INT_EXACTNESS_JACOBI
Exactness of Gauss-Jacobi Quadrature Rules

INT_EXACTNESS_JACOBI is a C++ program, using double precision arithmetic, which investigates the polynomial exactness of a Gauss-Jacobi quadrature rule for the interval [-1,1] with a weight function.

The Gauss-Jacobi quadrature rule is designed to approximate integrals on the interval [-1,1], with a weight function of the form (1-x)^ALPHA * (1+x)^BETA. ALPHA and BETA are real parameters that must be greater than -1.

Gauss-Jacobi quadrature assumes that the integrand we are considering has a form like:

        Integral ( -1 <= x <= +1 ) (1-x)^alpha (1+x)^beta f(x) dx
      

For a Gauss-Jacobi 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 ) (1-x)^alpha (1+x)^beta f(x) 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 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_jacobi prefix degree_max alpha beta

prefix

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.

alpha

the value of the exponent of (1-x) in the weight function; alpha should be a real number greater than -1.0.

beta

the value of the exponent of (1+x) in the weight function; beta should be a real number greater than -1.0.

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:

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

INT_EXACTNESS_GEN_HERMITE is an executable C++ program which tests the polynomial exactness of a generalized Gauss-Hermite quadrature rule.

INT_EXACTNESS_GEN_LAGUERRE is an executable C++ program which tests the polynomial exactness of a generalized Gauss-Laguerre quadrature rule.

INT_EXACTNESS_HERMITE is an executable C++ program which tests the polynomial exactness of a Gauss-Hermite quadrature rule.

INT_EXACTNESS_JACOBI is also available in a FORTRAN90 version and a MATLAB version

INT_EXACTNESS_LAGUERRE is an executable C++ program which tests the polynomial exactness of a Gauss-Laguerre quadrature rule.

INT_EXACTNESS_LEGENDRE is an executable C++ program which tests the polynomial exactness of a Gauss-Legendre quadrature rule.

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

JACOBI_RULE is an executable C++ program which can generate a Gauss-Jacobi quadrature rule on request.

QUADRATURE_RULES_JACOBI is a dataset directory which contains sets of files that define Gauss-Jacobi quadrature rules.

QUADRULE is a FORTRAN90 library which defines quadrature rules on a variety of intervals with different weight functions.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
2. Shanjie Zhang, Jianming Jin,
   Computation of Special Functions,
   Wiley, 1996,
   ISBN: 0-471-11963-6,
   LC: QA351.C45.

Source Code:

• int_exactness_jacobi.C, the source code.
• int_exactness_jacobi.csh, commands to compile the source code.

Examples and Tests:

JAC_O1_A0.5_B1.5 is a Gauss-Jacobi order 1 rule with ALPHA = 0.5, BETA = 1.5.

• jac_o1_a0.5_b1.5_x.txt, the abscissas of the rule.
• jac_o1_a0.5_b1.5_w.txt, the weights of the rule.
• jac_o1_a0.5_b1.5_r.txt, defines the region for the rule.
• jac_o1_a0.5_b1.5_exact.txt, the results of the command

  int_exactness_jacobi jac_o1_a0.5_b1.5 5 0.5 1.5

JAC_O2_A0.5_B1.5 is a Gauss-Jacobi order 2 rule with ALPHA = 0.5, BETA = 1.5.

• jac_o2_a0.5_b1.5_x.txt, the abscissas of the rule.
• jac_o2_a0.5_b1.5_w.txt, the weights of the rule.
• jac_o2_a0.5_b1.5_r.txt, defines the region for the rule.
• jac_o2_a0.5_b1.5_exact.txt, the results of the command

  int_exactness_jacobi jac_o2_a0.5_b1.5 5 0.5 1.5

JAC_O4_A0.5_B1.5 is a Gauss-Jacobi order 4 rule with ALPHA = 0.5, BETA = 1.5.

• jac_o4_a0.5_b1.5_x.txt, the abscissas of the rule.
• jac_o4_a0.5_b1.5_w.txt, the weights of the rule.
• jac_o4_a0.5_b1.5_r.txt, defines the region for the rule.
• jac_o4_a0.5_b1.5_exact.txt, the results of the command

  int_exactness_jacobi jac_o4_a0.5_b1.5 10 0.5 1.5

JAC_O8_A0.5_B1.5 is a Gauss-Jacobi order 8 rule with ALPHA = 0.5, BETA = 1.5.

• jac_o8_a0.5_b1.5_x.txt, the abscissas of the rule.
• jac_o8_a0.5_b1.5_w.txt, the weights of the rule.
• jac_o8_a0.5_b1.5_r.txt, defines the region for the rule.
• jac_o8_a0.5_b1.5_exact.txt, the results of the command

  int_exactness_jacobi jac_o8_a0.5_b1.5 18 0.5 1.5

JAC_O16_A0.5_B1.5 is a Gauss-Jacobi order 16 rule with ALPHA = 0.5, BETA = 1.5.

• jac_o16_a0.5_b1.5_x.txt, the abscissas of the rule.
• jac_o16_a0.5_b1.5_w.txt, the weights of the rule.
• jac_o16_a0.5_b1.5_r.txt, defines the region for the rule.
• jac_o16_a0.5_b1.5_exact.txt, the results of the command

  int_exactness_jacobi jac_o16_a0.5_b1.5 35 0.5 1.5

List of Routines:

• MAIN is the main program for INT_EXACTNESS_JACOBI.
• 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.
• JACOBI_INTEGRAL evaluates the integral of a monomial with Jacobi weight.
• MONOMIAL_QUADRATURE_JACOBI applies a quadrature rule to a monomial.
• R8_ABS returns the absolute value of an R8.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HUGE returns a "huge" R8.
• R8_HYPER_2F1 evaluates the hypergeometric function 2F1(A,B,C,X).
• R8_PSI evaluates the function Psi(X).
• 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.

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