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 N-dimensional 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:

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.
3. Dongbin Xiu,
   Numerical integration formulas of degree two,
   Applied Numerical Mathematics,
   Volume 58, 2008, pages 1515-1520.

Source Code:

• 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.
• 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.
• en_her_01_1.m a precision 1 rule for EN_HER.
• en_her_02_xiu.m a precision 2 rule for EN_HER.
• en_her_03_1.m a precision 3 rule for EN_HER.
• en_her_03_2.m a precision 3 rule for EN_HER.
• en_her_03_xiu.m a precision 3 rule for EN_HER.
• en_her_05_1.m a precision 5 rule for EN_HER.
• en_her_05_2.m a precision 5 rule for EN_HER.
• en_her_05_3.m a precision 5 rule for EN_HER.
• en_her_05_4.m a precision 5 rule for EN_HER.
• en_her_05_5.m a precision 5 rule for EN_HER.
• en_her_05_6.m a precision 5 rule for EN_HER.
• en_her_07_1.m a precision 7 rule for EN_HER.
• en_her_07_2.m a precision 7 rule for EN_HER.
• en_her_07_3.m a precision 7 rule for EN_HER.
• en_her_09_1.m a precision 9 rule for EN_HER.
• en_her_11_1.m a precision 11 rule for EN_HER.
• en_her_monomial_integral.m evaluates the exact integral of a monomial in Stroud's EN_HER 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.
• monomial_value.m, evaluates a monomial given the exponents.
• r8_factorial.m evaluates the factorial function.
• 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).
• timestamp.m, prints the YMDHMS date as a timestamp;

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.