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Tensor Products of Integration Rules


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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)$.

see also:


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  automatically generated 1/17/2017