Hermite-Gauss Quadrature

Eric W. Weisstein
January 31, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumericalMath/Hermite-GaussQuadrature.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Hermite-GaussQuadrature.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Quadrature`;

Check

x/.{ToRules@Roots[HermiteH[3,x]==0,x]}

%//FullForm

RootReduce@Table[{
(2^n)(n-1)! Sqrt[π]/(HermiteH[n-1,#]Derivative[0,1][HermiteH][n,#]),-(2^(n+1))n! Sqrt[π]/(HermiteH[n+1,#]Derivative[0,1][HermiteH][n,#]),
(2^(n+1))n! Sqrt[π]/(Derivative[0,1][HermiteH][n,#])^2,
2^(n+1)n!Sqrt[π]/HermiteH[n+1,#]^2,
2^(n-1)n!Sqrt[π]/(n^2 HermiteH[n-1,#]^2)
}&/@(x/.{ToRules@Roots[HermiteH[n,x]==0,x]}),{n,5}]

Equal/@%

Table

Table[Together//@GaussianQuadratureExact[HermiteH,Exp[-#^2]&,{-Infinity,Infinity},i],{i,2,5}]//TableForm

%//N//TableForm