PATTERSON_RULE
Gauss-Patterson Quadrature Rules

PATTERSON_RULE is a FORTRAN90 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. 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 is 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 FORTRAN90 program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, is a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE is a FORTRAN90 program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, is a FORTRAN90 program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, is a FORTRAN90 program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, is a FORTRAN90 program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, is a FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS_LEGENDRE, is a FORTRAN90 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 FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, is a FORTRAN90 program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is a FORTRAN90 program which can compute and print a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, is a FORTRAN90 program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

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

PRODUCT_FACTOR is a FORTRAN90 program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is a FORTRAN90 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_PATTERSON is a dataset directory which contains triples of files defining standard Gauss-Patterson quadrature rules.

QUADRULE is 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.

TEST_INT is a FORTRAN90 library which contains 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.f90, 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.
• 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.
• DTABLE_WRITE0 writes a DTABLE file with no headers.
• 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.
• S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• S_TO_I4 reads an I4 from a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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