STROUD
Numerical Integration
in M Dimensions

STROUD is a MATLAB 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 N-dimensional sphere;
• the interior of the circle, sphere, or N-dimensional sphere;
• the interior of the triangle, tetrahedron, or N-dimensional simplex;
• the interior of the the square, cube, or the N-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 MATLAB library which defines a Clenshaw Curtis quadrature grid in multiple dimensions.

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

FEKETE is a MATLAB library which defines Fekete rules for quadrature or interpolation on a triangle.

FELIPPA is a MATHEMATICA library which defines some quadrature rules for regions of interest in finite element calculations.

GM_RULES is a MATLAB 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 MATLAB library which defines quadrature rules for a tetrahedron.

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

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

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

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

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

NINT_EXACTBESS_TRI is a MATLAB program which investigates the polynomial exactness of a quadrature rule for the triangle.

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

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

PRODUCT_RULE is a MATLAB program which can create a multidimensional quadrature rule as a product of one dimensional 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 defines various quadrature rules on tetrahedrons.

QUADRATURE_RULES_TRI is a dataset directory which contains quadrature rules which can be applied to triangular regions.

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

QUADRULE_FAST is a MATLAB library which defines efficient versions of a few 1D quadrature rules.

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 FORTRAN90 version.

TEST_NINT is a MATLAB library which contains functions that can be used to test N-dimensional quadrature routines.

TEST_TRI_INT is a MATLAB library which can be used to test algorithms for quadrature over a triangle.

TESTPACK is a FORTRAN90 library which tests multidimensional quadrature.

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

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

Reference:

1. Milton Abramowitz, Irene Stegun,
   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.
4. Paul Bratley, Bennett Fox, Linus Schrage,
   A Guide to Simulation,
   Second Edition,
   Springer, 1987,
   ISBN: 0387964673,
   LC: QA76.9.C65.B73.
5. William Cody, Kenneth Hillstrom,
   Chebyshev Approximations for the Natural Logarithm of the Gamma Function, Mathematics of Computation,
   Volume 21, Number 98, April 1967, pages 198-203.
6. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
7. Elise deDoncker, Ian Robinson,
   Algorithm 612: Integration over a Triangle Using Nonlinear Extrapolation,
   ACM Transactions on Mathematical Software,
   Volume 10, Number 1, March 1984, pages 17-22.
8. Hermann Engels,
   Numerical Quadrature and Cubature,
   Academic Press, 1980,
   ISBN: 012238850X,
   LC: QA299.3E5.
9. Thomas Ericson, Victor Zinoviev,
   Codes on Euclidean Spheres,
   Elsevier, 2001,
   ISBN: 0444503293,
   LC: QA166.7E75
10. Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
    Engineering Computation,
    Volume 21, Number 8, 2004, pages 867-890.
11. Gerald Folland,
    How to Integrate a Polynomial Over a Sphere,
    American Mathematical Monthly,
    Volume 108, Number 5, May 2001, pages 446-448.
12. Bennett Fox,
    Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators,
    ACM Transactions on Mathematical Software,
    Volume 12, Number 4, December 1986, pages 362-376.
13. Axel Grundmann, Michael Moeller,
    Invariant Integration Formulas for the N-Simplex by Combinatorial Methods,
    SIAM Journal on Numerical Analysis,
    Volume 15, Number 2, April 1978, pages 282-290.
14. John Harris, Horst Stocker,
    Handbook of Mathematics and Computational Science,
    Springer, 1998,
    ISBN: 0-387-94746-9,
    LC: QA40.S76.
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,
    Dover, 2006,
    ISBN: 0486445798,
    LC: QA311.K713.
17. Dirk Laurie,
    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.
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,
    Integration Over Multidimensional Hypercubes, A Progressive Procedure,
    The Computer Journal,
    Volume 6, 1963, pages 264-270.
21. AD McLaren,
    Optimal Numerical Integration on a Sphere,
    Mathematics of Computation,
    Volume 17, Number 84, October 1963, pages 361-383.
22. Albert Nijenhuis, Herbert Wilf,
    Combinatorial Algorithms for Computers and Calculators,
    Second Edition,
    Academic Press, 1978,
    ISBN: 0-12-519260-6,
    LC: QA164.N54.
23. William Peirce,
    Numerical Integration Over the Planar Annulus,
    Journal of the Society for Industrial and Applied Mathematics,
    Volume 5, Number 2, June 1957, pages 66-73.
24. Hans Rudolf Schwarz,
    Finite Element Methods,
    Academic Press, 1988,
    ISBN: 0126330107,
    LC: TA347.F5.S3313.
25. Gilbert Strang, George Fix,
    An Analysis of the Finite Element Method,
    Cambridge, 1973,
    ISBN: 096140888X,
    LC: TA335.S77.
26. Arthur Stroud,
    Approximate Calculation of Multiple Integrals,
    Prentice Hall, 1971,
    ISBN: 0130438936,
    LC: QA311.S85.
27. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.
28. Stephen Wandzura, Hong Xiao,
    Symmetric Quadrature Rules on a Triangle,
    Computers and Mathematics with Applications,
    Volume 45, 2003, pages 1829-1840.
29. Stephen Wolfram,
    The Mathematica Book,
    Fourth Edition,
    Cambridge University Press, 1999,
    ISBN: 0-521-64314-7,
    LC: QA76.95.W65.
30. Olgierd Zienkiewicz,
    The Finite Element Method,
    Sixth Edition,
    Butterworth-Heinemann, 2005,
    ISBN: 0750663200,
    LC: TA640.2.Z54
31. Daniel Zwillinger, editor,
    CRC Standard Mathematical Tables and Formulae,
    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.

