CHEBYSHEV1_RULE
Gauss-Chebyshev Type 1 Quadrature Rules

CHEBYSHEV1_RULE is a C++ program which can generate a specific Gauss-Chebyshev type 1 quadrature rule, based on user input.

The rule can be output as text in a standard programming language, or the data can be written to three files for easy use as input to other programs.

The Gauss-Chebyshev type 1 quadrature rule is designed for the interval [-1,+1].

The Gauss-Chebyshev type 1 quadrature assumes that the integrand has the form:

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

The standard Gauss-Chevbyshev type 1 quadrature rule is used as follows:

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

        Sum ( 1 <= i <= order ) w(i) * f(x(i)) 
      

Usage:

chebyshev1_rule order output

order

the number of points in the quadrature rule. A typical value might be 4, 8, or 16.

output

specifies how the rule is to be reported:

• C++, print as C++ text;
• F77, print as FORTRAN77 text;
• F90, print as FORTRAN90 text;
• MAT, print as MATLAB text;
• file, written to three files, file_w.txt, file_x.txt, and file_r.txt, containing the weights, abscissas, and interval limits.

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:

CHEBYSHEV1_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

CHEBYSHEV2_RULE, is an executable C++ program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

GEGENBAUER_RULE, is an executable C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, is an executable C++ program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, is an executable C++ program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, is an executable C++ program which can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS_CHEBYSHEV1, is an executable C++ program which checks the polynomial exactness of a Gauss-Chebyshev type 1 quadrature rule.

INTLIB is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

JACOBI_RULE, is an executable C++ program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, is an executable C++ program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is an executable C++ program which can compute and print a Gauss-Legendre quadrature rule.

PRODUCT_FACTOR is an executable C++ program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is an executable C++ program which constructs a product rule from identical 1D factor rules.

QUADPACK is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

QUADRATURE_RULES is a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_CHEBYSHEV1 is a dataset directory of triples of files defining standard Gauss-Chebyshev type 1 quadrature rules.

QUADRULE is a C++ library which defines 1-dimensional quadrature rules.

TEST_INT is a FORTRAN90 library containing a number of functions that may be used as test integrands for quadrature rules in 1D.

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
3. Arthur Stroud, Don Secrest,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.

Source Code:

• chebyshev1_rule.C, the source code.
• chebyshev1_rule.csh, commands to compile the source code.

Examples and Tests:

• output_cpp.txt, printed output from the command

  
              chebyshev1_rule 4 C++
            

• output_f77.txt, printed output from the command

  
              chebyshev1_rule 4 F77
            
          

• output_f90.txt, printed output from the command

  
              chebyshev1_rule 4 F90
            

• output_mat.txt, printed output from the command

  
              chebyshev1_rule 8 MAT
            

• cheby1_o5_r.txt, the region file created by the command

  
              chebyshev1_rule 5 cheby1_o5
            

• cheby1_o5_w.txt, the weight file created by the command

  
              chebyshev1_rule 5 cheby1_o5
            

• cheby1_o5_x.txt, the abscissa file created by the command

  
              chebyshev1_rule 5 cheby1_o5
            

List of Routines:

• MAIN is the main program for CHEBYSHEV1_RULE.
• 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.
• CHEBYSHEV1_COMPUTE computes a Gauss-Chebyshev type 1 quadrature rule.
• CHEBYSHEV1_HANDLE handles the requested Gauss-Chebyshev type 1 rule.
• DTABLE_CLOSE_WRITE closes a file to which a DTABLE was to be written.
• DTABLE_DATA_WRITE writes data to a DTABLE file.
• DTABLE_HEADER_WRITE writes the header of a DTABLE file.
• DTABLE_WRITE writes information to a DTABLE file.
• R8_EPSILON returns the R8 roundoff unit.
• S_EQI reports whether two strings are equal, ignoring case.
• 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.