The Lagrange interpolating polynomial is the polynomial P(x) of degree <=(n - 1) that passes through the n points (x_1, y_1 = f(x_1)), (x_2, y_2 = f(x_2)), ..., (x_n, y_n = f(x_n)), and is given by P(x) = sum_(j = 1)^n P_j(x), where P_j(x) = y_j product_(k = 1 k!=j)^n (x - x_k)/(x_j - x_k). Written explicitly, P(x) | = | ((x - x_2)(x - x_3)...(x - x_n))/((x_1 - x_2)(x_1 - x_3)...(x_1 - x_n)) y_1 + ((x - x_1)(x - x_3)...(x - x_n))/((x_2 - x_1)(x_2 - x_3)...(x_2 - x_n)) y_2 + ... + ((x - x_1)(x - x_2)...(x - x_(n - 1)))/((x_n - x_1)(x_n - x_2)...(x_n - x_(n - 1))) y_n. The formula was first published by Waring, rediscovered by Euler in 1783, and published by Lagrange in 1795.
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