INT_EXACTNESS
Exactness of One Dimensional Quadrature Rules

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 + 27

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

• prefix_x.txt
• prefix_w.txt
• prefix_r.txt

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:

• prefix_exact.txt

Usage:

int_exactness prefix degree_max

• prefix is the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
• degree_max is the maximum monomial degree to check. This would normally be a relatively small nonnegative number, such as 5, 10 or 15.

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:

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

Related Data and Programs:

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.

Reference:

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

Source Code:

• int_exactness.cpp, the source code.

Examples and Tests:

CC_D1_O2 is a Clenshaw-Curtis order 2 rule for 1D.

• cc_d1_o2_x.txt, the abscissas of the rule.
• cc_d1_o2_w.txt, the weights of the rule.
• cc_d1_o2_r.txt, defines the region for the rule.
• cc_d1_o2_exact.txt, the results of the exactness test, up to degree 5.

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!

• cc_d1_o3_x.txt, the abscissas of the rule.
• cc_d1_o3_w.txt, the weights of the rule.
• cc_d1_o3_r.txt, defines the region for the rule.
• cc_d1_o3_exact.txt, the results of the exactness test, up to degree 5.

GL_D1_O3 is a Gauss-Legendre order 3 rule for 1D.

• gl_d1_o3_x.txt, the abscissas of the rule.
• gl_d1_o3_w.txt, the weights of the rule.
• gl_d1_o3_r.txt, defines the region for the rule.
• gl_d1_o3_exact.txt, the results of the exactness test, up to degree 5.

NCC_D1_O5 is a Newton-Cotes Closed order 5 rule for 1D.

• ncc_d1_o5_x.txt, the abscissas of the rule.
• ncc_d1_o5_w.txt, the weights of the rule.
• ncc_d1_o5_r.txt, defines the region for the rule.
• ncc_d1_o5_exact.txt, the results of the exactness test, up to degree 7.

List of Routines:

• MAIN is the main program for INT_EXACTNESS.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DTABLE_DATA_READ reads the data from a DTABLE file.
• DTABLE_HEADER_READ reads the header from a DTABLE file.
• 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.
• MONOMIAL_QUADRATURE applies a quadrature rule to a monomial.
• R8_ABS returns the absolute value of an R8.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• 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.
• TIMESTRING returns the current YMDHMS date as a string.

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