Word Pronunciation & Dictionary

lagrange's interpolation formula

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Frequency: Zipf frequency 0.00 in modern English

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noun

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.

Uncountable

What does lagrange's interpolation formula mean?: 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.
Where does lagrange's interpolation formula come from?: Named after Joseph Louis Lagrange (1736–1813), an Italian Enlightenment Era mathematician and astronomer.

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