Legendre rule for ( 1 + X ) * F(X) on [-1,1].
• LEGENDRE_SET_X2 sets a Gauss-Legendre rule for ( 1 + X )^2 * F(X) on [-1,1].
• LENS_HALF_2D approximates an integral in a circular half lens in 2D.
• LENS_HALF_AREA_2D returns the area of a circular half lens in 2D.
• LENS_HALF_H_AREA_2D returns the area of a circular half lens in 2D.
• LENS_HALF_W_AREA_2D returns the area of a circular half lens in 2D.
• OCTAHEDRON_UNIT_ND approximates integrals in the unit octahedron in ND.
• OCTAHEDRON_UNIT_VOLUME_ND returns the volume of the unit octahedron in ND.
• PARALLELIPIPED_VOLUME_3D returns the volume of a parallelipiped in 3D.
• PARALLELIPIPED_VOLUME_ND returns the volume of a parallelipiped in ND.
• POLYGON_1_2D integrates the function 1 over a polygon in 2D.
• POLYGON_X_2D integrates the function X over a polygon in 2D.
• POLYGON_XX_2D integrates the function X*X over a polygon in 2D.
• POLYGON_XY_2D integrates the function X*Y over a polygon in 2D.
• POLYGON_Y_2D integrates the function Y over a polygon in 2D.
• POLYGON_YY_2D integrates the function Y*Y over a polygon in 2D.
• PYRAMID_UNIT_O01_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_O05_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_O06_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_O08_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_O08B_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_O09_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_O13_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_O18_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_O27_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_O48_3D approximates an integral inside the unit pyramid in 3D.
• PYRAMID_UNIT_MONOMIAL_3D: monomial integral in a unit pyramid in 3D.
• PYRAMID_UNIT_VOLUME_3D: volume of a unit pyramid with square base in 3D.
• PYRAMID_VOLUME_3D returns the volume of a pyramid with square base in 3D.
• QMDPT carries out product midpoint quadrature for the unit cube in ND.
• QMULT_1D approximates an integral over an interval in 1D.
• QMULT_2D approximates an integral with varying Y dimension in 2D.
• QMULT_3D approximates an integral with varying Y and Z dimension in 3D.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_GAMMA_LOG calculates the natural logarithm of GAMMA ( X ) for positive X.
• R8_MOP returns the I-th power of -1 as an R8 value.
• R8_SWAP switches two R8's.
• R8_SWAP3 swaps three R8's.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8GE_DET computes the determinant of a matrix factored by R8GE_FA or R8GE_TRF.
• R8GE_FA performs a LINPACK style PLU factorization of a R8GE matrix.
• R8VEC_EVEN_SELECT returns the I-th of N evenly spaced values in [ XLO, XHI ].
• R8VEC_MIRROR_NEXT steps through all sign variations of an R8VEC.
• R8VEC_PRINT prints an R8VEC.
• RECTANGLE_3D approximates an integral inside a rectangular block in 3D.
• RECTANGLE_SUB_2D carries out a composite quadrature over a rectangle in 2D.
• RULE_ADJUST maps a quadrature rule from [A,B] to [C,D].
• SIMPLEX_ND approximates an integral inside a simplex in ND.
• SIMPLEX_UNIT_01_ND approximates an integral inside the unit simplex in ND.
• SIMPLEX_UNIT_03_ND approximates an integral inside the unit simplex in ND.
• SIMPLEX_UNIT_05_ND approximates an integral inside the unit simplex in ND.
• SIMPLEX_UNIT_05_2_ND approximates an integral inside the unit simplex in ND.
• SIMPLEX_UNIT_VOLUME_ND returns the volume of the unit simplex in ND.
• SIMPLEX_VOLUME_ND returns the volume of a simplex in ND.
• SIN_POWER_INT evaluates the sine power integral.
• SPHERE_05_ND approximates an integral on the surface of a sphere in ND.
• SPHERE_07_1_ND approximates an integral on the surface of a sphere in ND.
• SPHERE_AREA_3D computes the area of a sphere in 3D.
• SPHERE_AREA_ND computes the area of a sphere in ND.
• SPHERE_CAP_AREA_2D computes the surface area of a spherical cap in 2D.
• SPHERE_CAP_AREA_3D computes the surface area of a spherical cap in 3D.
• SPHERE_CAP_AREA_ND computes the area of a spherical cap in ND.
• SPHERE_CAP_VOLUME_2D computes the volume of a spherical cap in 2D.
• SPHERE_CAP_VOLUME_3D computes the volume of a spherical cap in 3D.
• SPHERE_CAP_VOLUME_ND computes the volume of a spherical cap in ND.
• SPHERE_K computes a factor useful for spherical computations.
• SPHERE_MONOMIAL_INT_ND integrates a monomial on the surface of a sphere in ND.
• SPHERE_SHELL_03_ND approximates an integral inside a spherical shell in ND.
• SPHERE_SHELL_VOLUME_ND computes the volume of a spherical shell in ND.
• SPHERE_UNIT_03_ND approximates an integral on the surface of the unit sphere in ND.
• SPHERE_UNIT_04_ND approximates an integral on the surface of the unit sphere in ND.
