PATTERSON_RULE
Gauss-Patterson Quadrature Rules

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
      

        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.

Usage:

patterson_rule order a b filename

• 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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

PATTERSON_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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].

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
3. Arthur Stroud, Don Secrest,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.

Source Code:

• patterson_rule.cpp, the source code.

Examples and Tests:

• 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
            

List of Routines:

• 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.