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### gaussian quadrature

Gaussian quadrature allows us to integrate polynomials up to order $2 n-1$ using only $n$ function values.
$\int_{x_0}^{x_1} ( \sum_{i=0}^{2 n-1} a_i x^i ) dx = f(a_0, \dots , a_{2 n-1}) $
thus, the ...