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 can be output as text in a standard programming language, or the data can be written to three files for easy use as input to other programs.

The Gauss-Patterson quadrature rule is designed for the interval [-1,+1].

The Gauss-Patterson quadrature assumes that the integrand has the form:

        Integral ( -1 <= x <= +1 ) f(x) dx
      

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, and 127.

The standard Gauss-Patterson quadrature rule is used as follows:

        Integral ( -1 <= x <= +1 ) 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.

The polynomial precision of a Gauss-Patterson rule can be checked numerically by the INT_EXACTNESS_LEGENDRE program. We should expect

#	Order	Free+Fixed	Precision
1	1	1 + 0	1
2	3	3 + 0	5
3	7	4 + 3	10 + 1
4	15	8 + 7	22 + 1
5	31	16 + 15	46 + 1
6	63	32 + 31	94 + 1
7	127	64 + 63	191 + 1

where the extra 1 degree of precision comes about because the rules are symmetric, and can integrate any odd monomial exactly. (So you never have a precision that is even; it always increases to the next odd number.)

Usage:

patterson_rule order output

order

the number of points in the quadrature rule. Acceptable values are 1, 3, 7, 15, 31, 63, or 127.

output

specifies how the rule is to be reported:

• C++, print as C++ text;
• F77, print as FORTRAN77 text;
• F90, print as FORTRAN90 text;
• MAT, print as MATLAB text;
• file, written to three files, file_w.txt, file_x.txt, and file_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.

Related Data and Programs:

CHEBYSHEV1_RULE, is a C++ program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, is a C++ program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE is a C++ program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, is a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, is a C++ program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, is a C++ program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, is a C++ program which can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS_LEGENDRE, is a C++ program which checks the polynomial exactness of a Gauss-Legendre quadrature rule.

INTLIB is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

JACOBI_RULE, is a C++ program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, is a C++ program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is a C++ program which can compute and print a Gauss-Legendre quadrature rule.

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

PRODUCT_FACTOR is a C++ program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is a C++ program which constructs a product rule from identical 1D factor rules.

QUADPACK is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

QUADRATURE_RULES is a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_LEGENDRE is a dataset directory of triples of files defining standard Gauss-Legendre quadrature rules.

QUADRULE is a C++ library which defines 1-dimensional quadrature rules.

TEST_INT is a FORTRAN90 library which defines functions that may be used as test integrands for quadrature rules in 1D.

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.C, the source code.
• patterson_rule.csh, commands to compile the source code.

Examples and Tests:

• output_cpp.txt, printed output from the command

  
              patterson_rule 7 C++
            

• output_f77.txt, printed output from the command

  
              patterson_rule 7 F77
            
          

• output_f90.txt, printed output from the command

  
              patterson_rule 7 F90
            

• output_mat.txt, printed output from the command

  
              patterson_rule 7 MAT
            

• 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.
• DTABLE_WRITE0 writes information to a DTABLE file.
• 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.
• R8_EPSILON returns the R8 roundoff unit.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.