CLENSHAW_CURTIS_RULE
Clenshaw Curtis Quadrature Rules

CLENSHAW_CURTIS_RULE is a MATLAB program which generates a Clenshaw Curtis 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 Clenshaw Curtise quadrature rule is designed for the interval [-1,+1].

The Clenshaw Curtis quadrature assumes that the integrand has the form:

        Integral ( -1 <= x <= +1 ) f(x) dx
      

The standard Clenshaw Curtis quadrature rule is used as follows:

        Integral ( -1 <= x <= +1 ) f(x) dx
      

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

Usage:

clenshaw_curtis_rule ( order, 'output' )

order

the number of points in the quadrature rule. A typical value might be 3, 5, 9, or 17.

'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 a MATLAB program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, is a MATLAB program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

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

GEGENBAUER_RULE, is a MATLAB program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, is a MATLAB program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, is a MATLAB program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, is a MATLAB program which can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS_LEGENDRE, is a MATLAB program which checks the polynomial exactness of a Gauss-Legendre quadrature rule.

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

JACOBI_RULE, is a MATLAB program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, is a MATLAB program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is a MATLAB program which can compute and print a Gauss-Legendre quadrature rule.

PATTERSON_RULE, is a MATLAB program which computes a Gauss-Patterson quadrature rule.

PRODUCT_FACTOR is a MATLAB program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is a MATLAB 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_CLENSHAW_CURTIS, a dataset directory which contains quadrature rules for integration on [-1,+1], using a Clenshaw Curtis rule.

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

TANH_QUAD, a MATLAB library which sets up the tanh quadrature rule;

TEST_INT is a FORTRAN90 library which defines 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:

• clenshaw_curtis_rule.m the source code.

Examples and Tests:

• output_cpp.txt, printed output from the command

  
              clenshaw_curtis_rule ( 4, 'C++' )
            

• output_f77.txt, printed output from the command

  
              clenshaw_curtis_rule ( 4, 'F77' )
            
          

• output_f90.txt, printed output from the command

  
              clenshaw_curtis_rule ( 4, 'F90' )
            

• output_mat.txt, printed output from the command

  
              legendre_rule ( 8, 'MAT' )
            

• cc_o5_r.txt, the region file created by the command

  
              clenshaw_curtis_rule ( 5, 'cc_o5' )
            

• cc_o5_w.txt, the weight file created by the command

  
              clenshaw_curtis_rule ( 5, 'cc_o5' )
            

• cc_o5_x.txt, the abscissa file created by the command

  
              clenshaw_curtis_rule ( 5, 'cc_o5' )
            

You can go up one level to the MATLAB source codes.