NINT_EXACTNESS_MIXED
Exactness of Multidimensional Quadrature Using Mixed Rules

NINT_EXACTNESS_MIXED is a MATLAB program which investigates the polynomial exactness of a multidimensional quadrature rule based on a direct product of one-dimensional rules. The one dimensional rules, in turn, may be any mixture of rules for 6 standard quadrature problems.

The six standard 1d quadrature problems that may be used as factors for the multidimensional problem are:

• Legendre integration
  interval: [-1,+1],
  weight: w(x) = 1,
  rules: Clenshaw Curtis, Fejer Type 2, Gauss Legendre, Gauss Patterson.
• Jacobi integration:
  interval: [-1,+1],
  weight: w(x) = (1-x)^alpha (1+x)^beta.
  rules: Gauss Jacobi.
• Laguerre integration
  interval: [0,+oo),
  weight: w(x) = exp(-x).
  rules: Gauss Laguerre.
• Generalized Laguerre integration
  interval: [0,+oo),
  weight: w(x) = x^alpha exp(-x).
  rules: Gauss Laguerre.
• Hermite integration
  interval: (-oo,+oo),
  weight: w(x) = exp(-x*x).
  rules: Gauss Hermite.
• Generalized Hermite integration
  interval: (-oo,+oo),
  weight: w(x) = |x|^alpha exp(-x*x).
  rules: generalized Gauss Hermite.

The M-dimensional quadrature region R based on mixed factors is formed by the direct product

R = R1 x R2 x ... x Rm

The M-dimensional weight function W based on mixed factors is formed by the dproduct

w(x1,x2,...xm) = w1(x1) x w2(x2) x ... x wm(xm)

For a quadrature region R based on mixed factors, the corresponding monomial integrand has the form

Mono(X,E) = X1^E1 x X2^E2 x ... x Xm^Em

The total degree of a monomial Mono(X,E) is:

TotalDegree(Mono(X,E)) = Sum ( 1 <= I <= M ) E(I)

Thus, for instance, the total degree of

x1² * x2 * x3⁵

The corresponding monomial integral is:

Integral ( X in R ) Mono(X,E) W(X) dX

The monomial exactness of a quadrature rule is the maximum number D such that, for every monomial of total degree D or less, the quadrature rule produces the exact value of the monomial integral.

The polynomial exactness of a quadrature rule is the maximum number D such that, for every polynomial of total degree D or less, the quadrature rule produces the exact value of the polynomial integral. The total degree of a polynomial is simply the maximum of the total degrees of the monomials that form the polynomial.

This program is given a quadrature rule based on mixed factors, and seeks to determine the polynomial exactness of the rule. It does this simply by applying the quadrature rule to all the monomials of a total degree 0 up to some limit specified by the user.

The program is very flexible and interactive. The quadrature rule is defined by five files, to be read at input, and the maximum degree is specified by the user as well.

The files that define the quadrature rule are assumed to have related names, of the form

• prefix_a.txt, the "ALPHA" file;
• prefix_b.txt, the "BETA" file;
• prefix_r.txt, the "REGION" file;
• prefix_w.txt, the "WEIGHT" file;
• prefix_x.txt, the "ABSCISSA" file.

Usage:

nint_exactness_mixed ( 'prefix', degree_max )

• 'prefix' is the common prefix for the files containing the alpha, beta, region, weight and abscissa information of the quadrature rule;
• degree_max is the maximum total monomial degree to check. This should be a relatively small nonnegative number, particularly if the spatial dimension is high. A value of 5 or 10 might be reasonable, but a value of 50 or 100 is probably never a good input!

If the arguments are not supplied on the command line, the program will prompt for them.

Licensing:

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

Languages:

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

Related Data and Programs:

INT_EXACTNESS, a MATLAB program which tests the polynomial exactness of one dimensional quadrature rules.

INTEGRAL_TEST, a FORTRAN90 program which uses test integrals to measure the effectiveness of certain sets of quadrature rules.

NINT_EXACTNESS, a MATLAB program which tests the polynomial exactness of integration rules for the unit hypercube.

NINTLIB, a MATLAB library which numerically estimates integrals in multiple dimensions.

PYRAMID_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the pyramid.

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRULE, a MATLAB library which defines quadrature rules on a variety of intervals with different weight functions.

SPARSE_GRID_MIXED, a dataset directory which contains multidimensional Smolyak sparse grids based on a mixed set of 1D factor rules.

SPARSE_GRID_MIXED, a MATLAB library which defines multidimensional quadrature rules using sparse grid techniques, based on a mixed set of 1D quadrature rules.

SPHERE_EXACTNESS, a MATLAB program which tests the polynomial exactness of a quadrature rule for the unit sphere;

TEST_NINT, a MATLAB library which defines integrand functions for testing multidimensional quadrature routines.

TESTPACK, a MATLAB library which defines a set of integrands used to test multidimensional quadrature.

