PATTERSON_RULE
Gauss-Patterson Quadrature Rules {#patterson_rule-gauss-patterson-quadrature-rules align="center"}
PATTERSON_RULE is a C++ program 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
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
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.
patterson_rule order a b filename
where
- 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.
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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version.
CCN_RULE, a C++ program which defines a nested Clenshaw Curtis quadrature rule.
CHEBYSHEV1_RULE, a C++ program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.
CHEBYSHEV2_RULE, a C++ program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.
CLENSHAW_CURTIS_RULE, a C++ program which defines a Clenshaw Curtis quadrature rule.
GEGENBAUER_RULE, a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.
GEN_HERMITE_RULE, a C++ program which can compute and print a generalized Gauss-Hermite quadrature rule.
GEN_LAGUERRE_RULE, a C++ program which can compute and print a generalized Gauss-Laguerre quadrature rule.
HERMITE_RULE, a C++ program which can compute and print a Gauss-Hermite quadrature rule.
INT_EXACTNESS_LEGENDRE, a C++ program which checks the polynomial exactness of a Gauss-Legendre quadrature rule.
JACOBI_RULE, a C++ program which can compute and print a Gauss-Jacobi quadrature rule.
KRONROD, a C++ library which can compute a Gauss and Gauss-Kronrod pair of quadrature rules of arbitrary order, by Robert Piessens, Maria Branders.
LAGUERRE_RULE, a C++ program which can compute and print a Gauss-Laguerre quadrature rule.
LATTICE_RULE, a C++ library which approximates M-dimensional integrals using lattice rules.
LEGENDRE_RULE, a C++ program which can compute and print a Gauss-Legendre quadrature rule.
LINE_FELIPPA_RULE, a C++ library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.
LINE_NCC_RULE, a C++ 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 C++ library which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.
QUADRULE, a C++ library which defines 1-dimensional quadrature rules.
TOMS699, a FORTRAN77 library which
implements a new representation of Patterson's quadrature formula;
this is ACM TOMS algorithm 699.
TRUNCATED_NORMAL_RULE, a C++ program which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].
- Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34. - Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28. - Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.
- patterson_rule.cpp, the source code.
-
gp_o15_r.txt, the region file created by the command
patterson_rule 15 gp_o15 -
gp_o15_w.txt, the weight file created by the command
patterson_rule 15 gp_o15 -
gp_o15_x.txt, the abscissa file created by the command
patterson_rule 15 gp_o15
- MAIN is the main program for PATTERSON_RULE.
- ORDER_CHECK checks the value of ORDER.
- PATTERSON_HANDLE computes the requested Gauss-Patterson rule and outputs it.
- PATTERSON_SET sets abscissas and weights for Gauss-Patterson quadrature.
- R8MAT_WRITE writes an R8MAT file with no header.
- RESCALE rescales a Legendre quadrature rule from [-1,+1] to [A,B].
- TIMESTAMP prints the current YMDHMS date as a time stamp.
You can go up one level to the C++ source codes.
Last revised on 16 February 2010.