Tar File:

A GZIP'ed TAR file of the contents of this directory is available. This is only done as a convenience for users who want ALL the files, and don't want to download them individually. This is not a convenience for me, so don't be surprised if the tar file is somewhat out of date.

• stroud_m_src.tar.csh the commands used to create the GZIP'ed TAR file.
• stroud_m_src.tar.gz the GZIP'ed TAR file.

Source Code:

• arc_sine.m, computes the inverse sine of an angle;
• ball_f1_nd.m, approximates an integral in a ball in ND;
• ball_f3_nd.m, approximates an integral in a ball in ND;
• ball_monomial_nd.m, approximates the integral of a monomial in a ball in ND;
• ball_unit_07_3d.m, approximates an integral in the unit ball in 3D;
• ball_unit_14_3d.m, approximates an integral in the unit ball in 3D;
• ball_unit_15_3d.m, approximates an integral in the unit ball in 3D;
• ball_unit_f1_nd.m, approximates an integral in the unit ball in ND;
• ball_unit_f3_nd.m, approximates an integral in the unit ball in ND;
• ball_unit_volume_3d.m, returns the volume of the unit ball in 3D;
• ball_unit_volume_nd.m, returns the volume of the unit ball in ND;
• ball_volume_3d.m, returns the volume of a ball in 3D;
• ball_volume_nd.m, returns the volume of a ball in ND;
• circle_annulus.m, approximates an integral inside a circular annulus in 2D;
• circle_annulus_area_2d.m, returns the area of a circular annulus in 2D;
• circle_annulus_sector.m, approximates an integral in a sector of a circular annulus in 2D;
• circle_annulus_sector_area_2d.m, returns the area of a sector of a circular annulus in 2D;
• circle_area_2d.m, returns the area of a circle in 2D;
• circle_cap_area_2d.m, computes the area of a circle cap in 2D.
• circle_cum.m, computes an integral on the circumference of a circle in 2D;
• circle_rt_set.m, sets an R-Theta product quadrature rule for inside a circle in 2D;
• circle_rt_size.m, sizes an R-Theta product quadrature rule for inside a circle in 2D;
• circle_rt_sum.m, applies an R-Theta product quadrature rule inside a circle in 2D;
• circle_sector.m, approximates an integral in a circular sector in 2D;
• circle_sector_area_2d.m, returns the area of a circular sector in 2D;
• circle_triangle_area_2d.m, returns the area of a circular triangle in 2D;
• circle_xy_set.m, sets an XY quadrature rule for inside a circle in 2D;
• circle_xy_size.m, sizes an XY product quadrature rule for inside a circle in 2D;
• circle_xy_sum.m, applies an XY quadrature rule inside a circle in 2D;
• cone_unit_3d.m, approximates an integral inside a cone in 3D;
• cone_volume_3d.m, returns the volume of a cone in 3D;
• cube_shell_nd.m, aproximates an integral inside a cubic shell in ND;
• cube_shell_volume_nd.m, returns the volume of a cubic shell in ND;
• cube_unit_3d.m, approximates an integral inside the unit cube in 3D;
• cube_unit_nd.m, approximates an integral inside the unit cube in ND;
• cube_unit_volume_nd.m, returns the volume of the unit cube in ND;
• ellipse_area_2d.m, computes the area of an ellipse in 2D;
• ellipse_circumference_2d.m, computes the circumference of an ellipse in 2D;
• ellipse_eccentricity_2d.m, computes the eccentricity of an ellipse in 2D;
• ellipsoid_volume_3d.m, computes the volume of an ellipsoid in 3D;
• hexagon_area_2d.m, returns the area of a regular hexagon in 2D;
• hexagon_sum.m, applies a quadrature rule for a hexagon in 2D;