TETRAHEDRON_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

Reference:

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

Source Code:

• nint_exactness_mixed.m, is the source code.

Examples and Tests:

SPARSE_GRID_MIXED_D2_L2_CCXCC is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Clenshaw Curtis].

• sparse_grid_mixed_d2_l2_ccxcc_a.txt, the A file.
• sparse_grid_mixed_d2_l2_ccxcc_b.txt, the B file.
• sparse_grid_mixed_d2_l2_ccxcc_r.txt, the R file.
• sparse_grid_mixed_d2_l2_ccxcc_w.txt, the W file.
• sparse_grid_mixed_d2_l2_ccxcc_x.txt, the X file.
• sparse_grid_mixed_d2_l2_ccxcc_exactness.txt, the output file for DEGREE_MAX = 7.

SPARSE_GRID_MIXED_D2_L2_CCXGL is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Legendre].

• sparse_grid_mixed_d2_l2_ccxgl_a.txt, the A file.
• sparse_grid_mixed_d2_l2_ccxgl_b.txt, the B file.
• sparse_grid_mixed_d2_l2_ccxgl_r.txt, the R file.
• sparse_grid_mixed_d2_l2_ccxgl_w.txt, the W file.
• sparse_grid_mixed_d2_l2_ccxgl_x.txt, the X file.
• sparse_grid_mixed_d2_l2_ccxgl_exactness.txt, the output file for DEGREE_MAX = 7.

SPARSE_GRID_MIXED_D2_L2_CCXGLG is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Generalized Gauss Laguerre]. The Generalized Gauss Laguerre rule uses ALPHA = 1.5.

• sparse_grid_mixed_d2_l2_ccxglg_a.txt, the A file.
• sparse_grid_mixed_d2_l2_ccxglg_b.txt, the B file.
• sparse_grid_mixed_d2_l2_ccxglg_r.txt, the R file.
• sparse_grid_mixed_d2_l2_ccxglg_w.txt, the W file.
• sparse_grid_mixed_d2_l2_ccxglg_x.txt, the X file.
• sparse_grid_mixed_d2_l2_ccxglg_exactness.txt, the output file for DEGREE_MAX = 7.

SPARSE_GRID_MIXED_D2_L2_CCXLG is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Laguerre].

• sparse_grid_mixed_d2_l2_ccxlg_a.txt, the A file.
• sparse_grid_mixed_d2_l2_ccxlg_b.txt, the B file.
• sparse_grid_mixed_d2_l2_ccxlg_r.txt, the R file.
• sparse_grid_mixed_d2_l2_ccxlg_w.txt, the W file.
• sparse_grid_mixed_d2_l2_ccxlg_x.txt, the X file.
• sparse_grid_mixed_d2_l2_ccxlg_exactness.txt, the output file for DEGREE_MAX = 7.

SPARSE_GRID_MIXED_D2_L2_F2XGJ is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Fejer Type 2, Gauss Jacobi]. The Gauss Jacobi rule uses ALPHA = 0.5, BETA = 1.5.

• sparse_grid_mixed_d2_l2_f2xgj_a.txt, the A file.
• sparse_grid_mixed_d2_l2_f2xgj_b.txt, the B file.
• sparse_grid_mixed_d2_l2_f2xgj_r.txt, the R file.
• sparse_grid_mixed_d2_l2_f2xgj_w.txt, the W file.
• sparse_grid_mixed_d2_l2_f2xgj_x.txt, the X file.
• sparse_grid_mixed_d2_l2_f2xgj_exactness.txt, the output file for DEGREE_MAX = 7.

SPARSE_GRID_MIXED_D2_L3_CCXGP is a level 3 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Patterson].

• sparse_grid_mixed_d2_l3_ccxgp_a.txt, the A file.
• sparse_grid_mixed_d2_l3_ccxgp_b.txt, the B file.
• sparse_grid_mixed_d2_l3_ccxgp_r.txt, the R file.
• sparse_grid_mixed_d2_l3_ccxgp_w.txt, the W file.
• sparse_grid_mixed_d2_l3_ccxgp_x.txt, the X file.
• sparse_grid_mixed_d2_l3_ccxgp_exactness.txt, the output file for DEGREE_MAX = 9.

SPARSE_GRID_MIXED_D3_L2_CCXF2XGH is a level 2 sparse grid quadrature rule for dimension 3, based on 1D factors of [Clenshaw Curtis, Fejer Type 2, Gauss Hermite].

• sparse_grid_mixed_d3_l2_ccxf2xgh_a.txt, the A file.
• sparse_grid_mixed_d3_l2_ccxf2xgh_b.txt, the B file.
• sparse_grid_mixed_d3_l2_ccxf2xgh_r.txt, the R file.
• sparse_grid_mixed_d3_l2_ccxf2xgh_w.txt, the W file.
• sparse_grid_mixed_d3_l2_ccxf2xgh_x.txt, the X file.
• sparse_grid_mixed_d3_l2_ccxf2xgh_exactness.txt, the output file.

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