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Mathematics-Online lexicon:

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.


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}) \

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


[Examples] [Links]

  automatically generated 1/17/2017