English Dictionary

Lagrange's interpolation formula

名詞

不可算名詞

(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.

日本語の意味

ラグランジュの補間公式：数学における公式で、与えられた n 個の点 (x_i, y_i) を通る一意の、最高次数 n − 1 以下の多項式（補間多項式）を構成する。公式は、各点に対応する補間基底多項式の加重和として表され、各補間基底多項式は対象の点で 1 となり他の点で 0 となるように設計されている。

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.

サンプリングされた点から多項式を復元する際、これらの点を通る次数が高々 n−1 の一意の多項式を得るためにラグランジュ補間公式をよく用います。

意味（1）

(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.