• hexagon_unit_area_2d.m, returns the area of the unit hexagon in 2D;
• hexagon_unit_set.m, sets a quadrature rule for the unit hexagon in 2D;
• hexagon_unit_size.m, sizes a quadrature rule for the unit hexagon in 2D;
• i4_factorial.m, computes N! for small values of N;
• i4_factorial2.m, computes N!!, the double factorial function;
• ksub_next2.m, computes the next K subset of an N set;
• legendre_set.m, sets a Gauss-Legendre rule to integrate F(X) on [-1,1];
• legendre_set_x1.m, sets a Gauss-Legendre rule to integrate ( 1 + X ) * F(X) on [-1,1];
• legendre_set_x2.m, sets a Gauss-Legendre rule to integrate ( 1 + X )**2 * F(X) on [-1,1];
• lens_half_area_2d.m, returns the area of a circular half lens in 2D;
• lens_half_area_h_2d.m, returns the area of a circular half lens in 2D;
• lens_half_area_w_2d.m, returns the area of a circular half lens in 2D;
• octahedron_volume_nd.m, approximates an integral in the unit octahedron in ND;
• octahedron_unit_volume_nd.m, returns the volume of the unit octahedron in ND;
• parallelipiped_volume_3d.m, returns the volume of a parallelipiped in 3D;
• parallelipiped_volume_nd.m, returns the volume of a parallelipiped in ND;
• polygon_1_2d.m, integrates the function 1 over a polygon in 2D;
• polygon_x_2d.m, integrates the function X over a polygon in 2D;
• polygon_xx_2d.m, integrates the function X^2 over a polygon in 2D;
• polygon_xy_2d.m, integrates the function X*Y over a polygon in 2D;
• polygon_y_2d.m, integrates the function Y over a polygon in 2D;
• polygon_yy_2d.m, integrates the function Y*Y over a polygon in 2D;
• pyramid_unit_o01_3d.m, approximates an integral inside a unit pyramid in 3d.
• pyramid_unit_o05_3d.m, approximates an integral inside a unit pyramid in 3d.
• pyramid_unit_o06_3d.m, approximates an integral inside a unit pyramid in 3d.
• pyramid_unit_o08_3d.m, approximates an integral inside a unit pyramid in 3d.
• pyramid_unit_o08b_3d.m, approximates an integral inside a unit pyramid in 3d.
• pyramid_unit_o09_3d.m, approximates an integral inside a unit pyramid in 3d.
• pyramid_unit_o13_3d.m, approximates an integral inside a unit pyramid in 3d.
• pyramid_unit_o18_3d.m, approximates an integral inside a unit pyramid in 3d.
• pyramid_unit_o27_3d.m, approximates an integral inside a unit pyramid in 3d.
• pyramid_unit_o48_3d.m, approximates an integral inside a unit pyramid in 3D;
• pyramid_unit_monomial_3d.m, returns the exact integral of a monomial in a unit pyramid in 3D;
• pyramid_unit_volume_3d.m, returns the volume of a unit pyramid in 3D;
• pyramid_volume_3d.m, returns the volume of a pyramid in 3D;
• qmdpt.m, carries out product midpoint quadrature inside the unit cube in ND;
• qmult_1d.m, approximates the integral of a function over an interval in 1D;
• qmult_2d.m, approximates the integral of a function over a region with varying Y dimensions in 2D;
• qmult_3d.m, approximates the integral of a function over a region with varying Y and Z dimensions in 3D;
• r8_choose.m, computes the binomial coefficient C(N,K).
• r8_epsilon.m, returns the R8 machine precision;
• r8_gamma.m, returns the Gamma function;
• r8_gamma_log.m, returns the logarithm of the Gamma function;
• r8_huge.m, returns a "huge" R8;
• r8_mop.m, returns an integer power of minus one;
• r8_swap.m, swaps two R8s;
• r8_swap3.m, swaps three R8s;
• r8_uniform_01.m, returns a unit pseudorandom number R8;
