LEGENDRE_RULE
Gauss-Legendre Quadrature Rules

LEGENDRE_RULE is a MATLAB program which generates a specific Gauss-Legendre 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-Legendre quadrature rule is designed for the interval [-1,+1].

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

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

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

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

        Sum ( 1 <= i <= order ) w(i) * f(x(i)) 
      

Usage:

legendre_rule ( order, output )

order

the number of points in the quadrature rule. A typical value might be 4, 8, or 16.

output

a quoted string which 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 an executable MATLAB program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

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

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

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

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

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

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

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

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

PRODUCT_FACTOR is an executable MATLAB program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is an executable MATLAB 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 MATLAB library which defines 1-dimensional quadrature rules.

TEST_INT is a FORTRAN90 library containing a number of 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:

• legendre_rule.m the main program for LEGENDRE_RULE.
• ch_cap.m
• ch_eqi.m is a case insensitive comparison of two characters for equality.
• dtable_close_write.m closes a file used to write a DTABLE.
• dtable_data_write.m writes DTABLE data to a file.
• dtable_header_write.m writes the header to a DTABLE file.
• dtable_write.m writes a DTABLE to a file.
• legendre_compute.m computes a Gauss-Legendre quadrature rule.
• legendre_handle.m computes the requested Gauss-Legendre rule and outputs it.
• s_eqi.m compares two string for equality, ignoring case.
• timestamp.m prints the current YMDHMS date as a time stamp.
• timestring.m writes the current YMDHMS date into a string.

Examples and Tests:

• output_cpp.txt, printed output from the command

  
              legendre_rule ( 4, 'C++' )
            

• output_f77.txt, printed output from the command

  
              legendre_rule ( 4, 'F77' )
            
          

• output_f90.txt, printed output from the command

  
              legendre_rule ( 4, 'F90' )
            

• output_mat.txt, printed output from the command

  
              legendre_rule ( 8, 'MAT' )
            

• leg_o4_r.txt, the region file created by the command

  
              legendre_rule ( 4, 'leg_o4' )
            

• leg_o4_w.txt, the weight file created by the command

  
              legendre_rule ( 4, 'leg_o4' )
            

• leg_o4_x.txt, the abscissa file created by the command

  
              legendre_rule ( 4, 'leg_o4' )
            

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