SANDIA_CUBATURE
Numerical Integration
in M Dimensions
SANDIA_CUBATURE
is a MATLAB library which
implements quadrature rules for certain multidimensional regions and weight functions.
We consider the following integration 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;

EN_HER, the Ndimensional product space (oo,+oo)^N,
with the Hermite weight function:
w(x) = product ( 1 <= i <= n ) exp (  x(i)^2 );

EPN_GLG, the positive product 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 product space [0,+oo)^N, with the exponential or
Laguerre weight function:
w(x) = product ( 1 <= i <= n ) exp (  x(i) );
The available rules for region EN_HER all have odd precision, ranging
from 1 to 11. Some of these rules are valid for any spatial dimension N.
However, many of these rules are restricted to a limited range, such as
2 <= N < 6. Some of the rules have two forms; in that case,
the particular form is selectable by setting an input argument OPTION
to 1 or 2. Finally, note that in multidimensional integration, the dependence
of the order O (number of abscissas) on the spatial dimension N
is critical. Rules for which the order is a multiple of 2^N are not
practical for large values of N. The source code for each rule lists its
formula for the order as a function of N.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
SANDIA_CUBATURE is available in
a C++ version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
SANDIA_RULES,
a MATLAB library which
produces 1D quadrature rules of
Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite,
generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.
STROUD,
a MATLAB library which
defines quadrature rules for a variety of multidimensional reqions.
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.

Dongbin Xiu,
Numerical integration formulas of degree two,
Applied Numerical Mathematics,
Volume 58, 2008, pages 15151520.
Source Code:
Examples and Tests:

sandia_cubature_test.m,
runs all the tests.

sandia_cubature_test_output.txt,
the output file.

cn_geg_tests.m,
tests the CN_GEG rules for various monomials.

cn_geg_test.m,
tests the CN_GEG rules for a particular monomial.

cn_jac_tests.m,
tests the CN_JAC rules for various monomials.

cn_jac_test.m,
tests the CN_JAC rules for a particular monomial.

cn_leg_tests.m,
tests the CN_LEG rules for various monomials.

cn_leg_test.m,
tests the CN_LEG rules for a particular monomial.

en_her_tests.m,
tests the EN_HER rules for various monomials.

en_her_test.m,
tests the EN_HER rules for a particular monomial.

epn_glg_tests.m,
tests the EPN_GLG rules for various monomials.

epn_glg_test.m,
tests the EPN_GLG rules for a particular monomial.

epn_lag_tests.m,
tests the EPN_LAG rules for various monomials.

epn_lag_test.m,
tests the EPN_LAG rules for a particular monomial.
You can go up one level to
the MATLAB source codes.
Last revised on 03 March 2010.