PATTERSON_RULE
Gauss-Patterson Quadrature Rules

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) 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.

Languages:

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

Licensing:

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

Related Data and Programs:

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].

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.f90, 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.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• GET_UNIT returns a free FORTRAN unit number.
• ORDER_CHECK checks the value of ORDER.
• PATTERSON_HANDLE looks up the requested Gauss-Patterson rule and outputs it.
• PATTERSON_SET sets abscissas and weights for Gauss-Patterson quadrature.
• R8MAT_WRITE writes an R8MAT file.
• RESCALE rescales a Legendre quadrature rule from [-1,+1] to [A,B].
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 value from a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.