stroud_rule


stroud_rule, a Fortran77 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 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 Fortran77 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).

dunavant, a Fortran90 code which defines Dunavant rules for quadrature on a triangle.

fekete, a Fortran90 code which defines a Fekete rule for quadrature or interpolation over a triangle.

felippa, a Fortran90 code which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

gm_rule, a Fortran90 code which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

keast, a Fortran90 code which defines a number of quadrature rules for a tetrahedron.

ncc_tetrahedron, a Fortran90 code which defines Newton-Cotes closed quadrature rules on a tetrahedron.

ncc_triangle, a Fortran90 code which defines Newton-Cotes closed quadrature rules on a triangle.

nco_tetrahedron, a Fortran90 code which defines Newton-Cotes open quadrature rules on a tetrahedron.

nco_triangle, a Fortran90 code which defines Newton-Cotes open quadrature rules on a triangle.

product_rule, a Fortran90 code which constructs a product rule from 1D factor rules.

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

quad_rule, a Fortran77 code which defines quadrature rules on a variety of intervals with different weight functions.

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 Fortran77 code which computes Lebedev quadrature rules for the unit sphere;

sphere_triangle_quad, a Fortran90 code which estimates the integral of a function over a spherical triangle.

test_nint, a Fortran90 code which tests N-dimensional quadrature routines.

test_tri_int, a Fortran90 code which tests algorithms for quadrature over a triangle.

testpack, a Fortran77 code which tests multidimensional quadrature.

wandzura, a Fortran90 code which defines Wandzura rules for quadrature on a triangle.

Reference:

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    Handbook of Mathematical Functions,
    National Bureau of Standards, 1964,
    ISBN: 0-486-61272-4,
    LC: QA47.A34.
  2. Jarle Berntsen, Terje Espelid,
    Algorithm 706: DCUTRI: an algorithm for adaptive cubature over a collection of triangles,
    ACM Transactions on Mathematical Software,
    Volume 18, Number 3, September 1992, pages 329-342.
  3. SF Bockman,
    Generalizing the Formula for Areas of Polygons to Moments,
    American Mathematical Society Monthly,
    Volume 96, Number 2, February 1989, pages 131-132.
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    A Guide to Simulation,
    Second Edition,
    Springer, 1987,
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    Algorithm 612: Integration over a Triangle Using Nonlinear Extrapolation,
    ACM Transactions on Mathematical Software,
    Volume 10, Number 1, March 1984, pages 17-22.
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    Volume 21, Number 8, 2004, pages 867-890.
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    Volume 108, Number 5, May 2001, pages 446-448.
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    Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
    ACM Transactions on Mathematical Software,
    Volume 12, Number 4, December 1986, pages 362-376.
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    SIAM Journal on Numerical Analysis,
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    Moderate Degree Tetrahedral Quadrature Formulas,
    Computer Methods in Applied Mechanics and Engineering,
    Volume 55, Number 3, May 1986, pages 339-348.
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    Dover, 2006,
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    LC: QA311.K713.
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    Algorithm 584: CUBTRI, Automatic Cubature Over a Triangle,
    ACM Transactions on Mathematical Software,
    Volume 8, Number 2, 1982, pages 210-218.
  18. Frank Lether,
    A Generalized Product Rule for the Circle,
    SIAM Journal on Numerical Analysis,
    Volume 8, Number 2, June 1971, pages 249-253.
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    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.
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    Volume 6, 1963, pages 264-270.
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    Optimal Numerical Integration on a Sphere,
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    Volume 17, Number 84, October 1963, pages 361-383.
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    Second Edition,
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    LC: TA335.S77.
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    Prentice Hall, 1971,
    ISBN: 0130438936,
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    Computers and Mathematics with Applications,
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    Fourth Edition,
    Cambridge University Press, 1999,
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  30. Dongbin Xiu,
    Numerical integration formulas of degree two,
    Applied Numerical Mathematics,
    Volume 58, 2008, pages 1515-1520.
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    Sixth Edition,
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    ISBN: 0750663200,
    LC: TA640.2.Z54
  32. Daniel Zwillinger, editor,
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    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.

Source Code:


Last revised on 15 December 2023.