PATTERSON_RULE is a FORTRAN90 program 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) 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 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 GNU LGPL license.
ALPERT_RULE, a FORTRAN90 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 FORTRAN90 program which defines a nested Clenshaw Curtis quadrature rule.
CHEBYSHEV1_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.
CHEBYSHEV2_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.
CLENSHAW_CURTIS_RULE, a FORTRAN90 program which defines a Clenshaw Curtis quadrature rule.
GEGENBAUER_RULE, a FORTRAN90 program which can compute and print a Gauss-Gegenbauer quadrature rule.
GEN_HERMITE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Hermite quadrature rule.
GEN_LAGUERRE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Laguerre quadrature rule.
HERMITE_RULE, a FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule.
INT_EXACTNESS_LEGENDRE, a FORTRAN90 program which checks the polynomial exactness of a Gauss-Legendre quadrature rule.
JACOBI_RULE, a FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.
LAGUERRE_RULE, a FORTRAN90 program which can compute and print a Gauss-Laguerre quadrature rule.
LEGENDRE_RULE, a FORTRAN90 program which can compute and print a Gauss-Legendre quadrature rule.
LEGENDRE_RULE_FAST, a FORTRAN90 program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.
LINE_NCC_RULE, a FORTRAN90 library which computes a Newton Cotes Closed (NCC) quadrature rule for the line, that is, for an interval of the form [A,B], using equally spaced points which include the endpoints.
LINE_NCO_RULE, a FORTRAN90 library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.
LOGNORMAL_RULE, a FORTRAN90 program which can compute and print a quadrature rule for functions of a variable whose logarithm is normally distributed.
PATTERSON_RULE_COMPUTE, a FORTRAN90 program which computes the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.
QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.
QUADRATURE_RULES_PATTERSON, a dataset directory which contains triples of files defining standard Gauss-Patterson quadrature rules.
QUADRULE, a FORTRAN90 library which defines 1-dimensional quadrature rules.
TANH_SINH_RULE, a FORTRAN90 program which computes and writes out a tanh-sinh quadrature rule of given order.
TOMS699,
a FORTRAN77 library which
implements a new representation of Patterson's quadrature formula;
this is ACM TOMS algorithm 699.
TRUNCATED_NORMAL_RULE, a FORTRAN90 program which computes a quadrature rule for a normal distribution that has been truncated to [A,+oo), (-oo,B] or [A,B].
patterson_rule 15 gp_o15
patterson_rule 15 gp_o15
patterson_rule 15 gp_o15
You can go up one level to the FORTRAN90 source codes.