patterson_rule

patterson_rule, a C code 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
      

        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

Index	Order	Free+Fixed	Expected Precision	Actual Precision
0	1	1 + 0	2*1+0-1=1	1
1	3	3 + 0	2*3+0-1=5	5
2	7	4 + 3	2*4+3-1=10	10 + 1 = 11
3	15	8 + 7	2*8+7-1=22	22 + 1 = 23
4	31	16 + 15	2*16+15-1=46	46 + 1 = 47
5	63	32 + 31	2*32+31-1=94	94 + 1 = 95
6	127	64 + 63	2*64+63-1=190	190 + 1 = 191
7	255	128 + 127	2*128+127-1=382	382 + 1 = 383
8	511	256 + 255	2*256+255-1=766	766 + 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

• 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 information on this web page is distributed under the MIT license.

Languages:

patterson_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

patterson_rule_test

c_rule, a C code which computes a quadrature rule which estimates the integral of a function f(x), which might be defined over a one dimensional region (a line) or more complex shapes such as a circle, a triangle, a quadrilateral, a polygon, or a higher dimensional region, and which might include an associated weight function w(x).

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:

• patterson_rule.c, the source code.
• patterson_rule.sh, compiles the source code.