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Let $S(x)$ be the natural cubic spline over the interval $\left[x_{0}, x_{n}\right]$ determined by the knots $\left\{\left(x_{0}, y_{0}\right),\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)\right\}$. Let $S_{i}(x)$ be the cubic polynomial for the spline over the interval $\left[x_{i}, x_{i+1}\right] .$ Give the equations to determine the coefficients for $S_{i}(x)=a_{i}+b_{i}\left(x-x_{i}\right)+c_{i}\left(x-x_{i}\right)^{2}+d_{i}\left(x-x_{i}\right)^{3}$ for $i \in\{0,1,2, \ldots, n-1\}$
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