patterson_rule, an Octave 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 MIT license
patterson_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.
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