| 释义 |
QuadratureThe word quadrature has (at least) three incompatible meanings. Integration by quadrature either means solving anIntegral analytically (i.e., symbolically in terms of known functions), or solving of an integral numerically (e.g.,Gaussian Quadrature, Quadrature Formulas). The word quadrature is also used to mean Squaring: theconstruction of a square using only Compass and Straightedge which has the same Area as a givengeometric figure. If quadrature is possible for a Plane figure, it is said to be Quadrable.
For a function tabulated at given values  (so the Abscissas cannot be chosen at will), write thefunction  as a sum of Orthonormal Functions  satisfying
 | (1) |
 | (2) |
giving
 | (4) |
 | (5) |
 | (6) |
 in (1) gives
 | (7) |
 , so (7) becomes
where (6) has been used in the last step. We therefore have the Matrix equation
 | (10) |
 s (Press et al. 1992).See also Calculus, Chebyshev-Gauss Quadrature, Chebyshev Quadrature, Derivative, FundamentalTheorem of Gaussian Quadrature, Gauss-Jacobi Mechanical Quadrature, Gaussian Quadrature, Hermite-GaussQuadrature, Hermite Quadrature, Jacobi-Gauss Quadrature, Jacobi Quadrature, Laguerre-GaussQuadrature, Laguerre Quadrature, Legendre-Gauss Quadrature, Legendre Quadrature, LobattoQuadrature, Mechanical Quadrature, Mehler Quadrature, Newton-Cotes Formulas, NumericalIntegration, Radau Quadrature, Recursive Monotone Stable Quadrature
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Integration.'' §25.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 885-897, 1972.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 365-366, 1992. |
