STROUD
Numerical Integration
in M Dimensions

STROUD is a FORTRAN77 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 FORTRAN77 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).

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

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

FELIPPA, a FORTRAN90 library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

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

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

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

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

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

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

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

PYRAMID_RULE, a FORTRAN90 program which computes a quadrature rule for a pyramid.

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 contains triples of files defining various quadrature rules on tetrahedrons.

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

QUADRULE, a FORTRAN77 library which defines quadrature rules on a variety of intervals with different weight functions.

SPHERE_DESIGN_RULE, a FORTRAN90 library 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 library which computes Lebedev quadrature rules for the unit sphere;

SPHERE_TRIANGLE_QUAD, a FORTRAN90 library which estimates the integral of a function over a spherical triangle.

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

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

TESTPACK, a FORTRAN77 library which tests multidimensional quadrature.

WANDZURA, a FORTRAN90 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. Dongbin Xiu,
    Numerical integration formulas of degree two,
    Applied Numerical Mathematics,
    Volume 58, 2008, pages 1515-1520.
31. Olgierd Zienkiewicz,
    The Finite Element Method,
    Sixth Edition,
    Butterworth-Heinemann, 2005,
    ISBN: 0750663200,
    LC: TA640.2.Z54
32. Daniel Zwillinger, editor,
    CRC Standard Mathematical Tables and Formulae,
    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.

Source Code:

• stroud.f, the source code;
• stroud.sh, commands to compile the source code.

Examples and Tests:

• stroud_prb.f, 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: 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_SECTOR_RULE approximates an integral in a circular sector.
• 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 precision 2 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 precision 2 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 precision 2 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 precision 2 rule for region EN_R2.
• EN_R2_02_XIU_SIZE sizes the Xiu rule 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 precision 2 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.
• 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 precision 2 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 precision 2 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.
• GW_02_XIU implements the Golub-Welsch version of the Xiu rule.
• GW_02_XIU_SIZE sizes the Golub Welsch version of the Xiu rule.
• 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 the factorial of N.
• I4_FACTORIAL2 computes the double factorial function.
• I4VEC_SUM returns the sum of the entries of 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 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.
• 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_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_ADD adds two R8's.
• 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 evaluates log ( Gamma ( X ) ) for a real argument.
• R8_HYPER_2F1 evaluates the hypergeometric function F(A,B,C,X).
• R8_MOP returns the I-th power of -1 as an R8.
• R8_PSI evaluates the function Psi(X).
• 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 an R8GE matrix.
• R8VEC_DOT_PRODUCT finds 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.
• 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 surface area of an implicit sphere in 3D.
• SPHERE_AREA_ND computes the surface area of an implicit 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 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 integral on surface of the unit sphere in ND.
• SPHERE_UNIT_04_ND approximates integral on surface of the unit sphere in ND.
• SPHERE_UNIT_05_ND approximates integral on surface of the unit sphere in ND.
• SPHERE_UNIT_07_3D approximates integral on surface of the unit sphere in 3D.
• SPHERE_UNIT_07_1_ND approximates integral on surface of unit sphere in ND.
• SPHERE_UNIT_07_2_ND approximates integral on surface of unit sphere in ND.
• SPHERE_UNIT_11_3D approximates integral on surface of unit sphere in 3D.
• SPHERE_UNIT_11_ND approximates integral on surface of unit sphere in ND.
• SPHERE_UNIT_14_3D approximates integral on surface of unit sphere in 3D.
• SPHERE_UNIT_15_3D approximates integral on surface of 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 a unit sphere in ND.
• SPHERE_UNIT_AREA_VALUES returns some areas of the unit sphere in ND.
• SPHERE_UNIT_MONOMIAL_ND integrate monomial on surface of 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 rules 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 out the current YMDHMS date as a timestamp.
• 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_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 evenly spaced 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 from THETA1 to 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 FORTRAN77 source codes.