jburkardt

SANDIA_CUBATURE\ Numerical Integration\ in M Dimensions {#sandia_cubature-numerical-integration-in-m-dimensions align=”center”} ======================

---

SANDIA_CUBATURE is a C++ 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: {#licensing align=”center”}

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages: {#languages align=”center”}

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

Related Data and Programs: {#related-data-and-programs align=”center”}

SANDIA_RULES, a C++ 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 C++ library which defines quadrature rules for a variety of multidimensional reqions.

Reference: {#reference align=”center”}

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: {#source-code align=”center”}

• sandia_cubature.cpp, the source code.
• sandia_cubature.hpp, the include file.

Examples and Tests: {#examples-and-tests align=”center”}

• sandia_cubature_prb.cpp, a sample calling program.
• sandia_cubature_prb_output.txt, the output file.

List of Routines: {#list-of-routines align=”center”}

• C1_GEG_MONOMIAL_INTEGRAL: integral of monomial with Gegenbauer weight on C1.
• C1_JAC_MONOMIAL_INTEGRAL: integral of a monomial with Jacobi weight over C1.
• C1_LEG_MONOMIAL_INTEGRAL: integral of monomial with Legendre weight on C1.
• CN_GEG_01_1 implements a precision 1 rule for region CN_GEG.
• CN_GEG_01_1_SIZE sizes a precision 1 rule for region CN_GEG.
• CN_GEG_02_XIU implements the Xiu precision 2 rule for region CN_GEG.
• CN_GEG_02_XIU_SIZE sizes the Xiu precision 2 rule for region CN_GEG.
• CN_GEG_03_XIU implements the Xiu precision 3 rule for region CN_GEG.
• CN_GEG_03_XIU_SIZE sizes the Xiu precision 3 rule for region CN_GEG.
• CN_GEG_MONOMIAL_INTEGRAL: integral of monomial with Gegenbauer weight on CN.
• CN_JAC_01_1 implements a precision 1 rule for region CN_JAC.
• CN_JAC_01_1_SIZE sizes a precision 1 rule for region CN_JAC.
• CN_JAC_02_XIU implements the Xiu precision 2 rule for region CN_JAC.
• CN_JAC_02_XIU_SIZE sizes the Xiu precision 2 rule for region CN_JAC.
• CN_JAC_MONOMIAL_INTEGRAL: integral of a monomial with Jacobi weight over CN.
• CN_LEG_01_1 implements the midpoint rule for region CN_LEG.
• CN_LEG_01_1_SIZE sizes the midpoint rule for region CN_LEG.
• CN_LEG_02_XIU implements the Xiu precision 2 rule for region CN_LEG.
• CN_LEG_02_XIU_SIZE sizes the Xiu precision 2 rule for region CN_LEG.
• CN_LEG_03_1 implements the Stroud rule CN:3-1 for region CN_LEG.
• CN_LEG_03_1_SIZE sizes the Stroud rule CN:3-1 for region CN_LEG.
• CN_LEG_03_XIU implements the Xiu precision 3 rule for region CN_LEG.
• CN_LEG_03_XIU_SIZE sizes the Xiu precision 3 rule for region CN_LEG.
• CN_LEG_05_1 implements the Stroud rule CN:5-1 for region CN_LEG.
• CN_LEG_05_1_SIZE sizes the Stroud rule CN:5-1 for region CN_LEG.
• CN_LEG_05_2 implements the Stroud rule CN:5-2 for region CN_LEG.
• CN_LEG_05_2_SIZE sizes the Stroud rule CN:5-2 for region CN_LEG.
• CN_LEG_MONOMIAL_INTEGRAL: integral of monomial with Legendre weight on CN.
• EN_HER_01_1 implements the Stroud rule 1.1 for region EN_HER.
• EN_HER_01_1_SIZE sizes the Stroud rule 1.1 for region EN_HER.
• EN_HER_02_XIU implements the Xiu precision 2 rule for region EN_HER.
• EN_HER_02_XIU_SIZE sizes the Xiu precision 2 rule for region EN_HER.
• EN_HER_03_1 implements the Stroud rule 3.1 for region EN_HER.
• EN_HER_03_1_SIZE sizes the Stroud rule 3.1 for region EN_HER.
• EN_HER_03_XIU implements the Xiu precision 3 rule for region EN_HER.
• EN_HER_03_XIU_SIZE sizes the Xiu precision 3 rule for region EN_HER.
• EN_HER_05_1 implements the Stroud rule 5.1 for region EN_HER.
• EN_HER_05_1_SIZE sizes the Stroud rule 5.1 for region EN_HER.
• EN_HER_05_2 implements the Stroud rule 5.2 for region EN_HER.
• EN_HER_05_2_SIZE sizes the Stroud rule 5.2 for region EN_HER.
• EN_HER_MONOMIAL_INTEGRAL evaluates monomial integrals in EN_HER.
• EP1_GLG_MONOMIAL_INTEGRAL: integral of monomial with GLG weight on EP1.
• EP1_LAG_MONOMIAL_INTEGRAL: integral of monomial with Laguerre weight on EP1.
• EPN_GLG_01_1 implements a precision 1 rule for region EPN_GLG.
• EPN_GLG_01_1_SIZE sizes a precision 1 rule for region EPN_GLG.
• EPN_GLG_02_XIU implements the Xiu precision 2 rule for region EPN_GLG.
• EPN_GLG_02_XIU_SIZE sizes the Xiu precision 2 rule for region EPN_GLG.
• EPN_GLG_MONOMIAL_INTEGRAL: integral of monomial with GLG weight on EPN.
• EPN_LAG_01_1 implements a precision 1 rule for region EPN_LAG.
• EPN_LAG_01_1_SIZE sizes a precision 1 rule for region EPN_LAG.
• EPN_LAG_02_XIU implements the Xiu precision 2 rule for region EPN_LAG.
• EPN_LAG_02_XIU_SIZE sizes the Xiu precision 2 rule for region EPN_LAG.
• EPN_LAG_MONOMIAL_INTEGRAL: integral of monomial with Laguerre weight on EPN.
• GW_02_XIU implements the Golub-Welsch version of the Xiu rule.
• GW_02_XIU_SIZE sizes the Golub Welsch version of the Xiu rule.
• MONOMIAL_VALUE evaluates a monomial.

You can go up one level to the C++ source codes.

---

Last revised on 05 March 2010.