patterson_rule, a FORTRAN90 code which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511, based on interaction with the user.
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
| 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 filenamewhere
patterson_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
The computer code and data files described and made available on this web page are distributed under the MIT license
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