• SPHERE_UNIT_05_ND approximates an integral on the surface of the unit sphere in ND.
• SPHERE_UNIT_07_3D approximates an integral on the surface of the unit sphere in 3D.
• SPHERE_UNIT_07_1_ND approximates an integral on the surface of the unit sphere in ND.
• SPHERE_UNIT_07_2_ND approximates an integral on the surface of the unit sphere in ND.
• SPHERE_UNIT_11_3D approximates an integral on the surface of the unit sphere in 3D.
• SPHERE_UNIT_11_ND approximates an integral on the surface of the unit sphere in ND.
• SPHERE_UNIT_14_3D approximates an integral on the surface of the unit sphere in 3D.
• SPHERE_UNIT_15_3D approximates an integral on the surface of the unit sphere in 3D.
• SPHERE_UNIT_AREA_3D computes the surface area of the unit sphere in 3D.
• SPHERE_UNIT_AREA_ND computes the surface area of the unit sphere in ND.
• SPHERE_UNIT_AREA_VALUES returns some areas of the unit sphere in ND.
• SPHERE_UNIT_MONOMIAL_ND integrates a monomial on the surface of the unit sphere in ND.
• SPHERE_UNIT_VOLUME_ND computes the volume of a unit sphere in ND.
• SPHERE_UNIT_VOLUME_VALUES returns some volumes of the unit sphere in ND.
• SPHERE_VOLUME_2D computes the volume of an implicit sphere in 2D.
• SPHERE_VOLUME_3D computes the volume of an implicit sphere in 3D.
• SPHERE_VOLUME_ND computes the volume of an implicit sphere in ND.
• SQUARE_SUM carries out a quadrature rule over a square.
• SQUARE_UNIT_SET sets quadrature weights and abscissas in the unit square.
• SQUARE_UNIT_SIZE sizes a quadrature rule in the unit square.
• SQUARE_UNIT_SUM carries out a quadrature rule over the unit square.
• SUBSET_GRAY_NEXT generates all subsets of a set of order N, one at a time.
• TETRA_07 approximates an integral inside a tetrahedron in 3D.
• TETRA_SUM carries out a quadrature rule in a tetrahedron in 3D.
• TETRA_TPRODUCT approximates an integral in a tetrahedron in 3D.
• TETRA_UNIT_SET sets quadrature weights and abscissas in the unit tetrahedron.
• TETRA_UNIT_SIZE sizes quadrature weights and abscissas in the unit tetrahedron.
• TETRA_UNIT_SUM carries out a quadrature rule in the unit tetrahedron in 3D.
• TETRA_UNIT_VOLUME returns the volume of the unit tetrahedron in 3D.
• TETRA_VOLUME computes the volume of a tetrahedron in 3D.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TORUS_1 approximates an integral on the surface of a torus in 3D.
• TORUS_14S approximates an integral inside a torus in 3D.
• TORUS_5S2 approximates an integral inside a torus in 3D.
• TORUS_6S2 approximates an integral inside a torus in 3D.
• TORUS_AREA_3D returns the area of a torus in 3D.
• TORUS_SQUARE_14C approximates an integral in a "square" torus in 3D.
• TORUS_SQUARE_5C2 approximates an integral in a "square" torus in 3D.
• TORUS_SQUARE_AREA_3D returns the area of a square torus in 3D.
• TORUS_SQUARE_VOLUME_3D returns the volume of a square torus in 3D.
• TORUS_VOLUME_3D returns the volume of a torus in 3D.
• TRIANGLE_RULE_ADJUST adjusts a unit quadrature rule to an arbitrary triangle.
• TRIANGLE_SUB carries out quadrature over subdivisions of a triangular region.
• TRIANGLE_SUM carries out a unit quadrature rule in an arbitrary triangle.
• TRIANGLE_SUM_ADJUSTED carries out an adjusted quadrature rule in a triangle.
• TRIANGLE_UNIT_PRODUCT_SET: product rule on the unit triangle.
• TRIANGLE_UNIT_SET sets a quadrature rule in the unit triangle.
• TRIANGLE_UNIT_SIZE returns the "size" of a unit triangle quadrature rule.
• TRIANGLE_UNIT_SUM carries out a quadrature rule in the unit triangle.
• TRIANGLE_UNIT_VOLUME returns the "volume" of the unit triangle in 2D.
• TRIANGLE_VOLUME returns the "volume" of a triangle in 2D.
• TVEC_EVEN computes an evenly spaced set of angles between 0 and 2*PI.
• TVEC_EVEN2 computes evenly spaced angles between 0 and 2*PI.
• TVEC_EVEN3 computes evenly spaced angles between 0 and 2*PI.
• TVEC_EVEN_BRACKET computes evenly spaced angles between THETA1 and THETA2.
• TVEC_EVEN_BRACKET2 computes evenly spaced angles between THETA1 and THETA2.
• TVEC_EVEN_BRACKET3 computes evenly spaced angles between THETA1 and THETA2.
• VEC_LEX_NEXT generates vectors in lex order.

You can go up one level to the FORTRAN90 source codes.