Mathematik-Online lexicon: Trapezoidal Rule for a Rectangle

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	Mathematik-Online lexicon:
	Trapezoidal Rule for a Rectangle

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overview

The weights of the trapezoidal rule with step size are equal to at inner knots and equal to at the endpoints and of the integration interval. Hence, for a rectangle

$\displaystyle [a_1,b_1] \times [a_2,b_2] \,,$

we obtain three different weights, as is illustrated in the figure.

$\includegraphics[width=0.5\linewidth]{Bild_Gewichte_Trapezregel}$

More generally, for an -dimensional rectangle with step sizes $h_\nu$ with respect to the $\nu$ -th coordinate,

$\displaystyle w_{k_1,\ldots,k_m} = h^m 2^{-\alpha_1} \cdots 2^{-\alpha_m} \,,$

where $\alpha_\nu=0$ (

) for an inner (a boundary) index of the $\nu$ -th coordinate.

As in the univariate case, the error of the multivariate trapezoidal rule has a quadratic expansion, and, consequently, Romberg extrapolation is applicable.

see also:

Tensor Products of Integration Rules

automatisch erstellt am 17. 1. 2017