Anybody knows how to find the polynomial function with evaluated values, where if the degree is $n$ I have $n+1$ values of the function like $f(0) = a_0, f(1) = a_1, \ldots, f(n) = a_n$.
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$\begingroup$ Lagrange interpolation. en.wikipedia.org/wiki/Lagrange_polynomial and mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html $\endgroup$– Eric TowersJul 10, 2014 at 5:28
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$\begingroup$ its essentially equivalent to solving system of $n+1$ linear equations $\endgroup$– user160738Jul 10, 2014 at 5:34
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$\begingroup$ I thought that, but I can't to solve how to represent the constant with the $f(k) = a_k$ $\endgroup$– user3145102Jul 10, 2014 at 5:38
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1 Answer
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Use $$p_i(x)=\frac {\prod_{i\neq j} (x-x_i)} {\prod_{i\neq j} (x_j - x_i)}$$ as a basis for the vector space, and notice that $p_i(x_j)=\delta_{ij}$.