Lagrange's interpolation formula

English

Named after Joseph Louis Lagrange (1736–1813), an Italian Enlightenment Era mathematician and astronomer.

Lagrange's interpolation formula (uncountable)

(mathematics) A formula which when given a set of n points ${\\displaystyle (x_{i},y_{i})}$ , 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: ${\\displaystyle \\sum _{i}^{n}y_{i}\\prod _{j\eq i}{x-x_{j} \\over x_{i}-x_{j}}}$ . When ${\\displaystyle x=x_{i}}$ then all terms in the sum other than the i^th contain a factor ${\\displaystyle x-x_{i}}$ in the numerator, which becomes equal to zero, thus all terms in the sum other than the i^th vanish, and the i^th term has factors ${\\displaystyle x_{i}-x_{j}}$ both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return ${\\displaystyle y_{i}}$ as the function of ${\\displaystyle x_{i}}$ for any i in the set ${\\displaystyle \\{1,...,n\\}}$ .

单词	Lagrange's interpolation formula
释义	Lagrange's interpolation formula English Etymology Named after Joseph Louis Lagrange (1736–1813), an Italian Enlightenment Era mathematician and astronomer. Noun Lagrange's interpolation formula (uncountable) (mathematics) A formula which when given a set of n points ${\\displaystyle (x_{i},y_{i})}$ , 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: ${\\displaystyle \\sum _{i}^{n}y_{i}\\prod _{j\eq i}{x-x_{j} \\over x_{i}-x_{j}}}$ . When ${\\displaystyle x=x_{i}}$ then all terms in the sum other than the i^th contain a factor ${\\displaystyle x-x_{i}}$ in the numerator, which becomes equal to zero, thus all terms in the sum other than the i^th vanish, and the i^th term has factors ${\\displaystyle x_{i}-x_{j}}$ both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return ${\\displaystyle y_{i}}$ as the function of ${\\displaystyle x_{i}}$ for any i in the set ${\\displaystyle \\{1,...,n\\}}$ .
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