stroud

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

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

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

stroud_test

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

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

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

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

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

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

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

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

TRIANGLE_WANDZURA_RULE, a C++ code 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.cpp, the source code;
stroud.sh, compiles the source code;
stroud.hpp, the include file;

Last revised on 20 April 2020.