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In [[numerical analysis]] '''Gauss–Laguerre quadrature''' is an extension of [[Gaussian quadrature]] method for approximating the value of integrals of the following kind:


:<math>\int_{0}^{+\infty} e^{-x} f(x)\,dx.</math>


In this case
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:<math>\int_{0}^{+\infty} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)</math>
 
where ''x''<sub>''i''</sub> is the ''i''-th root of [[Laguerre polynomial]] ''L''<sub>''n''</sub>(''x'') and the weight ''w''<sub>''i''</sub> is given by
<ref name="AS">[[Milton Abramowitz|Abramowitz, M.]] & [[Irene Stegun|Stegun, I. A.]], ''Handbook of Mathematical Functions'', 10th printing with corrections (1972), Dover, ISBN 978-0-486-61272-0. Equation 25.4.45.[http://www.nrbook.com/abramowitz_and_stegun/page_890.htm access online]</ref>
 
:<math>w_i = \frac {x_i} {(n+1)^2[L_{n+1}(x_i)]^2}.</math>
 
==For more general functions==
 
To integrate the function <math>f</math> we apply the following transformation
:<math>\int_{0}^{\infty}f\left(x\right)dx=\int_{0}^{\infty}f\left(x\right)e^{x}e^{-x}dx=\int_{0}^{\infty}g\left(x\right)e^{-x}dx</math>
where <math>g\left(x\right) := e^{x} f\left(x\right)</math>. For the last integral
one then uses Gauss-Laguerre quadrature. Note, that while this approach works
from an analytical perspective, it is not always numerically stable.
 
==Generalized Gauss–Laguerre quadrature==
 
More generally, one can also consider integrands that have a known <math>x^\alpha</math> power-law singularity at ''x''=0, for some real number <math>\alpha > -1</math>, leading to integrals of the form:
:<math>\int_{0}^{+\infty} x^\alpha e^{-x} f(x)\,dx.</math>
This allows one to efficiently evaluate such integrals for polynomial or smooth ''f''(''x'') even when α is not an integer.<ref>{{cite journal|first1=Philip |last1=Rabinowitz |authorlink1= Philip Rabinowitz (mathematician) |first2=George |last2=Weiss | title=Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form <math>\int_0^{\infty} \exp(-x) x^n f(x) dx</math> | journal=Mathematical Tables and Other Aids to Computation |volume=13 | pages=285–294 | year=1959 | doi=10.1090/S0025-5718-1959-0107992-3 }}</ref>
 
==References==
<references/>
*{{cite journal | first1=Herbert E. |last1=Salzer
|first2=Ruth |last2=Zucker
|title=Table of zeros and weight factors of the first fifteen Laguerre polynomials
|journal=Bull. Am. Math. Soc.
|year=1949 | volume=55
|pages=1004–1012 | doi=
10.1090/S0002-9904-1949-09327-8 }}
*{{cite journal|first1=P. |last1=Concus | first2=D. |last2=Cassatt
|first3=G. |last3=Jaehnig | first4=E. | last4=Melby
|title=Tables for the evaluation of int_0^infinity x^beta exp(-x) f(x) dx by Gauss-Laguerre quadrature
|journal=Math. Comp.
|volume=17
|pages=245–256 |year=1963 | doi=10.1090/S0025-5718-1963-0158534-9 }}
*{{cite journal
|first1=T. S. | last1=Shao | first2=T. C. | last2=Chen
|first3=R. M. | last3=Frank
|title=Table of zeros and Gaussian Weights of certain Associated Laguerre Polynomials and the related Hermite Polynomials
|year=1964
|jstor=2002946
|journal=Math. Comput.
|volume=18 |issue=88 | pages=598–616 | mr=0166397 |doi=10.1090/S0025-5718-1964-0166397-1 }}
*{{cite journal
|first1=Sven | last1=Ehrich
|title=On stratified extensions of Gauss-Laguerre and Gauss-Hermite quadrature formulas
|year=2002 | journal=J. Comput. Appl. Math. | volume=140 | issue=1-2
|pages=291–299 | doi=
10.1016/S0377-0427(01)00407-1
}}
*{{ cite web
|first1= R. J. | last1 = Mathar
|url=http://vixra.org/abs/1303.0013
|title=Gauss-Laguerre and Gauss-Hermite Quadrature on 64, 96 and 128 Nodes
|year=2013
}}
 
==External links==
* [http://www.mathworks.com/matlabcentral/fileexchange/8067-gauss-laguerre Matlab routine for Gauss–Laguerre quadrature]
* [http://people.sc.fsu.edu/~jburkardt/m_src/gen_laguerre_rule/gen_laguerre_rule.html generalized Gauss–Laguerre quadrature], [[free software]] in Matlab, C++, and Fortran.
 
{{DEFAULTSORT:Gauss-Laguerre quadrature}}
[[Category:Numerical integration (quadrature)]]

Latest revision as of 21:56, 6 December 2014


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