GEN_LAGUERRE_RULE_SS is a C++ program which generates a specific generalized Gauss-Laguerre 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.
Both the basic and generalized Gauss-Laguerre quadrature rules are designed to approximate integrals on semi-infinite intervals. By default, this interval is [0,oo). If this default is not suitable, it is possible to redefine the abscissas and weights for the interval [A,oo) instead.
Generalized Gauss-Laguerre quadrature assumes that the integrand we are considering has a form like:
Integral ( 0 <= x < +oo ) x^alpha * exp(-x) f(x) dxwhere the factor x^alpha * exp(-x) is regarded as a weight factor.
The standard generalized Gauss-Laguerre quadrature rule is used as follows:
Integral ( 0 <= x < +oo ) x^alpha * exp(-x) f(x) dxis to be approximated by
Sum ( 1 <= i <= order ) w(i) * f(x(i))
Although the standard rule is defined in terms of the product of the integral weight function x^alpha * exp(-x) and a function f(x), there may be cases when it is more convenient to think that we are simply approximating
Integral ( 0 <= x < +oo ) f(x) dx
The standard rule can easily be modified, by adjusting the weights, so that the computation can be done in this form. The program allows the user to specify, through the parameter OPTION, whether the standard rule is to be computed (OPTION=0), or the modified rule (OPTION=1).
The modified generalized Gauss-Laguerre quadrature rule is used as follows:
Integral ( 0 <= x < +oo ) f(x) dxis to be approximated by
Sum ( 1 <= i <= order ) w(i) * f(x(i))
Note that when the real paramater alpha is equal to 0.0, we recover the basic (that is, "non-generalized") Gauss-Laguerre quadrature rule. Also note that the generalized Gauss-Laguerre rule is only defined for parameter values alpha which are greater than -1.0.
gen_laguerre_rule_ss order alpha option outputwhere
Integral ( 0 <= x < +oo ) x^ALPHA exp(-x) f(x) dxa 1 value requests the "modified" integration rule for
Integral ( 0 <= x < +oo ) f(x) dx
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
GEN_LAGUERRE_RULE_SS is available in a C++ version and a FORTRAN90 version and a MATLAB version.
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CHEBYSHEV2_RULE, is a C++ program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.
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GEN_LAGUERRE_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.
HERMITE_RULE is a C++ program which computes a Gauss-Hermite quadrature rule.
INT_EXACTNESS, is a C++ program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.
INT_EXACTNESS_LAGUERRE is a C++ program which checks the polynomial exactness of a Gauss-Laguerre quadrature rule.
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LAGUERRE_RULE is a C++ program which computes a Gauss-Laguerre quadrature rule.
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QUADRATURE_RULES is a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.
QUADRATURE_RULES_LAGUERRE is a dataset directory which contains triples of files defining Gauss-Laguerre quadrature rules.
QUADRULE is a C++ library which contains 1-dimensional quadrature rules.
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TEST_INT is a FORTRAN90 library which defines functions that may be used as test integrands for quadrature rules in 1D.
TEST_INT_LAGUERRE is a C++ library which defines test integrands for Gauss-Laguerre rules.
gen_laguerre_rule_ss 4 0.5 0 C++
gen_laguerre_rule_ss 4 -0.5 0 F77
gen_laguerre_rule_ss 4 0.0 0 F90
gen_laguerre_rule_ss 4 0.0 1 F90
gen_laguerre_rule_ss 8 1.5 0 MAT
gen_laguerre_rule_ss 4 0.5 0 gen_lag_o4_a0.5
gen_laguerre_rule_ss 4 0.5 0 gen_lag_o4_a0.5
gen_laguerre_rule_ss 4 0.5 0 gen_lag_o4_a0.5
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