• r8ge_det.m, computes the determinant of a matrix factored by R8GE_FA;
• r8ge_fa.m, factors a general matrix;
• r8vec_even_select.m, returns the I-th of N even spaced R8s;
• r8vec_print.m, prints an R8VEC;
• rectangle_3d.m, approximates an integral inside a rectangular block in 3D;
• rectangle_sub_2d.m, carries out composite quadrature in a rectangle in 2D;
• rule_adjust.m, maps a quadrature rule from [A,B] to [C,D];
• s_len_trim.m, returns the length of a string to the last nonblank;
• simplex_nd.m, approximates an integral in a simplex in ND;
• simplex_unit_01_nd.m, approximates an integral in the unit simplex in ND;
• simplex_unit_03_nd.m, approximates an integral in the unit simplex in ND;
• simplex_unit_05_nd.m, approximates an integral in the unit simplex in ND;
• simplex_unit_05_2_nd.m, approximates an integral in the unit simplex in ND;
• simplex_unit_volume_nd.m, returns the volume of the unit simplex in ND;
• simplex_volume_nd.m, returns the volume of a simplex in ND;
• sin_power_int.m, computes the sine power integral;
• sphere_05_nd.m, approximates an integral on the surface of a sphere in ND;
• sphere_07_1_nd.m, approximates an integral on the surface of a sphere in ND;
• sphere_area_3d.m, computes the surface area of a sphere in 3D;
• sphere_area_nd.m, computes the surface area of a sphere in ND;
• sphere_cap_area_2d.m, computes the surface area of a spherical cap in 2D;
• sphere_cap_area_3d.m, computes the surface area of a spherical cap in 3D;
• sphere_cap_area_nd.m, computes the surface area of a spherical cap in ND;
• sphere_cap_volume_2d.m, computes the volume of a spherical cap in 2D;
• sphere_cap_volume_3d.m, computes the volume of a spherical cap in 3D;
• sphere_cap_volume_nd.m, computes the volume of a spherical cap in ND;
• sphere_k.m, returns a factor useful in spherical calculations;
• sphere_monomial_int_nd.m, integrates a monomial over the surface of a sphere in ND;
• sphere_shell_03_nd.m, approximates an integral inside a spherical shell in ND;
• sphere_shell_volume_nd.m, computes the volume of a spherical shell in ND;
• sphere_unit_07_3d.m, approximates an integral on the surface of the unit sphere in 3D;
• sphere_unit_11_3d.m, approximates an integral on the surface of the unit sphere in 3D;
• sphere_unit_14_3d.m, approximates an integral on the surface of the unit sphere in 3D;
• sphere_unit_15_3d.m, approximates an integral on the surface of the unit sphere in 3D;
• sphere_unit_03_nd.m, approximates an integral on the surface of the unit sphere in ND;
• sphere_unit_04_nd.m, approximates an integral on the surface of the unit sphere in ND;
• sphere_unit_05_nd.m, approximates an integral on the surface of the unit sphere in ND;
• sphere_unit_07_1_nd.m, approximates an integral on the surface of the unit sphere in ND;
• sphere_unit_07_2_nd.m, approximates an integral on the surface of the unit sphere in ND;
• sphere_unit_11_nd.m, approximates an integral on the surface of the unit sphere in ND;
• sphere_unit_area_3d.m, computes the surface area of the unit sphere in 3D;
• sphere_unit_area_nd.m, computes the surface area of the unit sphere in ND;
• sphere_unit_area_values.m, returns some values of the area of the unit sphere in ND.
• sphere_unit_monomial_nd.m, integrates a monomial on the surface of the unit sphere in ND;
• sphere_unit_volume_nd.m, computes the volume of the unit sphere in ND.
