Gaussian 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:
...
5-12-2021
2097
Gaussian Quadrature
Seeks to obtain the best numerical estimate of an integral by picking optimal abscissas 
at which to evaluate the function 
. The fundamental theorem of Gaussian quadrature states that the optimal abscissas of the 
-point Gaussian quadrature formulas are precisely the roots of the orthogonal polynomial for the same interval and weighting function. Gaussian quadrature is optimal because it fits all polynomials up to degree 
exactly. Slightly less optimal fits are obtained from Radau quadrature and Laguerre-Gauss quadrature.
To determine the weights corresponding to the Gaussian abscissas 
, compute a Lagrange interpolating polynomial for 
by letting
 |
(1)
|
(where Chandrasekhar 1967 uses 
instead of 
), so
 |
(2)
|
Then fitting a Lagrange interpolating polynomial through the 
points gives
 |
(3)
|
for arbitrary points 
. We are therefore looking for a set of points 
and weights 
such that for a weighting function 
,
with weight
 |
(6)
|
The weights 
are sometimes also called the Christoffel numbers (Chandrasekhar 1967). For orthogonal polynomials 
with 
, ..., 
,
 |
(7)
|
(Hildebrand 1956, p. 322), where 
is the coefficient of 
in 
, then
where
![gamma_m=int[phi_m(x)]^2W(x)dx.](https://mathworld.wolfram.com/images/equations/GaussianQuadrature/NumberedEquation6.gif) |
(10)
|
Using the relationship
 |
(11)
|
(Hildebrand 1956, p. 323) gives
 |
(12)
|
(Note that Press et al. 1992 omit the factor 
.) In Gaussian quadrature, the weights are all positive. The error is given by
where 
(Hildebrand 1956, pp. 320-321).
Other curious identities are
 |
(15)
|
and
(Hildebrand 1956, p. 323).
In the notation of Szegö (1975), let 
be an ordered set of points in ![[a,b]](https://mathworld.wolfram.com/images/equations/GaussianQuadrature/Inline70.gif)
, and let 
, ..., 
be a set of real numbers. If 
is an arbitrary function on the closed interval ![[a,b]](https://mathworld.wolfram.com/images/equations/GaussianQuadrature/Inline74.gif)
, write the Gaussian quadrature as
 |
(18)
|
Here 
are the abscissas and 
are the Cotes numbers.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 887-888, 1972.
Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 103, 1990.
Arfken, G. "Appendix 2: Gaussian Quadrature." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 968-974, 1985.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 461, 1987.
Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, 1967.
Gauss, C. F. "Methodus nova integralium valores per approximationem inveniendi." Commentationes Societatis regiae scientarium Gottingensis recentiores 3, 39-76, 1814. Reprinted in Werke, Vol. 3. New York: George Olms, p. 163, 1981.
Golub, G. H. and Welsh, J. H. "Calculation of Gauss Quadrature Rules." Math. Comput. 23, 221-230, 1969.
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319-323, 1956.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Gaussian Quadratures and Orthogonal Polynomials." §4.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 140-155, 1992.
Stroud, A. H. and Secrest, D. Gaussian Quadrature Formulas. Englewood Cliffs, NJ: Prentice-Hall, 1966.
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 37-48 and 340-349, 1975.
Whittaker, E. T. and Robinson, G. "Gauss's Formula of Numerical Integration." §80 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 152-163, 1967.
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