Mathematics-Online lexicon: Tensor Products of Integration Rules

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	Mathematics-Online lexicon:
	Tensor Products of Integration Rules

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overview

Integration rules for rectangles

$\displaystyle Q = [a_1,b_1] \times \cdots \times [a_m,b_m]$

can be obtained by forming tensor products of univariate formulas.

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

If the formulas $\sum_k w_{k,\nu} f(t_{k,\nu})$ for approximating $\int_{a_\nu}^{b_\nu} f$ are exact for polynomials of degree $\le n_\nu$ , then the product rule

$\displaystyle \int_Q f \approx \sum_{k_1} \cdots \sum_{k_m} (w_{k_1,1}\cdots w_{k_m,m}) \ f(t_{k_1,1},\ldots,t_{k_m,m})$

is exact for polynomials of coordinate degree $\le (n_1,\ldots,n_m)$ .

Annotation:

Proof

[Examples] [Links]

automatically generated 1/17/2017