• sphere_unit_volume_values.m, returns some values of the volume of the unit sphere in ND.
• sphere_volume_2d.m, computes the volume of a sphere in 2D;
• sphere_volume_3d.m, computes the volume of a sphere in 3D;
• sphere_volume_nd.m, computes the volume of a sphere in ND;
• square_sum.m, evaluates a quadrature rule for the square;
• square_unit_set.m, sets one of several quadrature rules for the unit square;
• square_unit_size.m, sizes one of several quadrature rules for the unit square;
• square_unit_sum.m, evaluates a quadrature rule for the unit square;
• subset_gray_next.m, generates all subsets of a set, one at a time;
• tetra_07.m, approximates an integral inside a tetrahedron in 3D;
• tetra_sum.m, evaluates a quadrature rule in a tetrahedron in 3D;
• tetra_tproduct.m, approximates an integral inside a tetrahedron in 3D;
• tetra_unit_set.m, sets one of several quadrature rules for the unit tetrahedron in 3D;
• tetra_unit_size.m, sizes one of several quadrature rules for the unit tetrahedron in 3D;
• tetra_unit_sum.m, evaluates a quadrature rules for the unit tetrahedron in 3D;
• tetra_unit_volume.m, prints the volume of the unit tetrahedron in 3D;
• tetra_volume.m, prints the volume of a tetrahedron in 3D;
• timestamp.m, prints the YMDHMS date as a timestamp;
• torus_1.m, approximates an integral on the surface of a torus in 3D;
• torus_5s2.m, approximates an integral inside a torus in 3D;
• torus_6s2.m, approximates an integral inside a torus in 3D;
• torus_14s.m, approximates an integral inside a torus in 3D;
• torus_area_3d.m, returns the area of a torus in 3D;
• torus_square_5c2.m, approximates an integral in a square torus in 3D;
• torus_square_14c.m, approximates an integral in a square torus in 3D;
• torus_square_area_3d.m, returns the area of a square torus in 3D;
• torus_square_volume_3d.m, returns the volume of a square torus in 3D;
• torus_volume_3d.m, returns the volume of a torus in 3D;
• triangle_rule_adjust.m, adjusts a quadrature rule defined for a unit triangle to be used on a general triangle;
• triangle_sub.m, carries out a quadrature rule over subdivisions of a triangle;
• triangle_sum.m, carries out a quadrature rule in a triangle;
• triangle_sum_adjusted.m, carries out an adjusted quadrature rule in a triangle;
• triangle_unit_product_set.m, sets a product quadrature rule for the unit triangle;
• triangle_unit_product_size.m, sizes a product quadrature rule for the unit triangle;
• triangle_unit_set.m, sets any one of a set of quadrature rules for the unit triangle;
• triangle_unit_size.m, returns the order of any one of a set of quadrature rules for the unit triangle;
• triangle_unit_sum.m, carries out a quadrature rule in the unit triangle;
• triangle_unit_volume.m, returns the volume (area) of the unit triangle;
• triangle_volume.m, returns the volume (area) of a triangle;
• tvec_even.m, computes evenly spaced angles between 0 and 2*PI;
• tvec_even2.m, computes evenly spaced angles between 0 and 2*PI;
• tvec_even3.m, computes evenly spaced angles between 0 and 2*PI;
• tvec_even_bracket.m, computes evenly spaced angles between THETA1 and THETA2;
• tvec_even_bracket2.m, computes evenly spaced angles between THETA1 and THETA2;
• tvec_even_bracket3.m, computes evenly spaced angles between THETA1 and THETA2;
• vec_lex_next.m, generates all N-vectors of integers modulo a given base;

