Chebyshev Quadrature
المؤلف:
Beyer, W. H.
المصدر:
CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press
الجزء والصفحة:
...
2-12-2021
787
Chebyshev Quadrature
A Gaussian quadrature-like formula for numerical estimation of integrals. It uses weighting function 
in the interval ![[-1,1]](https://mathworld.wolfram.com/images/equations/ChebyshevQuadrature/Inline2.gif)
and forces all the weights to be equal. The general formula is
 |
(1)
|
where the abscissas 
are found by taking terms up to 
in the Maclaurin series of
![s_n(y)=exp<span style=]() {1/2n[-2+ln(1-y)(1-1/y)+ln(1+y)(1+1/y)]}, " src="https://mathworld.wolfram.com/images/equations/ChebyshevQuadrature/NumberedEquation2.gif" style="height:38px; width:368px" /> |
(2)
|
and then defining
 |
(3)
|
The roots of 
then give the abscissas. The first few values are
(OEIS A002680 and A101270).
Because the roots are all real for 
and 
only (Hildebrand 1956), these are the only permissible orders for Chebyshev quadrature. The error term is
![E_n=<span style=]() {c_n(f^((n+1))(xi))/((n+1)!) n odd; c_n(f^((n+2))(xi))/((n+2)!) n even, " src="https://mathworld.wolfram.com/images/equations/ChebyshevQuadrature/NumberedEquation4.gif" style="height:90px; width:170px" /> |
(14)
|
where
![c_n=<span style=]() {int_(-1)^1xG_n(x)dx n odd; int_(-1)^1x^2G_n(x)dx n even. " src="https://mathworld.wolfram.com/images/equations/ChebyshevQuadrature/NumberedEquation5.gif" style="height:82px; width:188px" /> |
(15)
|
The first few values of 
are 2/3, 8/45, 1/15, 32/945, 13/756, and 16/1575 (Hildebrand 1956). Beyer (1987) gives abscissas up to 
and Hildebrand (1956) up to 
.
The abscissas and weights can be computed analytically for small 
.
REFERENCES:
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 466, 1987.
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 345-351, 1956.
Salzer, H. E. "Tables for Facilitating the Use of Chebyshev's Quadrature Formula." J. Math. Phys. 26, 191-194, 1947.
Sloane, N. J. A. Sequences A002680/M2261 and A101270 in "The On-Line Encyclopedia of Integer Sequences."
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