STROUD
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:

• C2, the interior of the square;
• C3, the interior of the cube;
• CN, the interior of the N-dimensional hypercube;
• CN:C2, a 3-dimensional pyramid;
• CN:S2, a 3-dimensional cone;
• CN_SHELL, the region contained between two concentric N-dimensional hypercubes;
• ELP, the interior of the 2-dimensional ellipse with weight function 1/sqrt((x-c)^2+y^2)/(sqrt((x+c)^2+y^2);
• EN_R, all of N-dimensional space, with the weight function:
  w(x) = exp ( - sqrt ( sum ( 1 <= i < n ) x(i)^2 ) );
• EN_R2, all of N-dimensional space, with the Hermite weight function: w(x) = product ( 1 <= i <= n ) exp ( - x(i)^2 );
• GN, the interior of the N-dimensional octahedron;
• H2, the interior of the 2-dimensional hexagon;
• PAR, the first parabolic region;
• PAR2, the second parabolic region;
• PAR3, the third parabolic region;
• S2, the interior of the circle;
• S3, the interior of the sphere;
• SN, the interior of the N-dimensional hypersphere;
• SN_SHELL, the region contained between two concentric N-dimensional hyperspheres;
• T2, the interior of the triangle;
• T3, the interior of the tetrahedron;
• TN, the interior of the N-dimensional simplex;
• TOR3:S2, the interior of a 3-dimensional torus with circular cross-section;
• TOR3:C2, the interior of a 3-dimensional torus with square cross-section;
• U2, the "surface" of the circle;
• U3, the surface of the sphere;
• UN, the surface of the N-dimensional sphere;

We have added a few new terms for regions:

• CN_GEG, the N dimensional hypercube [-1,+1]^N, with the Gegenbauer weight function:
  w(alpha;x) = product ( 1 <= i <= n ) ( 1 - x(i)^2 )^alpha;
• CN_JAC, the N dimensional hypercube [-1,+1]^N, with the Beta or Jacobi weight function:
  w(alpha,beta;x) = product ( 1 <= i <= n ) ( 1 - x(i) )^alpha * ( 1 + x(i) )^beta;
• CN_LEG, the N dimensional hypercube [-1,+1]^N, with the Legendre weight function:
  w(x) = 1;
• EPN_GLG, the positive space [0,+oo)^N, with the generalized Laguerre weight function:
  w(alpha;x) = product ( 1 <= i <= n ) x(i)^alpha exp ( - x(i) );
• EPN_LAG, the positive space [0,+oo)^N, with the exponential or Laguerre weight function:
  w(x) = product ( 1 <= i <= n ) exp ( - x(i) );

Licensing:

The computer code and data files described and 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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

DISK_RULE, a C++ library which computes quadrature rules for the unit disk in 2D, that is, the interior of the circle of radius 1 and center (0,0).

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

PYRAMID_RULE, a C++ program which computes a quadrature rule for a pyramid.

SIMPLEX_GM_RULE, a C++ library 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_LEBEDEV_RULE, a C++ library which computes Lebedev quadrature rules on the surface of the unit sphere in 3D.

TETRAHEDRON_ARBQ_RULE, a C++ library 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 C++ library which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

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_DUNAVANT_RULE, a C++ library which defines Dunavant rules for quadrature over the interior of a triangle in 2D.

TRIANGLE_FEKETE, a C++ library which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_LYNESS_RULE, a C++ library which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2D.

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.

TRIANGLE_WANDZURA_RULE, a C++ library which returns quadrature rules of exactness 5, 10, 15, 20, 25 and 30 over the interior of the triangle in 2D.

Reference:

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   Handbook of Mathematical Functions,
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   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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11. Gerald Folland,
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17. Dirk Laurie,
    Algorithm 584: CUBTRI, Automatic Cubature Over a Triangle,
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    Volume 8, Number 2, 1982, pages 210-218.
18. Frank Lether,
    A Generalized Product Rule for the Circle,
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    Volume 8, Number 2, June 1971, pages 249-253.
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    Moderate Degree Symmetric Quadrature Rules for the Triangle,
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    Volume 15, Number 1, February 1975, pages 19-32.
20. James Lyness, BJJ McHugh,
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    Volume 6, 1963, pages 264-270.
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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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    Volume 58, 2008, pages 1515-1520.
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    Sixth Edition,
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Source Code:

• stroud.cpp, the source code;
• stroud.hpp, the include file;
• stroud.sh, commands to compile the source code.

