NINTLIB
Multi-dimensional quadrature

NINTLIB is a library of MATLAB routines, using double precision arithmetic, for estimating integrals over multi-dimensional regions.

Please note that these routines are simple and academic. A good program for computing an integral in multiple dimensions must include error estimation and adaptivity. Simple straightforward approaches to reducing the error will cause a ruinous explosion in the number of function evaluations required.

Related Data and Programs:

CLENSHAW_CURTIS is a MATLAB library which can set up a Clenshaw Curtis quadrature grid in multiple dimensions.

FEKETE is a MATLAB library of routines for defining a Fekete rule for quadrature or interpolation over a triangle.

INTEGRAL_TEST is a FORTRAN90 executable program which tests the suitability of a set of N points for use in an equal-weight quadrature rule over the multi-dimensional unit hypercube.

INTLIB is a FORTRAN90 library of routines for estimating the integral of a function over a one-dimensional interval.

NINT_EXACTNESS is a MATLAB program which demonstrates how to measure the polynomial exactness of a multidimensional quadrature rule.

NINTLIB is also available in a C++ version and a FORTRAN77 version and a FORTRAN90 version.

PRODUCT_RULE is an executable MATLAB program which can create a multidimensional quadrature rule as a product of one dimensional rules.

QUADRATURE_TEST an executable MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.

QUADRULE is a MATLAB library of routines which define a variety of (mostly 1-dimensional) quadrature rules.

QUADRULE_FAST is a MATLAB library of routines defining efficient versions of a few 1D quadrature rules.

STROUD is a MATLAB library of routines which define a variety of quadrature rules over various "interesting" geometric shapes.

TEST_NINT is a MATLAB library of routines that can be used to test multi-dimensional quadrature routines.

TEST_TRI_INT is a MATLAB library of functions that can be used to test algorithms for quadrature over a triangle.

TESTPACK is a MATLAB library of functions which define a set of integrands used to test multidimensional quadrature.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.

Source Code:

• box_nd.m, computes the integral of F(X) over a product box.
• i4_huge.m, returns a huge I4.
• monte_carlo_nd.m, computes a Monte Carlo estimate of an integral in a box.
• p5_nd.m, approximates the integral of F(X) over a product region.
• r8_epsilon.m, returns the R8 roundoff unit.
• r8vec_uniform_01.m, returns a unit pseudorandom R8VEC.
• romberg_nd.m, approximates the integral of a F(X) over a product region.
• sample_nd.m, estimates the integral of F(X) over the multi-dimensional unit box.
• sum2_nd.m, computes the integral of F(X) over a multi-dimensional product region.
• timestamp.m, prints the current YMDHMS date as a time stamp.
• tuple_next.m, computes the next element of a tuple space.

Examples and Tests:

• nintlib_test.m, calls all the tests;
• nintlib_testnd.m, calls all the integrators for one test integrand.
• nintlib_test01.m, tests BOX_ND.
• nintlib_test02.m, tests P5_ND.
• nintlib_test03.m, tests ROMBERG_ND.
• nintlib_test04.m, tests SAMPLE_ND;
• nintlib_test05.m, demonstrates how to refine multi-dimensional integration results.
• nintlib_test06.m, tests MONTE_CARLO_ND.
• nintlib_prb.out, sample problem output.
• f1dn.m, evaluates integrand F(X(1:DIM_NUM)) = 1;
• fbdn.m, evaluates integrand F(X(1:DIM_NUM)) = 1/(1+(X(1:DIM_NUM)**2));
• fedn.m, evaluates integrand F(X(1:DIM_NUM)) = exp (sum(X(1:DIM_NUM));
• fxdn.m, evaluates integrand F(X(1:DIM_NUM)) = sum ( X(1:DIM_NUM) );
• fx2dn.m, evaluates integrand F(X(1:DIM_NUM)) = sum ( X(1:DIM_NUM)^2 );
• fx3dn.m, evaluates integrand F(X(1:DIM_NUM)) = sum ( X(1:DIM_NUM)^3 );

You can go up one level to the MATLAB source codes.