patterson_rule, a MATLAB 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) dxis 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
|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|
patterson_rule ( order, a, b, 'filename' )where
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
patterson_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
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