INT_EXACTNESS is a C++ program which investigates the polynomial exactness of a one dimensional quadrature rule defined on a finite interval.
The polynomial exactness of a quadrature rule is defined as the highest degree D such that the quadrature rule is guaranteed to integrate exactly all polynomials of degree DEGREE_MAX or less, ignoring roundoff. The degree of a polynomial is the maximum of the degrees of all its monomial terms. The degree of a monomial term is the exponent. Thus, for instance, the DEGREE of
3*x5 - 7*x9 + 27is the maximum of 5, 9 and 0, so it is 9.
To be thorough, the program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates the corresponding monomial term, applies the quadrature rule to it, and determines the quadrature error. The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.
The program is very flexible and interactive. The quadrature rule is defined by three files, to be read at input, and the maximum degree is specified by the user as well.
Note that the three files that define the quadrature rule are assumed to have related names, of the form
For information on the form of these files, see the QUADRATURE_RULES directory listed below.
The exactness results are written to an output file with the corresponding name:
int_exactness prefix degree_maxwhere
If the arguments are not supplied on the command line, the program will prompt for them.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
INT_EXACTNESS is available in a C++ version and a FORTRAN90 version and a MATLAB version.
EXACTNESS, a C++ library which investigates the exactness of quadrature rules that estimate the integral of a function with a density, such as 1, exp(-x) or exp(-x^2), over an interval such as [-1,+1], [0,+oo) or (-oo,+oo).
HERMITE_EXACTNESS, a C++ program which tests the polynomial exactness of Gauss-Hermite quadrature rules.
INT_EXACTNESS_CHEBYSHEV1, a C++ program which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.
INT_EXACTNESS_CHEBYSHEV2, a C++ program which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.
INT_EXACTNESS_GEGENBAUER, a C++ program which tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.
INT_EXACTNESS_GEN_HERMITE, a C++ program which tests the polynomial exactness of a generalized Gauss-Hermite quadrature rule.
INT_EXACTNESS_GEN_LAGUERRE, a C++ program which tests the polynomial exactness of a generalized Gauss-Laguerre quadrature rule.
INT_EXACTNESS_JACOBI, a C++ program which tests the polynomial exactness of a Gauss-Jacobi quadrature rule.
LAGUERRE_EXACTNESS, a C++ program which tests the polynomial exactness of Gauss-Laguerre quadrature rules for integration over [0,+oo) with density function exp(-x).
LEGENDRE_EXACTNESS, a C++ program which tests the monomial exactness of quadrature rules for the Legendre problem of integrating a function with density 1 over the interval [-1,+1].
NINT_EXACTNESS, a C++ program which tests the polynomial exactness of multidimensional integration rules.
CC_D1_O2 is a Clenshaw-Curtis order 2 rule for 1D.
CC_D1_O3 is a Clenshaw-Curtis order 3 rule for 1D. If you are paying attention, you may be surprised to see that a Clenshaw Curtis rule of odd order has one more degree of accuracy than you'd expect!
GL_D1_O3 is a Gauss-Legendre order 3 rule for 1D.
NCC_D1_O5 is a Newton-Cotes Closed order 5 rule for 1D.
You can go up one level to the C++ source codes.