STROUD
Numerical Integration
in M Dimensions

STROUD, a MATLAB 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 FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

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

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

stroud_test

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

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

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

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

TRIANGLE_FEKETE, a MATLAB library which defines Fekete rules for quadrature or interpolation 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 MATLAB library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a triangle in 2D.

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

TRIANGLE_WANDZURA_RULE, a MATLAB 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:

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;
c1_geg_monomial_integral.m integral of a monomial with Gegenbauer weight over C1.
c1_jac_monomial_integral.m integral of a monomial with Jacobi weight over C1.
c1_leg_monomial_integral.m integral of a monomial with Legendre weight over C1.
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;
cn_geg_00_1.m implements the midpoint rule for region CN_GEG.
cn_geg_01_1.m implements a precision 1 rule for region CN_GEG.
cn_geg_02_xiu.m implements the Xiu rule for region CN_GEG.
cn_geg_03_xiu.m implements the Xiu rule for region CN_GEG.
cn_geg_monoial_integral.m integral of a monomial with Gegenbauer weight over CN.
cn_jac_00_1.m implements the midpoint rule for region CN_JAC.
cn_jac_01_1.m implements a precision 1 rule for region CN_JAC.
cn_jac_02_xiu.m implements the Xiu rule for region CN_JAC.
cn_jac_monoial_integral.m integral of a monomial with Jacobi weight over CN.
cn_leg_01_1.m implements the midpoint rule for region CN_LEG.
cn_leg_02_xiu.m implements the Xiu rule for region CN_LEG.
cn_leg_03_1.m implements the Stroud CN:3-1 rule for region CN_LEG.
cn_leg_03_xiu.m implements the Xiu rule for region CN_LEG.
cn_leg_05_1.m implements the Stroud CN:5-1 rule for region CN_LEG.
cn_leg_05_2.m implements the Stroud CN:5-2 rule for region CN_LEG.
cn_leg_monoial_integral.m integral of a monomial with Legendre weight over CN.
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;
en_r2_01_1.m a precision 1 rule for EN_R2.
en_r2_02_xiu.m a precision 2 rule for EN_R2.
en_r2_03_1.m a precision 3 rule for EN_R2.
en_r2_03_2.m a precision 3 rule for EN_R2.
en_r2_03_xiu.m a precision 3 rule for EN_R2.
en_r2_05_1.m a precision 5 rule for EN_R2.
en_r2_05_2.m a precision 5 rule for EN_R2.
en_r2_05_3.m a precision 5 rule for EN_R2.
en_r2_05_4.m a precision 5 rule for EN_R2.
en_r2_05_5.m a precision 5 rule for EN_R2.
en_r2_05_6.m a precision 5 rule for EN_R2.
en_r2_07_1.m a precision 7 rule for EN_R2.
en_r2_07_2.m a precision 7 rule for EN_R2.
en_r2_07_3.m a precision 7 rule for EN_R2.
en_r2_09_1.m a precision 9 rule for EN_R2.
en_r2_11_1.m a precision 11 rule for EN_R2.
en_r2_monomial_integral.m evaluates the exact integral of a monomial in Stroud's EN_R2 region.
ep1_glg_monomial_integral.m integral of monomial with generalized Laguerre weight on EP1.
ep1_lag_monomial_integral.m integral of monomial with Laguerre weight on EP1.
epn_glg_001.m implements the midpoint rule for region EPN_GLG.
epn_glg_01_1.m implements a precision 1 rule for region EPN_GLG.
epn_glg_02_xiu.m implements the Xiu rule for region EPN_GLG.
epn_glg_monomial_integral.m integral of monomial with generalized Laguerre weight on EPN.
epn_lag_001.m implements the midpoint rule for region EPN_LAG.
epn_lag_01_1.m implements a precision 1 rule for region EPN_LAG.
epn_lag_02_xiu.m implements the Xiu rule for region EPN_LAG.
epn_lag_monomial_integral.m integral of monomial with Laguerre weight on EPN.
gw_02_xiu.m implements the Golub-Welsch version of the Xiu rule.
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;
monomial_value.m, evaluates a monomial given the exponents.
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_asin.m, computes the inverse sine of an angle;
r8_choose.m, computes the binomial coefficient C(N,K).
r8_factorial.m evaluates the factorial function.
r8_huge.m, returns a "huge" R8;
r8_hyper_2f1.m evaluates the hypergeometric function 2F1(A,B,C,X).
r8_mop.m returns the I-th power of -1 as an R8 value.
r8_psi.m evaluates the function Psi(X).
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, computes the volume of the unit tetrahedron;
tetra_volume.m, computes the volume of a tetrahedron;
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 of the unit triangle;
triangle_volume.m, returns the volume 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;