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Gaussian quadrature
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====Proof that the weights are positive==== Consider the following polynomial of degree <math>2n - 2</math> <math display="block">f(x) = \prod_{\begin{smallmatrix} 1 \leq j \leq n \\ j \neq i \end{smallmatrix}}\frac{\left(x - x_j\right)^2}{\left(x_i - x_j\right)^2}</math> where, as above, the {{mvar|x<sub>j</sub>}} are the roots of the polynomial <math>p_{n}(x)</math>. Clearly <math>f(x_j) = \delta_{ij}</math>. Since the degree of <math>f(x)</math> is less than <math>2n - 1</math>, the Gaussian quadrature formula involving the weights and nodes obtained from <math>p_{n}(x)</math> applies. Since <math>f(x_{j}) = 0</math> for {{mvar|j}} not equal to {{mvar|i}}, we have <math display="block">\int_{a}^{b}\omega(x)f(x)dx=\sum_{j=1}^{n}w_{j}f(x_{j}) = \sum_{j=1}^{n} \delta_{ij} w_j = w_{i} > 0.</math> Since both <math>\omega(x)</math> and <math>f(x)</math> are non-negative functions, it follows that <math>w_{i} > 0</math>.
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