# 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) );

### Languages:

stroud is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_TRI, a dataset directory which defines quadrature rules to be applied to triangular regions.

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

SPHERE_LEBEDEV_RULE, a dataset directory which contains sets of points on a sphere which can be used for quadrature rules of a known precision;

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_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.

### Reference:

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### Source Code:

Last revised on 11 August 2019.