Quadrature Rules of Gauss-Laguerre Type

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

Gauss-Laguerre quadrature rules are designed to approximate integrals on semi-infinite intervals. By default, this interval is [0,oo) but in some cases, the interval may be specified with a different initial point, which we will suggest by [A,oo).

In the following discussion, we will represent the left endpoint by A, while understanding that it is usually taken to be 0.

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

        Integral ( A <= x < oo ) exp(-x) f(x) dx
      

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

        Integral ( A <= x < oo ) exp(-x) f(x) dx
      

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

It is often convenient to consider approximating integrals in which the weighting factor exp(-x) is implicit. In that case, we are looking at approximating

        Integral ( A <= x < oo ) f(x) dx
      

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

        Integral ( A <= x < oo ) f(x) dx
      

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

When using a Gauss-Laguerre quadrature rule, it's important to know whether the rule has been developed for the standard or modified cases. Basically, the only change is that the weights of the modified rule have been multiplied by an exponential factor evaluated at the corresponding abscissa.

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 modified Gauss-Laguerre quadrature rule of order 4.

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

        0.60315
        0.35741
        0.03888
        0.00053
      

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

        0.32254
        1.74576
        4.53662
        9.39507
      

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

        0.0
        1.0E+30
      

Related Data and Programs:

• INT_EXACTNESS_LAGUERRE is a FORTRAN90 program which can read a set of files defining a quadrature rule in 1D for a semi-infinite interval, and test it for exactness against monomial integrands.
• LAGUERRE_RULE is an executable FORTRAN90 program which can generate a Gauss-Laguerre rule of the desired parameters.
• 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_RULES_GEN_LAGUERRE is a collection of generalized Gauss-Laguerre quadrature rules, for the interval [A,oo) with weight x^alpha*exp(-x).
• 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_LAGUERRE is a FORTRAN90 libary of routines which defines integrands that can be integrated using a Gauss-Laguerre rule.

Sample Files:

Standard Gauss-Laguerre Rule, Order 1:

• lag_o1_x.txt, the abscissas for the order 1 rule.
• lag_o1_w.txt, the weights.
• lag_o1_r.txt, the range of the integration region.

Standard Gauss-Laguerre Rule, Order 2:

• lag_o2_x.txt, the abscissas for the order 2 rule.
• lag_o2_w.txt, the weights.
• lag_o2_r.txt, the range of the integration region.

Standard Gauss-Laguerre Rule, Order 4:

• lag_o4_x.txt, the abscissas for the order 4 rule.
• lag_o4_w.txt, the weights.
• lag_o4_r.txt, the range of the integration region.

Standard Gauss-Laguerre Rule, Order 8:

• lag_o8_x.txt, the abscissas for the order 8 rule.
• lag_o8_w.txt, the weights.
• lag_o8_r.txt, the range of the integration region.

Standard Gauss-Laguerre Rule, Order 16:

• lag_o16_x.txt, the abscissas for the order 16 rule.
• lag_o16_w.txt, the weights.
• lag_o16_r.txt, the range of the integration region.

Standard Gauss-Laguerre Rule, Order 32:

• lag_o32_x.txt, the abscissas for the order 32 rule.
• lag_o32_w.txt, the weights.
• lag_o32_r.txt, the range of the integration region.

Standard Gauss-Laguerre Rule, Order 64:

• lag_o64_x.txt, the abscissas for the order 64 rule.
• lag_o64_w.txt, the weights.
• lag_o64_r.txt, the range of the integration region.

Modified Gauss-Laguerre Rule, Order 1:

• lag_o1_modified_x.txt, the abscissas for the order 1 rule.
• lag_o1_modified_w.txt, the weights.
• lag_o1_modified_r.txt, the range of the integration region.

Modified Gauss-Laguerre Rule, Order 2:

• lag_o2_modified_x.txt, the abscissas for the order 2 rule.
• lag_o2_modified_w.txt, the weights.
• lag_o2_modified_r.txt, the range of the integration region.

Modified Gauss-Laguerre Rule, Order 4:

• lag_o4_modified_x.txt, the abscissas for the order 4 rule.
• lag_o4_modified_w.txt, the weights.
• lag_o4_modified_r.txt, the range of the integration region.

Modified Gauss-Laguerre Rule, Order 8:

• lag_o8_modified_x.txt, the abscissas for the order 8 rule.
• lag_o8_modified_w.txt, the weights.
• lag_o8_modified_r.txt, the range of the integration region.

Modified Gauss-Laguerre Rule, Order 16:

• lag_o16_modified_x.txt, the abscissas for the order 16 rule.
• lag_o16_modified_w.txt, the weights.
• lag_o16_modified_r.txt, the range of the integration region.

Modified Gauss-Laguerre Rule, Order 32:

• lag_o32_modified_x.txt, the abscissas for the order 32 rule.
• lag_o32_modified_w.txt, the weights.
• lag_o32_modified_r.txt, the range of the integration region.

Modified Gauss-Laguerre Rule, Order 64:

• lag_o64_modified_x.txt, the abscissas for the order 64 rule.
• lag_o64_modified_w.txt, the weights.
• lag_o64_modified_r.txt, the range of the integration region.

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