NINT_EXACTNESS_MIXED
Exactness of Multidimensional Quadrature Using Mixed Rules

NINT_EXACTNESS_MIXED is a FORTRAN90 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

where each factor region Ri is the region associated with one of the six rules. Thus, R is a sort of generalized hyperrectangle, with the understanding that in some coordinate directions the region may be semi-infinite or infinite.

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)

where each factor weight wi(xi) is the weight function associated with one of the six rules. Some weight functions include parameters alpha and beta, and these parameters may be specified independently in each dimension.

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

where each exponent Ei is a nonnegative integer.

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

is 2+1+5=8.

The corresponding monomial integral is:

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

where each exponent Ei is a nonnegative integer.

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.

When running the program, the user only enters the common prefix part of the file names, which is enough information for the program to find all the files.

Usage:

nint_exactness_mixed prefix degree_max

where

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 FORTRAN90 program which tests the polynomial exactness of one dimensional quadrature rules.

NINT_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of integration rules for the unit hypercube.

NINTLIB, a FORTRAN90 library which numerically estimates integrals in multiple dimensions.

PYRAMID_EXACTNESS, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 library which defines multidimensional quadrature rules using sparse grid techniques, based on a mixed set of 1D quadrature rules.

SPHERE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule for the unit sphere;

STROUD, a FORTRAN90 library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and multiple dimensions.

TEST_NINT, a FORTRAN90 library which defines integrand functions for testing multidimensional quadrature routines.

TESTPACK, a FORTRAN90 library which defines a set of integrands used to test multidimensional quadrature.

TETRAHEDRON_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the triangle.

Reference:

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

Source Code:

nint_exactness_mixed.f90, 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.

List of Routines:

MAIN is the main program for NINT_EXACTNESS_MIXED.
CH_CAP capitalizes a single character.
CH_EQI is a case insensitive comparison of two characters for equality.
CH_TO_DIGIT returns the integer value of a base 10 digit.
COMP_NEXT computes the compositions of the integer N into K parts.
FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
FILE_ROW_COUNT counts the number of row records in a file.
GET_UNIT returns a free FORTRAN unit number.
MONOMIAL_INTEGRAL_GENERALIZED_HERMITE evaluates a 1D monomial generalized Hermite integral.
MONOMIAL_INTEGRAL_GENERALIZED_LAGUERRE evaluates a 1D monomial generalized Laguerre integral.
MONOMIAL_INTEGRAL_HERMITE evaluates a 1D monomial Hermite integral.
MONOMIAL_INTEGRAL_JACOBI evaluates the integral of a monomial with Jacobi weight.
MONOMIAL_INTEGRAL_LAGUERRE evaluates a 1D monomial Laguerre integral.
MONOMIAL_INTEGRAL_LEGENDRE evaluates a 1D monomial Legendre integral.
MONOMIAL_INTEGRAL_MIXED evaluates a multi-D monomial mixed integral.
MONOMIAL_QUADRATURE applies a quadrature rule to a monomial.
MONOMIAL_VALUE evaluates a monomial.
R8_FACTORIAL computes the factorial of N.
R8_FACTORIAL2 computes the double factorial function N!!
R8_GAMMA evaluates Gamma(X) for a real argument.
R8_HUGE returns a very large R8.
R8_HYPER_2F1 evaluates the hypergeometric function 2F1(A,B,C,X).
R8_PSI evaluates the function Psi(X).
R8MAT_DATA_READ reads data from an R8MAT file.
R8MAT_HEADER_READ reads the header from an R8MAT file.
S_TO_I4 reads an I4 from a string.
S_TO_R8 reads an R8 from a string.
S_TO_R8VEC reads an R8VEC from a string.
S_WORD_COUNT counts the number of "words" in a string.
TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.

Last revised on 14 October 2009.

NINT_EXACTNESS_MIXED Exactness of Multidimensional Quadrature Using Mixed Rules