Gauss–Laguerre quadrature

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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:

In this case

where xi is the i-th root of Laguerre polynomial Ln(x) and the weight wi is given by [1]

For more general functions

To integrate the function we apply the following transformation

where . 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 power-law singularity at x=0, for some real number , leading to integrals of the form:

This allows one to efficiently evaluate such integrals for polynomial or smooth f(x) even when α is not an integer.[2]

References

  1. Equation 25.4.45 in {{#invoke:citation/CS1|citation |CitationClass=book }} 10th reprint with corrections.
  2. {{#invoke:Citation/CS1|citation |CitationClass=journal }}

Further reading

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External links