# PATTERSON_RULE Gauss-Patterson Quadrature Rules

PATTERSON_RULE, a MATLAB program which generates a specific Gauss-Patterson quadrature rule, based on user input.

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

The Gauss-Patterson quadrature is a nested family which begins with the Gauss-Legendre rules of orders 1 and 3, and then succesively inserts one new abscissa in each subinterval. Thus, after the second rule, the Gauss-Patterson rules do not have the super-high precision of the Gauss-Legendre rules. They trade this precision in exchange for the advantages of nestedness. This means that Gauss-Patterson rules are only available for orders of 1, 3, 7, 15, 31, 63, 127, 255 or 511.

The standard Gauss-Patterson 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))
```

The polynomial precision of a Gauss-Patterson rule can be checked numerically by the INT_EXACTNESS_LEGENDRE program. We should expect
IndexOrderFree+FixedExpected PrecisionActual Precision
011 + 02*1+0-1=11
133 + 02*3+0-1=55
274 + 32*4+3-1=1010 + 1 = 11
3158 + 72*8+7-1=2222 + 1 = 23
43116 + 152*16+15-1=4646 + 1 = 47
56332 + 312*32+31-1=9494 + 1 = 95
612764 + 632*64+63-1=190190 + 1 = 191
7255128 + 1272*128+127-1=382382 + 1 = 383
8511256 + 2552*256+255-1=766766 + 1 = 767
where the extra 1 degree of precision comes about because the rules are symmetric, and can integrate any odd monomial exactly. Thus, after the first rule, the precision is 3*2^index - 1.

### Usage:

patterson_rule ( order, a, b, 'filename' )
where
• order is the number of points in the quadrature rule. Acceptable values are 1, 3, 7, 15, 31, 63, 127, 255 or 511.
• a is the left endpoint;
• b is the right endpoint;
• 'filename' specifies the output filenames: filename_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:

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

CCN_RULE, a MATLAB program which defines a nested Clenshaw Curtis quadrature rule.

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 defines a Clenshaw Curtis 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.

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

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.

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

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

QUADRATURE_RULES_LEGENDRE, a dataset directory which contains triples of files defining standard Gauss-Legendre quadrature rules.

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

TOMS699, a FORTRAN77 library which implements a new representation of Patterson's quadrature formula;
this is ACM TOMS algorithm 699.

TRUNCATED_NORMAL_RULE, a MATLAB program 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,