“Gaussian Quadrature”的概念、定义、习题解题方法及翻译-数学辞典

单词

Gaussian Quadrature

释义

Gaussian Quadrature

Seeks to obtain the best numerical estimate of an integral by picking optimal Abscissas at whichto evaluate the function . The Fundamental Theorem of Gaussian Quadrature states that the optimalAbscissas of the -point Gaussian Quadrature Formulas are precisely theroots of the orthogonal Polynomial for the same interval and Weighting Function. Gaussian quadrature is optimalbecause it fits all Polynomials up to degree exactly. Slightly less optimal fits are obtainedfrom Radau Quadrature and Laguerre Quadrature.

	Interval	Are Roots Of
1

To determine the weights corresponding to the Gaussian Abscissas, compute a Lagrange InterpolatingPolynomial 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 aWeighting Function


			(4)

with Weight

(5)

The weights

are sometimes also called the Christoffel Number (Chandrasekhar 1967).For orthogonal Polynomials

with

=1, ...,

(6)

(Hildebrand 1956, p. 322), where

is the Coefficient of

, then


			(7)

where

(8)

Using the relationship

(9)

(Hildebrand 1956, p. 323) gives

(10)

(Note that Press et al. 1992 omit the factor

.) In Gaussian quadrature, the weights are allPositive. The error is given by

(11)

where

(Hildebrand 1956, pp. 320-321).

Other curious identities are

(12)

and

(13)

(Hildebrand 1956, p. 323).

In the Notation of Szegö (1975), let be an ordered set of points in , and let, ..., be a set of Real Numbers. If is an arbitrary function on the Closed Interval , writethe Mechanical Quadrature as

(14)

Here

are the Abscissas and

are the Cotes Numbers.

See also Chebyshev Quadrature, Chebyshev-Gauss Quadrature, Chebyshev-Radau Quadrature, FundamentalTheorem of Gaussian Quadrature, Hermite-Gauss Quadrature, Jacobi-Gauss Quadrature, Laguerre-GaussQuadrature, Legendre-Gauss Quadrature, Lobatto Quadrature, Mehler Quadrature, Radau Quadrature

References

Abramowitz, M. and Stegun, C. 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.

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.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 37-48 and 340-349, 1975.

Whittaker, E. T. and Robinson, G. The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 152-163, 1967.

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