Quadrature Rules of Gauss-Legendre Type

These are some examples of quadrature rules of Gauss-Legendre type.

Gauss-Legendre quadrature rules are designed to approximate integrals on the interval [-1,1].

Standard Gauss-Legendre quadrature assumes that the integrand we are considering has a form like:

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

A standard Gauss-Legendre quadrature rule is a set of n positive weights w and abscissas x so that

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

        Sum ( 1 <= I <= N ) w(i) * f(x(i))
      

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the endpoints of the region. The first endpoint is either 0 or A; the second endpoint is really infinity, and we will simply place a "large" value here, trusting that the user will already know that this is merely a placeholder for infinity.

Example:

We consider a standard Gauss-Legendre quadrature rule of order 4.

Here is the text of the "W" file storing the weights of such a rule:

        0.3478548451374538    
        0.6521451548625461    
        0.6521451548625461    
        0.3478548451374538    
       

Here is the text of the "X" file storing the abscissas of such a rule:

 
       -0.8611363115940526    
       -0.3399810435848563    
        0.3399810435848563    
        0.8611363115940526    
      

Here is the text of the "R" file storing the lower and upper limits of the region:

        -1.0
         1.0
      

Related Data and Programs:

• INT_EXACTNESS_LEGENDRE is a FORTRAN90 program which can read a set of files defining a Gauss-Legendre quadrature rule, and test it for exactness against monomial integrands.
• LEGENDRE_RULE is an executable FORTRAN90 program which generates a Gauss-Legendre rule of the desired order.
• PRODUCT_FACTOR is a FORTRAN90 program which creates a multidimensional product rule whose factors are distinct 1D quadrature rules.
• PRODUCT_RULE is a FORTRAN90 program which creates a multidimensional product rule whose factors are identical 1D quadrature rules.
• QUADRATURE_TEST is an executable MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.
• QUADRULE is a FORTRAN90 library of routines which defines various quadrature rules.
• TABLE is the file format used to store this data;
• TEST_INT is a FORTRAN90 library of routines which defines several test integrands which may be integrated.

Sample Files:

Standard Gauss-Legendre Rule, Order 1:

• leg_o1_x.txt, the abscissas for the order 1 rule.
• leg_o1_w.txt, the weights.
• leg_o1_r.txt, the range of the integration region.

Standard Gauss-Legendre Rule, Order 2:

• leg_o2_x.txt, the abscissas for the order 2 rule.
• leg_o2_w.txt, the weights.
• leg_o2_r.txt, the range of the integration region.

Standard Gauss-Legendre Rule, Order 4:

• leg_o4_x.txt, the abscissas for the order 4 rule.
• leg_o4_w.txt, the weights.
• leg_o4_r.txt, the range of the integration region.

Standard Gauss-Legendre Rule, Order 8:

• leg_o8_x.txt, the abscissas for the order 8 rule.
• leg_o8_w.txt, the weights.
• leg_o8_r.txt, the range of the integration region.

Standard Gauss-Legendre Rule, Order 16:

• leg_o16_x.txt, the abscissas for the order 16 rule.
• leg_o16_w.txt, the weights.
• leg_o16_r.txt, the range of the integration region.

Standard Gauss-Legendre Rule, Order 32:

• leg_o32_x.txt, the abscissas for the order 32 rule.
• leg_o32_w.txt, the weights.
• leg_o32_r.txt, the range of the integration region.

Standard Gauss-Legendre Rule, Order 64:

• leg_o64_x.txt, the abscissas for the order 64 rule.
• leg_o64_w.txt, the weights.
• leg_o64_r.txt, the range of the integration region.

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