Chebyshev-Gauss Quadrature
المؤلف:
Abramowitz, M. and Stegun, I. A
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover,
الجزء والصفحة:
...
2-12-2021
2039
Chebyshev-Gauss Quadrature
Chebyshev-Gauss quadrature, also called Chebyshev quadrature, is a Gaussian quadrature over the interval ![[-1,1]](https://mathworld.wolfram.com/images/equations/Chebyshev-GaussQuadrature/Inline1.gif)
with weighting function 
(Abramowitz and Stegun 1972, p. 889). The abscissas for quadrature order 
are given by the roots of the Chebyshev polynomial of the first kind 
, which occur symmetrically about 0. The weights are
where 
is the coefficient of 
in 
,
 |
(3)
|
and 
the order-
Lagrange interpolating polynomial for 
.
For Chebyshev polynomials of the first kind,
 |
(4)
|
so
 |
(5)
|
Additionally,
 |
(6)
|
so
 |
(7)
|
Since
 |
(8)
|
the abscissas are given explicitly by
![x_i=cos[((2i-1)pi)/(2n)].](https://mathworld.wolfram.com/images/equations/Chebyshev-GaussQuadrature/NumberedEquation7.gif) |
(9)
|
Since
where
 |
(12)
|
all the weights are
 |
(13)
|
The explicit formula is then
![int_(-1)^1(f(x)dx)/(sqrt(1-x^2))=pi/nsum_(k=1)^nf[cos((2k-1)/(2n)pi)]+(2pi)/(2^(2n)(2n)!)f^((2n))(xi).](https://mathworld.wolfram.com/images/equations/Chebyshev-GaussQuadrature/NumberedEquation10.gif) |
(14)
|
The following two tables give the numerical and analytic values for the first few points and weights.
 |
 |
 |
| 2 |
 |
1.5708 |
| 3 |
0 |
1.0472 |
| |
 |
1.0472 |
| 4 |
 |
0.785398 |
| |
 |
0.785398 |
| 5 |
0 |
0.628319 |
| |
 |
0.628319 |
| |
 |
0.628319 |
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 889, 1972.
Bronwin, B. "On the Determination of the Coefficients in Any Series of Sines and Cosines of Multiples of a Variable Angle from Particular Values of that Series." Phil. Mag. 34, 260-268, 1849.
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 330-331, 1956.
Tchebicheff, P. "Sur les quadratures." J. de math. pures appliq. 19, 19-34, 1874.
Whittaker, E. T. and Robinson, G. "Chebyshef's Formulae." §79 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 158-159, 1967.
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