Examples and Tests:

• stroud_prb.cpp, the calling program;
• stroud_prb.sh, 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.
• C1_GEG_MONOMIAL_INTEGRAL: integral of monomial with Gegenbauer weight on C1.
• C1_JAC_MONOMIAL_INTEGRAL: integral of a monomial with Jacobi weight over C1.
• C1_LEG_MONOMIAL_INTEGRAL: integral of monomial with Legendre weight on C1.
• 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.
• CN_GEG_00_1 implements the midpoint rule for region CN_GEG.
• CN_GEG_00_1_SIZE sizes the midpoint rule for region CN_GEG.
• CN_GEG_01_1 implements a precision 1 rule for region CN_GEG.
• CN_GEG_01_1_SIZE sizes a precision 1 rule for region CN_GEG.
• CN_GEG_02_XIU implements the Xiu rule for region CN_GEG.
• CN_GEG_02_XIU_SIZE sizes the Xiu rule for region CN_GEG.
• CN_GEG_03_XIU implements the Xiu precision 3 rule for region CN_GEG.
• CN_GEG_03_XIU_SIZE sizes the Xiu precision 3 rule for region CN_GEG.
• CN_GEG_MONOMIAL_INTEGRAL: integral of monomial with Gegenbauer weight on CN.
• CN_JAC_00_1 implements the midpoint rule for region CN_JAC.
• CN_JAC_00_1_SIZE sizes the midpoint rule for region CN_JAC.
• CN_JAC_01_1 implements a precision 1 rule for region CN_JAC.
• CN_JAC_01_1_SIZE sizes a precision 1 rule for region CN_JAC.
• CN_JAC_02_XIU implements the Xiu rule for region CN_JAC.
• CN_JAC_02_XIU_SIZE sizes the Xiu rule for region CN_JAC.
• CN_JAC_MONOMIAL_INTEGRAL: integral of a monomial with Jacobi weight over CN.
• CN_LEG_01_1 implements the midpoint rule for region CN_LEG.
• CN_LEG_01_1_SIZE sizes the midpoint rule for region CN_LEG.
• CN_LEG_02_XIU implements the Xiu rule for region CN_LEG.
• CN_LEG_02_XIU_SIZE sizes the Xiu rule for region CN_LEG.
• CN_LEG_03_1 implements the Stroud rule CN:3-1 for region CN_LEG.
• CN_LEG_03_1_SIZE sizes the Stroud rule CN:3-1 for region CN_LEG.
• CN_LEG_03_XIU implements the Xiu precision 3 rule for region CN_LEG.
• CN_LEG_03_XIU_SIZE sizes the Xiu precision 3 rule for region CN_LEG.
• CN_LEG_05_1 implements the Stroud rule CN:5-1 for region CN_LEG.
• CN_LEG_05_1_SIZE sizes the Stroud rule CN:5-1 for region CN_LEG.
• CN_LEG_05_2 implements the Stroud rule CN:5-2 for region CN_LEG.
• CN_LEG_05_2_SIZE sizes the Stroud rule CN:5-2 for region CN_LEG.
• CN_LEG_MONOMIAL_INTEGRAL: integral of monomial with Legendre weight on CN.
• 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.
• EN_R2_01_1 implements the Stroud rule 1.1 for region EN_R2.
• EN_R2_01_1_SIZE sizes the Stroud rule 1.1 for region EN_R2.
• EN_R2_02_XIU implements the Xiu rule for region EN_R2.
• EN_R2_02_XIU_SIZE sizes the Xiu for region EN_R2.
• EN_R2_03_1 implements the Stroud rule 3.1 for region EN_R2.
• EN_R2_03_1_SIZE sizes the Stroud rule 3.1 for region EN_R2.
• EN_R2_03_2 implements the Stroud rule 3.2 for region EN_R2.
• EN_R2_03_2_SIZE sizes the Stroud rule 3.2 for region EN_R2.
• EN_R2_03_XIU implements the Xiu precision 3 rule for region EN_R2.
• EN_R2_03_XIU_SIZE sizes the Xiu precision 3 rule for region EN_R2.
• EN_R2_05_1 implements the Stroud rule 5.1 for region EN_R2.
• EN_R2_05_1_SIZE sizes the Stroud rule 5.1 for region EN_R2.
• EN_R2_05_2 implements the Stroud rule 5.2 for region EN_R2.
• EN_R2_05_2_SIZE sizes the Stroud rule 5.2 for region EN_R2.
• EN_R2_05_3 implements the Stroud rule 5.3 for region EN_R2.
• EN_R2_05_3_SIZE sizes the Stroud rule 5.3 for region EN_R2.
• EN_R2_05_4 implements the Stroud rule 5.4 for region EN_R2.
• EN_R2_05_4_SIZE sizes the Stroud rule 5.4 for region EN_R2.
• EN_R2_05_5 implements the Stroud rule 5.5 for region EN_R2.
• EN_R2_05_5_SIZE sizes the Stroud rule 5.5 for region EN_R2.
• EN_R2_05_6 implements the Stroud rule 5.6 for region EN_R2.
• EN_R2_05_6_SIZE sizes the Stroud rule 5.6 for region EN_R2.
• EN_R2_07_1 implements the Stroud rule 7.1 for region EN_R2.
• EN_R2_07_1_SIZE sizes the Stroud rule 7.1 for region EN_R2.
