STROUD
Numerical Integration
in M Dimensions

STROUD is a FORTRAN90 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 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 FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

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

SIMPLEX_GM_RULE, a FORTRAN90 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_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 FORTRAN90 library which computes Lebedev quadrature rules on the surface of the unit sphere in 3D.

TETRAHEDRON_ARBQ_RULE, a FORTRAN90 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 FORTRAN90 library which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_DUNAVANT_RULE, a FORTRAN90 library which defines Dunavant rules for quadrature over the interior of a triangle in 2D.

TETRAHEDRON_KEAST_RULE, a FORTRAN90 library which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

TRIANGLE_DUNAVANT_RULE, a FORTRAN90 library which defines Dunavant rules for quadrature over the interior of a triangle in 2D.

TRIANGLE_FEKETE, a FORTRAN90 library which defines Fekete rules for quadrature or interpolation over the interior of a triangle in 2D.

TRIANGLE_LYNESS_RULE, a FORTRAN90 library which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a triangle in 2D.

TRIANGLE_WANDZURA_RULE, a FORTRAN90 library which returns quadrature rules of exactness 5, 10, 15, 20, 25 and 30 over the interior of the triangle in 2D.

Reference:

Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
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.
SF Bockman,
Generalizing the Formula for Areas of Polygons to Moments,
American Mathematical Society Monthly,
Volume 96, Number 2, February 1989, pages 131-132.
Paul Bratley, Bennett Fox, Linus Schrage,
A Guide to Simulation,
Second Edition,
Springer, 1987,
ISBN: 0387964673,
LC: QA76.9.C65.B73.
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.
Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
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.
Hermann Engels,
Numerical Quadrature and Cubature,
Academic Press, 1980,
ISBN: 012238850X,
LC: QA299.3E5.
Thomas Ericson, Victor Zinoviev,
Codes on Euclidean Spheres,
Elsevier, 2001,
ISBN: 0444503293,
LC: QA166.7E75
Carlos Felippa,
A compendium of FEM integration formulas for symbolic work,
Engineering Computation,
Volume 21, Number 8, 2004, pages 867-890.
Gerald Folland,
How to Integrate a Polynomial Over a Sphere,
American Mathematical Monthly,
Volume 108, Number 5, May 2001, pages 446-448.
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.
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.
John Harris, Horst Stocker,
Handbook of Mathematics and Computational Science,
Springer, 1998,
ISBN: 0-387-94746-9,
LC: QA40.S76.
Patrick Keast,
Moderate Degree Tetrahedral Quadrature Formulas,
Computer Methods in Applied Mechanics and Engineering,
Volume 55, Number 3, May 1986, pages 339-348.
Vladimir Krylov,
Approximate Calculation of Integrals,
Dover, 2006,
ISBN: 0486445798,
LC: QA311.K713.
Dirk Laurie,
Algorithm 584: CUBTRI, Automatic Cubature Over a Triangle,
ACM Transactions on Mathematical Software,
Volume 8, Number 2, 1982, pages 210-218.
Frank Lether,
A Generalized Product Rule for the Circle,
SIAM Journal on Numerical Analysis,
Volume 8, Number 2, June 1971, pages 249-253.
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.
James Lyness, BJJ McHugh,
Integration Over Multidimensional Hypercubes, A Progressive Procedure,
The Computer Journal,
Volume 6, 1963, pages 264-270.
AD McLaren,
Optimal Numerical Integration on a Sphere,
Mathematics of Computation,
Volume 17, Number 84, October 1963, pages 361-383.
Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
Academic Press, 1978,
ISBN: 0-12-519260-6,
LC: QA164.N54.
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.
Hans Rudolf Schwarz,
Finite Element Methods,
Academic Press, 1988,
ISBN: 0126330107,
LC: TA347.F5.S3313.
Gilbert Strang, George Fix,
An Analysis of the Finite Element Method,
Cambridge, 1973,
ISBN: 096140888X,
LC: TA335.S77.
Arthur Stroud,
Approximate Calculation of Multiple Integrals,
Prentice Hall, 1971,
ISBN: 0130438936,
LC: QA311.S85.
Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.
Stephen Wandzura, Hong Xiao,
Symmetric Quadrature Rules on a Triangle,
Computers and Mathematics with Applications,
Volume 45, 2003, pages 1829-1840.
Stephen Wolfram,
The Mathematica Book,
Fourth Edition,
Cambridge University Press, 1999,
ISBN: 0-521-64314-7,
LC: QA76.95.W65.
Dongbin Xiu,
Numerical integration formulas of degree two,
Applied Numerical Mathematics,
Volume 58, 2008, pages 1515-1520.
Olgierd Zienkiewicz,
The Finite Element Method,
Sixth Edition,
Butterworth-Heinemann, 2005,
ISBN: 0750663200,
LC: TA640.2.Z54
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.f90, the source code;

Examples and Tests:

stroud_test.f90, the calling program;
stroud_test.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_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 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 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.
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 N! (for small values of N).
I4_FACTORIAL2 computes the double factorial function.
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.
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_CHOOSE computes the binomial coefficient C(N,K) as an R8.
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_HYPER_2F1 evaluates the hypergeometric function F(A,B,C,X).
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 returns a unit pseudorandom R8.
R8GE_DET: 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 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 the 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 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_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.

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Last revised on 12 July 2014.

STROUD Numerical Integration in M Dimensions