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Chebyshev-Gauss Quadrature

Also called Chebyshev Quadrature. A Gaussian Quadrature over the interval $[-1,1]$ with Weighting Function $W(x)=1/\sqrt{1-x^2}$. The Abscissas for quadrature order $n$ are given by the roots of the Chebyshev Polynomial of the First Kind $T_n(x)$, which occur symmetrically about 0. The Weights are

\begin{displaymath}
w_i=-{A_{n+1}\gamma_n\over A_nT_n'(x_i)T_{n+1}(x_i)}={A_n\over A_{n-1}}{\gamma_{n-1}\over T_{n-1}(x_i)T_n'(x_i)},
\end{displaymath} (1)

where $A_n$ is the Coefficient of $x^n$ in $T_n(x)$. For Hermite Polynomials,
\begin{displaymath}
A_n=2^{n-1},
\end{displaymath} (2)

so
\begin{displaymath}
{A_{n+1}\over A_n}=2.
\end{displaymath} (3)

Additionally,
\begin{displaymath}
\gamma_n={\textstyle{1\over 2}}\pi,
\end{displaymath} (4)

so
\begin{displaymath}
w_i=-{\pi\over T_{n+1}(x_i)T_n'(x_i)}.
\end{displaymath} (5)

Since
\begin{displaymath}
T_n(x)=\cos(n\cos^{-1}x),
\end{displaymath} (6)

the Abscissas are given explicitly by
\begin{displaymath}
x_i=\cos\left[{(2i-1)\pi\over 2n}\right].
\end{displaymath} (7)

Since
$\displaystyle T_n'(x_i)$ $\textstyle =$ $\displaystyle {(-1)^{i+1}n\over\alpha_i}$ (8)
$\displaystyle T_{n+1}(x_i)$ $\textstyle =$ $\displaystyle (-1)^i\sin\alpha_i,$ (9)

where
\begin{displaymath}
\alpha_i={(2i-1)\pi\over 2n},
\end{displaymath} (10)

all the Weights are
\begin{displaymath}
w_i={\pi\over n}.
\end{displaymath} (11)

The explicit Formula is then


\begin{displaymath}
\int_{-1}^1 {f(x)\,dx\over\sqrt{1-x^2}}={\pi\over n}\sum_{k=...
...2n}\pi}\right)}\right]+ {2\pi\over 2^{2n}(2n)!} f^{(2n)}(\xi).
\end{displaymath} (12)

$n$ $x_i$ $w_i$
2 ± 0.707107 1.5708
3 0 1.0472
  ± 0.866025 1.0472
4 ± 0.382683 0.785398
  ± 0.92388 0.785398
5 0 0.628319
  ± 0.587785 0.628319
  ± 0.951057 0.628319


References

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 330-331, 1956.



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© 1996-9 Eric W. Weisstein
1999-05-26