• EN_R2_07_2 implements the Stroud rule 7.2 for region EN_R2.
• EN_R2_07_2_SIZE sizes the Stroud rule 7.2 for region EN_R2.
• EN_R2_07_3 implements the Stroud rule 7.3 for region EN_R2.
• EN_R2_07_3_SIZE sizes the Stroud rule 7.3 for region EN_R2.
• EN_R2_09_1 implements the Stroud rule 9.1 for region EN_R2.
• EN_R2_09_1_SIZE sizes the Stroud rule 9.1 for region EN_R2.
• EN_R2_11_1 implements the Stroud rule 11.1 for region EN_R2.
• EN_R2_11_1_SIZE sizes the Stroud rule 11.1 for region EN_R2.
• EN_R2_MONOMIAL_INTEGRAL evaluates monomial integrals in EN_R2.
• EP1_GLG_MONOMIAL_INTEGRAL: integral of monomial with GLG weight on EP1.
• EP1_LAG_MONOMIAL_INTEGRAL: integral of monomial with Laguerre weight on EP1.
• EPN_GLG_00_1 implements the "midpoint rule" for region EPN_GLG.
• EPN_GLG_00_1_SIZE sizes the midpoint rule for region EPN_GLG.
• EPN_GLG_01_1 implements a precision 1 rule for region EPN_GLG.
• EPN_GLG_01_1_SIZE sizes a precision 1 rule for region EPN_GLG.
• EPN_GLG_02_XIU implements the Xiu rule for region EPN_GLG.
• EPN_GLG_02_XIU_SIZE sizes the Xiu rule for region EPN_GLG.
• EPN_GLG_MONOMIAL_INTEGRAL: integral of monomial with GLG weight on EPN.
• EPN_LAG_00_1 implements the "midpoint rule" for region EPN_LAG.
• EPN_LAG_00_1_SIZE sizes the midpoint rule for region EPN_LAG.
• EPN_LAG_01_1 implements a precision 1 rule for region EPN_LAG.
• EPN_LAG_01_1_SIZE sizes a precision 1 rule for region EPN_LAG.
• EPN_LAG_02_XIU implements the Xiu rule for region EPN_LAG.
• EPN_LAG_02_XIU_SIZE sizes the Xiu rule for region EPN_LAG.
• EPN_LAG_MONOMIAL_INTEGRAL: integral of monomial with Laguerre weight on EPN.
• 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 returns N!.
• I4_FACTORIAL2 computes the double factorial function.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• I4_POWER returns the value of I^J.
• I4VEC_SUM sums the entries of an I4VEC.
• I4VEC_ZERO zeroes an I4VEC.
• KSUB_NEXT2 generates the subsets of size K from a set of size N.
• 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 Gauss-Legendre rules 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.
• MONOMIAL_VALUE evaluates a monomial.
• 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_ABS returns the absolute value of an R8.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_FACTORIAL computes the factorial of N.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_GAMMA_LOG calculates the natural logarithm of GAMMA ( X ) for positive X.
• R8_HUGE returns a "huge" R8.
• R8_HYPER_2F1 evaluates the hypergeometric function 2F1(A,B,C,X).
• R8_MAX returns the maximum of two R8's.
• R8_MIN returns the minimum of two R8's.
• R8_MOP returns the I-th power of -1 as an R8 value.
• R8_PSI evaluates the function Psi(X).
• R8_SWAP switches two R8's.
• R8_SWAP3 swaps three R8's.
• R8_UNIFORM_01 is 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_DOT_PRODUCT computes the dot product of a pair of R8VEC's.
• 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.
• R8VEC_ZERO zeroes 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].
• S_LEN_TRIM returns the length of a string to the last nonblank.
• 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.
• TETRA_VOLUME computes the volume of a tetrahedron.
• 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 sets a product rule on the unit triangle.
• TRIANGLE_UNIT_PRODUCT_SIZE sizes a 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_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 an evenly spaced set of angles between 0 and 2*PI.
• TVEC_EVEN_BRACKET computes evenly spaced angles between THETA1 and THETA2.
• TVEC_EVEN_BRACKET2 computes evenly spaced angles from THETA1 to THETA2.
• TVEC_EVEN_BRACKET3 computes evenly spaced angles from THETA1 to THETA2.
• VEC_LEX_NEXT generates vectors in lex order.

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