CCN_RULE
Nested Clenshaw Curtis Quadrature Rules

CCN_RULE is a MATLAB program which generates a quadrature rule based on a nested set of points inspired by the Clenshaw Curtis quadrature rule.

The data defining the rule is written to three files for easy use as input to other programs.

The nested Clenshaw Curtis quadrature rule is used as follows:

        Integral ( A <= x <= B ) f(x) dx
      

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

If the order of the CCN rule is 1, 3, 5, 9, 17, 33, or in general 2^L+1, then the rule is identical to the Clenshaw Curtis rule.

Otherwise, the rule is based on a subset of the points in the Clenshaw Curtis rule of next highest order in the sequence 2^L+1.

The CCN rule has no special accuracy properties, except that the rules of odd order are symmetric, and hence get one extra degree of precision. Moreover, the rules of even order have a single unpaired point which is assigned weight zero, so that it is equivalent to the immediately preceding rule of odd order.

Usage:

ccn_rule ( n, a, b, 'filename' )

• n is the order, or number of points in the quadrature rule;
• a is the left endpoint;
• b is the right endpoint;
• filename specifies the output filenames: namefile_w.txt, filename_x.txt, and filename_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.

Languages:

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

Related Data and Programs:

CHEBYSHEV1_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

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

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

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

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

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

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

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

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

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

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

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

QUADRATURE_RULES_CCN, a dataset directory which contains quadrature rules for integration on [-1,+1], using a nested Clenshaw Curtis rule.

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

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:

• ccn_rule.m the source code.

Examples and Tests:

CCN_09 is a nested Clenshaw Curtis rule of order 9, which will exactly match the standard Clenshaw Curtis rule.

• ccn_o9_r.txt, the region file created by the command

  
              ccn_rule ( 9, -1, +1, 'ccn_o9' )
            

• ccn_o9_w.txt, the weight file created by the command

  
              ccn_rule ( 9, -1, +1, 'ccn_o9' )
            

• ccn_o9_x.txt, the abscissa file created by the command

  
              ccn_rule ( 9, -1, +1, 'ccn_o9' )
            

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