Lagrange's interpolation formula
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(mathematics) A formula which when given a set of n points (x_i,y_i), gives back the unique polynomial of degree (at most) n − 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: ∑ᵢⁿy_i∏_(j ne i)x-x_j/x_i-x_j. When x=x_i then all terms in the sum other than the iᵗʰ contain a factor x-x_i in the numerator, which becomes equal to zero, thus all terms in the sum other than the iᵗʰ vanish, and the iᵗʰ term has factors x_i-x_j both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return y_i as the function of x_i for any i in the set 1,...,n.
Lagrange's interpolation formula
When reconstructing a polynomial from sampled points, we often apply Lagrange's interpolation formula to obtain the unique polynomial of degree at most n−1 that passes through those points.
When reconstructing a polynomial from sampled points, we often apply Lagrange's interpolation formula to obtain the unique polynomial of degree at most n−1 that passes through those points.
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