# ccn_rule

ccn_rule, a C++ code 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
```
is to be approximated by
```        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
where
• 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.

### 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:

CHEBYSHEV1_RULE, a C++ code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a C++ code which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a C++ code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

GEGENBAUER_RULE, a C++ code which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a C++ code which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a C++ code which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a C++ code which can compute and print a Gauss-Hermite quadrature rule.

JACOBI_RULE, a C++ code which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a C++ code which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a C++ code which can compute and print a Gauss-Legendre quadrature rule.

LINE_FELIPPA_RULE, a C++ code which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

LINE_NCC_RULE, a C++ code 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 C++ code which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

PATTERSON_RULE, a C++ code 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.

TRUNCATED_NORMAL_RULE, a C++ code which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian 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,