Examples and Tests:

• stroud_test.m, runs all the tests.
• stroud_test_output.txt, the output file.
• stroud_test01.m, tests BALL_F1_ND and BALL_F3_ND.
• stroud_test02.m, tests SPHERE_UNIT_MONOMIAL_ND.
• stroud_test03.m, tests BALL_UNIT_07_3D, BALL_UNIT_14_3D and BALL_UNIT_15_3D.
• stroud_test04.m, tests BALL_UNIT_F1_ND, BALL_UNIT_F3_ND.
• stroud_test045.m, tests BALL_UNIT_VOLUME_3D, BALL_UNIT_VOLUME_ND.
• stroud_test05.m, tests BALL_UNIT_VOLUME_ND.
• stroud_test052.m, tests BALL_VOLUME_3D, BALL_VOLUME_ND.
• stroud_test054.m, tests BALL_VOLUME_ND.
• stroud_test07.m, tests CIRCLE_ANNULUS.
• stroud_test08.m, tests CIRCLE_ANNULUS, CIRCLE_RT_SET, and CIRCLE_RT_SUM.
• stroud_test085.m, tests CIRCLE_ANNULUS_AREA_2D.
• stroud_test09.m, tests CIRCLE_ANNULUS_SECTOR, CIRCLE_RT_SET, and CIRCLE_RT_SUM.
• stroud_test10.m, tests CIRCLE_CUM.
• stroud_test11.m, tests LENS_HALF_AREA_2D, CIRCLE_SECTOR_AREA_2D and CIRCLE_TRIANGLE_AREA_2D.
• stroud_test12.m, tests LENS_HALF_AREA_2D, LENS_H_HALF_AREA_2D, LENS_W_HALF_AREA_2D.
• stroud_test13.m, tests CIRCLE_SECTOR and CIRCLE_SECTOR_AREA_2D.
• stroud_test14.m, tests CIRCLE_SECTOR, CIRCLE_RT_SET and CIRCLE_RT_SUM.
• stroud_test15.m, tests CIRCLE_RT_SET and CIRCLE_RT_SUM.
• stroud_test16.m, tests CIRCLE_XY_SET and CIRCLE_XY_SUM.
• stroud_test17.m, tests CONE_UNIT_3D and CONE_VOLUME_3D.
• stroud_test18.m, tests CUBE_SHELL_ND.
• stroud_test19.m, tests CUBE_UNIT_3D, CUBE_UNIT_ND, QMULT_3D and RECTANGLE_3D.
• stroud_test20.m, tests CUBE_UNIT_ND.
• stroud_test205.m, tests ELLIPSE_AREA_2D, ELLIPSE_CIRCUMFERENCE_2D and ELLIPSE_ECCENTRICITY_2D.
• stroud_test21.m, tests HEXAGON_UNIT_SET and HEXAGON_SUM.
• stroud_test215.m, tests LENS_HALF_2D.
• stroud_test22.m, tests OCTAHEDRON_UNIT_ND.
• stroud_test23.m, tests PARALLELIPIPED_VOLUME_ND.
• stroud_test24.m, tests POLYGON_1_2D, POLYGON_X_2D, POLYGON_Y_2D, POLYGON_XX_2D, POLYGON_XY_2D and POLYGON_YY_2D.
• stroud_test25.m, tests PYRAMID_UNIT_O**_3D and PYRAMID_UNIT_VOLUME_3D.
• stroud_test255.m, tests PYRAMID_UNIT_MONOMIAL_3D.
• stroud_test26.m, tests QMULT_1D.
• stroud_test27.m, tests SIMPLEX_ND.
• stroud_test28.m, tests SIMPLEX_VOLUME_ND.
• stroud_test29.m, tests SIMPLEX_UNIT_01_ND, SIMPLEX_UNIT_03_ND, SIMPLEX_UNIT_05_ND and SIMPLE_UNIT_05_2_ND.
• stroud_test30.m, tests SPHERE_UNIT_07_3D, SPHERE_UNIT_11_3D, SPHERE_UNIT_14_3D and SPHERE_UNIT_15_3D.
• stroud_test31.m, tests SPHERE_UNIT_03_ND, SPHERE_UNIT_04_ND, SPHERE_UNIT_05_ND, SPHERE_UNIT_07_1_ND, SPHERE_UNIT_07_2_ND and SPHERE_UNIT_11_ND.
• stroud_test32.m, tests SPHERE_05_ND and SPHERE_07_1_ND.
• stroud_test322.m, tests SPHERE_CAP_AREA_3D and SPHERE_CAP_AREA_ND.
• stroud_test324.m, tests SPHERE_CAP_VOLUME_2D and SPHERE_CAP_VOLUME_ND.
• stroud_test326.m, tests SPHERE_CAP_VOLUME_3D and SPHERE_CAP_VOLUME_ND.
• stroud_test33.m, tests SPHERE_CAP_AREA_ND.
