Chebyshev-Gauss Quadrature

Eric W. Weisstein
January 31, 2002

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

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

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

Initialization

<<Quadrature`;

γ

Table[Integrate[1/Sqrt[1-x^2]ChebyshevT[n,x]^2,{x,-1,1}],{n,4}]

Integrate[1/Sqrt[1-x^2]ChebyshevT[n,x]^2,{x,-1,1}]

FunctionExpand[%]

A

Table[Coefficient[ChebyshevT[n,x],x,n]-2^(n-1),{n,5}]

FullSimplify[(# /. n->n+1)/#&[2^(n-1)]]

Table

Together//@Simplify@Table[GaussianQuadratureExact[ChebyshevT,1/Sqrt[1-#^2]&,{-1,1},i],{i,2,5}]

Together//@Simplify@Table[GaussianQuadratureExact[ChebyshevT,1/Sqrt[1-#^2]&,{-1,1},i],{i,2,5}]//TableForm

%//N//TableForm