INT_EXACTNESS_CHEBYSHEV2
Exactness of Gauss-Chebyshev Type 2 Quadrature Rules

INT_EXACTNESS_CHEBYSHEV2 is a C++ program which investigates the polynomial exactness of a Gauss-Chebyshev type 2 quadrature rule for the interval [-1,+1].

Standard Gauss-Chebyshev type 2 quadrature assumes that the integrand we are considering has a form like:

        Integral ( -1 <= x <= +1 ) f(x) * sqrt ( 1 - x^2 ) dx
      

A standard Gauss-Chebyshev type 2 quadrature rule is a set of n positive weights w and abscissas x so that

        Integral ( -1 <= x <= +1 ) f(x) * ( sqrt ( 1 - x^2 ) dx
      

        Sum ( 1 <= I <= N ) w(i) * f(x(i))
      

For a standard Gauss-Chebyshev type 2 rule, polynomial exactness is defined in terms of the function f(x). That is, we say the rule is exact for polynomials up to degree DEGREE_MAX if, for any polynomial f(x) of that degree or less, the quadrature rule will produce the exact value of

        Integral ( -1 <= x <= +1 ) f(x) * sqrt ( 1 - x^2 ) dx
      

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 top be checked 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_chebyshev2 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_CHEBYSHEV2 is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CHEBYSHEV_POLYNOMIAL, a C++ library which evaluates the Chebyshev polynomial and associated functions.

CHEBYSHEV2_RULE, a C++ program which generates a Gauss-Chebyshev type 2 quadrature rule.

HERMITE_EXACTNESS, a C++ program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

INT_EXACTNESS, a C++ program which tests the polynomial exactness of a quadrature rule for a finite interval.

INT_EXACTNESS_CHEBYSHEV1, a C++ program which tests the polynomial exactness of Gauss-Chebyshev type 1 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 generalized Gauss-Hermite quadrature rules.

INT_EXACTNESS_GEN_LAGUERRE, a C++ program which tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.

INT_EXACTNESS_JACOBI, a C++ program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

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].

Reference:

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

Source Code:

• int_exactness_chebyshev2.cpp, the source code.

Examples and Tests:

CHEBY2_O1 is a standard Gauss-Chebyshev type 2 order 1 rule.

• cheby2_o1_x.txt, the abscissas of the rule.
• cheby2_o1_w.txt, the weights of the rule.
• cheby2_o1_r.txt, defines the region for the rule.
• cheby2_o1_exact.txt, the results of the command

  int_exactness_chebyshev2 cheby2_o1 5

CHEBY2_O2 is a standard Gauss-Chebyshev type 2 order 2 rule.

• cheby2_o2_x.txt, the abscissas of the rule.
• cheby2_o2_w.txt, the weights of the rule.
• cheby2_o2_r.txt, defines the region for the rule.
• cheby2_o2_exact.txt, the results of the command

  int_exactness_chebyshev2 cheby2_o2 5

CHEBY2_O4 is a standard Gauss-Chebyshev type 2 order 4 rule.

• cheby2_o4_x.txt, the abscissas of the rule.
• cheby2_o4_w.txt, the weights of the rule.
• cheby2_o4_r.txt, defines the region for the rule.
• cheby2_o4_exact.txt, the results of the command

  int_exactness_chebyshev2 cheby2_o4 10

CHEBY2_O8 is a standard Gauss-Chebyshev type 2 order 8 rule.

• cheby2_o8_x.txt, the abscissas of the rule.
• cheby2_o8_w.txt, the weights of the rule.
• cheby2_o8_r.txt, defines the region for the rule.
• cheby2_o8_exact.txt, the results of the exactness test.
the results of the command

int_exactness_chebyshev2 cheby2_o8 18

CHEBY2_O16 is a standard Gauss-Chebyshev type 2 order 16 rule.

• cheby2_o16_x.txt, the abscissas of the rule.
• cheby2_o16_w.txt, the weights of the rule.
• cheby2_o16_r.txt, defines the region for the rule.
• cheby2_o16_exact.txt, the results of the command

  int_exactness_chebyshev2 cheby2_o16 35

List of Routines:

• MAIN is the main program for INT_EXACTNESS_CHEBYSHEV2.
• 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.
• CHEBYSHEV2_INTEGRAL evaluates a Chebyshev type 2 monomial integral.
• DTABLE_DATA_READ reads 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_CHEBYSHEV2 applies quadrature to a Chebyshev type 2 monomial.
• 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 writes the current YMDHMS date into a string.

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