• stroud_test335.m, tests SPHERE_SHELL_03_ND.
• stroud_test34.m, tests SPHERE_UNIT_AREA_ND.
• stroud_test345.m, tests SPHERE_UNIT_VOLUME_ND, SPHERE_UNIT_VOLUME_VALUES.
• stroud_test35.m, tests SQUARE_UNIT_SET and RECTANGLE_SUB_2D.
• stroud_test36.m, tests SQUARE_UNIT_SET and SQUARE_SUM.
• stroud_test37.m, tests SQUARE_UNIT_SET and SQUARE_SUM.
• stroud_test38.m, tests TETRA_07 and TETRA_TPRODUCT.
• stroud_test39.m, tests TETRA_UNIT_SET and TETRA_UNIT_SUM.
• stroud_test40.m, tests TETRA_UNIT_SET and TETRA_SUM.
• stroud_test41.m, tests TRIANGLE_UNIT_SET and TRIANGLE_SUB.
• stroud_test42.m, tests TRIANGLE_UNIT_SET and TRIANGLE_UNIT_SUM.
• stroud_test425.m, tests TRIANGLE_UNIT_SET.
• stroud_test43.m, tests TRIANGLE_UNIT_PRODUCT_SET and TRIANGLE_UNIT_SUM.
• stroud_test44.m, tests TRIANGLE_UNIT_SET and TRIANGLE_SUM.
• stroud_test45.m, tests TORUS_1.
• stroud_test46.m, tests TORUS_5S2, TORUS_6S2 and TORUS_14S.
• stroud_test47.m, tests TORUS_SQUARE_5C2 and TORUS_SQUARE_14C.
• stroud_test48.m, tests TVEC_EVEN, TVEC_EVEN2, TVEC_EVEN3.
• stroud_test49.m, tests TVEC_EVEN_BRACKET, TVEC_EVEN_BRACKET2, TVEC_EVEN_BRACKET3.
• f_1_2d.m, returns the function 1 in 2D.
• f_x_2d.m, returns the function X in 2D.
• f_r_2d.m, returns the function R in 2D.
• fl_18.m, returns the lower limit of integration as a function of X.
• fl_28.m, returns the lower limit of integration as a function of X and Y.
• fu_18.m, returns the upper limit of integration as a function of X.
• fu_28.m, returns the upper limit of integration as a function of X and Y.
• function_1d.m, evaluates the current 1D function.
• function_1d_name.m, returns the name of the current 1D function.
• function_1d_num.m, returns the number of 1D functions.
• function_1d_set.m, sets or gets the current 1D function.
• function_2d.m, evaluates the current 2D function.
• function_2d_name.m, returns the name of the current 2D function.
• function_2d_num.m, returns the number of 2D functions.
• function_2d_set.m, sets or gets the current 2D function.
• function_3d.m, evaluates the current 3D function.
• function_3d_name.m, returns the name of the current 3D function.
• function_3d_num.m, returns the number of 3D functions.
• function_3d_set.m, sets or gets the current 3D function.
• func_nd.m, evaluates the current ND function.
• function_nd_name.m, returns the name of the current ND function.
• function_nd_num.m, returns the number of ND functions.
• funcset.m, sets or gets the index of the current ND function.
• mono_000_3d.m, evaluates the monomial F(X,Y,Z)=1.
• mono_111_3d.m, evaluates the monomial F(X,Y,Z)=X*Y*Z.
• mono_202_3d.m, evaluates the monomial F(X,Y,Z)=X^2*Z^2.
• mono_422_3d.m, evaluates the monomial F(X,Y,Z)=X^4*Y^2*Z^2.
• setsim.m, returns the vertices of a simplex in ND.

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