INT_EXACTNESS_CHEBYSHEV2
Exactness of Gauss-Chebyshev Type 2 Quadrature Rules

INT_EXACTNESS_CHEBYSHEV2 is a MATLAB program, using double precision arithmetic, 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'

a quoted string, the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;

degree_max

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.

Related Data and Programs:

CHEBYSHEV2_RULE is an executable MATLAB program which generates a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS is a MATLAB library which sets up a Clenshaw Curtis quadrature grid in multiple dimensions.

INT_EXACTNESS is an executable MATLAB program which tests the polynomial exactness of a quadrature rule for a finite interval.

INT_EXACTNESS_CHEBYSHEV1 is an executable MATLAB program which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

INT_EXACTNESS_CHEBYSHEV2 is also available in a C++ version and a FORTRAN90 version.

INT_EXACTNESS_GEGENBAUER is an executable MATLAB program which tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.

INT_EXACTNESS_GEN_HERMITE is an executable MATLAB program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.

INT_EXACTNESS_GEN_LAGUERRE is an executable MATLAB program which tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.

INT_EXACTNESS_HERMITE is an executable MATLAB program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

INT_EXACTNESS_JACOBI is an executable MATLAB program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

INT_EXACTNESS_LAGUERRE is an executable MATLAB program which tests the polynomial exactness of Gauss-Laguerre quadrature rules.

INT_EXACTNESS_LEGENDRE is an executable MATLAB program which tests the polynomial exactness of Gauss-Legendre quadrature rules.

INTEGRAL_TEST is an executable FORTRAN90 program which uses test integrals to measure the effectiveness of certain sets of quadrature rules.

INTLIB is a FORTRAN90 library which numerically estimates integrals in one dimension.

QUADRATURE_RULES_CHEBYSHEV2 is a dataset directory which contains sets of files that define Gauss-Chebyshev type 2 quadrature rules.

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

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

TEST_INT is a FORTRAN90 library which defines integrand functions that can be approximately integrated by a Gauss-Legendre rule.

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.m, is the main program.
• ch_cap.m, capitalizes a single character.
• ch_eqi.m, is TRUE if two characters are equal, ignoring case.
• ch_to_digit.m, converts a character to a digit.
• chebyshev2_integral.m, evaluates a monomial Chebyshev type 2 integral.
• dtable_data_read.m, reads the data from a TABLE file.
• dtable_header_read.m, reads the header from a TABLE file.
• file_column_count.m, counts the columns in a file.
• file_row_count.m, counts the rows in a file.
• monomial_quadrature_chebyshev2.m, returns the integral of a monomial.
• s_len_trim.m, returns the length of a string to the last nonblank.
• s_to_i4.m, converts a string to an I4.
• s_to_r8.m, converts a string to an R8.
• s_to_r8vec.m, converts a string to an R8VEC.
• s_word_count.m, returns the number of words in a string.
• timestamp.m, prints the current YMDHMS date as a time stamp.
• timestring.m, returns the current YMDHMS date as a string.

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 )

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