CCN_RULE
Nested Clenshaw Curtis Quadrature Rules

CCN_RULE is a FORTRAN90 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 made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

ALPERT_RULE, a FORTRAN90 library which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

CHEBYSHEV1_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

GEGENBAUER_RULE, a FORTRAN90 program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule.

JACOBI_RULE, a FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a FORTRAN90 program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a FORTRAN90 program which can compute and print a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, a FORTRAN90 program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

LINE_NCC_RULE, a FORTRAN90 library which computes a Newton Cotes Closed (NCC) quadrature rule for the line, that is, for an interval of the form [A,B], using equally spaced points which include the endpoints.

LINE_NCO_RULE, a FORTRAN90 library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

LOGNORMAL_RULE, a FORTRAN90 program which can compute and print a quadrature rule for functions of a variable whose logarithm is normally distributed.

PATTERSON_RULE, a FORTRAN90 program which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

PATTERSON_RULE_COMPUTE, a FORTRAN90 program which computes the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

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

TANH_SINH_RULE, a FORTRAN90 program which computes and writes out a tanh-sinh quadrature rule of given order.

TRUNCATED_NORMAL_RULE, a FORTRAN90 program which computes a quadrature rule for a normal distribution that has been truncated to [A,+oo), (-oo,B] or [A,B].

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.f90, 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
            

List of Routines:

• MAIN is the main program for CCN_RULE.
• CCN_COMPUTE_POINTS: compute Clenshaw Curtis Nested points.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• GET_UNIT returns a free FORTRAN unit number.
• NC_COMPUTE computes a Newton-Cotes quadrature rule.
• R8MAT_WRITE writes an R8MAT file.
• RESCALE rescales a Legendre quadrature rule from [-1,+1] to [A,B].
• RULE_WRITE writes a quadrature rule to a file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 value from a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.