Gauss–Hermite quadrature

Consider a function h(y), where the variable y is Normally distributed: $y\; \backslash sim\; \backslash mathcal\{N\}(\backslash mu,\backslash sigma^2)$. The expectation of h corresponds to the following integral:

$E[h(y)]\; =\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; \backslash frac\{1\}\{\backslash sigma\; \backslash sqrt\{2\backslash pi\}\}\; \backslash exp\; \backslash left(\; -\backslash frac\{(y-\backslash mu)^2\}\{2\backslash sigma^2\}\; \backslash right)\; h(y)\; dy$

As this does not exactly correspond to the Hermite polynomial, we need to change variables:

$x\; =\; \backslash frac\{y-\backslash mu\}\{\backslash sqrt\{2\}\; \backslash sigma\}\; \backslash Leftrightarrow\; y\; =\; \backslash sqrt\{2\}\; \backslash sigma\; x\; +\; \backslash mu$

Coupled with the integration by substitution, we obtain:

$E[h(y)]\; =\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; \backslash frac\{1\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash exp(-x^2)\; h(\backslash sqrt\{2\}\; \backslash sigma\; x\; +\; \backslash mu)\; dx$

leading to:

$E[h(y)]\; \backslash approx\; \backslash frac\{1\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash sum\_\{i=1\}^n\; w\_i\; h(\backslash sqrt\{2\}\; \backslash sigma\; x\_i\; +\; \backslash mu)$

As an illustration, in the simplest non-trivial case, with $n\; =\; 2$, we have $x\_1\; =\; -\backslash frac\{1\}\{\backslash sqrt\{2\}\},\; x\_2\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}$ and $w\_1\; =\; w\_2\; =\; \backslash frac\{\backslash sqrt\{\backslash pi\}\}\{2\}$, so the estimate reduces to:

$E[h(y)]\; \backslash approx\; \backslash frac\{1\}\{2\}(h(\backslash mu\; -\; \backslash sigma)\; +\; h(\backslash mu\; +\; \backslash sigma))$

– i.e. the average of the function's values one standard deviation below and above the mean.

• {{dlmf| id = 3.5.E28| title=Quadrature: Gauss–Hermite Formula}}
• {{cite journal|first1=T. S. | last1=Shao | first2=T. C. | last2=Chen | first3= R. M. | last3=Frank

|title=Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials

|year=1964

|journal=Math. Comp. | mr=0166397 | volume=18 | number=88 | pages=598–616 |doi=10.1090/S0025-5718-1964-0166397-1| doi-access=free }}

• {{cite journal|first1=N. M. | last1=Steen | first2=G. D. | last2=Byrne

|first3=E. M. | last3=Gelbard | title = Gaussian quadratures for the integrals $\backslash textstyle\backslash int\_0^\backslash infty\; e^\{-x^2\}\; f(x)\; dx$ and $\backslash textstyle\backslash int\_0^b\; e^\{-x^2\}\; f(x)\; dx$

|journal = Math. Comp. | year=1969

|volume=23 | number=107 | pages=661–671 | mr=0247744 | doi=10.1090/S0025-5718-1969-0247744-3 | doi-access=free }}

• {{cite journal|first1= B. | last1=Shizgal | title = A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems

|journal=J. Comput. Phys. | volume=41 | pages=309–328 | year=1981

| issue=2 |doi=10.1016/0021-9991(81)90099-1 | bibcode=1981JCoPh..41..309S }}

• For tables of Gauss-Hermite abscissae and weights up to order n = 32 see http://www.efunda.com/math/num_integration/findgausshermite.cfm.
• [http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_hermite_rule/gen_hermite_rule.html Generalized Gauss–Hermite quadrature], free software in C++, Fortran, and Matlab

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Category:Numerical integration

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