stroud_rule

stroud_rule, an Octave 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 information on this web page is distributed under the MIT license.

Languages:

stroud_rule is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

stroud_rule_test

disk_rule, an Octave 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).

hexagon_stroud_rule, an Octave code which computes one of four Stroud quadrature rules over the interior of the unit hexagon.

pyramid_rule, an Octave code which computes a quadrature rule for a pyramid.

simplex_gm_rule, an Octave 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, an Octave code which computes Lebedev quadrature rules on the surface of the unit sphere in 3d.

tetrahedron_arbq_rule, an Octave 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, an Octave code which defines Keast quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3d.

tetrahedron_ncc_rule, an Octave code which defines Newton-Cotes closed (NCC) quadrature rules over the interior of a tetrahedron in 3d.

tetrahedron_nco_rule, an Octave code which defines Newton-Cotes open (NCO) quadrature rules over the interior of a tetrahedron in 3d.

triangle_dunavant_rule, an Octave code which defines Dunavant rules for quadrature over the interior of a triangle in 2d.

triangle_fekete_rule, an Octave code which defines Fekete rules for quadrature or interpolation over the interior of a triangle in 2d.

triangle_lyness_rule, a MATLAB library which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2d.

triangle_ncc_rule, an Octave code which defines Newton-Cotes closed (NCC) quadrature rules over the interior of a triangle in 2d.

triangle_nco_rule, an Octave code which defines Newton-Cotes open (NCO) quadrature rules over the interior of a triangle in 2d.

triangle_wandzura_rule, an Octave code which returns quadrature rules of exactness 5, 10, 15, 20, 25 and 30 over the interior of the triangle in 2d.

Reference:

1. Arthur Stroud,
   Approximate Calculation of Multiple Integrals,
   Prentice Hall, 1971,
   ISBN: 0130438936,
   LC: QA311.S85.
2. Arthur Stroud, Don Secrest,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.

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_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_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;
• 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;