stroud_rule
stroud_rule,
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 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
c_rule,
a C code which
computes a quadrature rule which
estimates the integral of a function f(x), which might be defined over
a one dimensional region (a line) or more complex shapes such as
a circle, a triangle, a quadrilateral, a polygon, or a higher dimensional
region, and which might include an associated weight function w(x).
Reference:
-
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.
Source Code:
Last revised on 30 August 2024.