Lagrange's interpolation formula
التعريفات والمعاني
== 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
(
x
i
,
y
i
)
{\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:
∑
i
n
y
i
∏
j
≠
i
x
−
x
j
x
i
−
x
j
{\displaystyle \sum _{i}^{n}y_{i}\prod _{j\neq i}{x-x_{j} \over x_{i}-x_{j}}}
. When
x
=
x
i
{\displaystyle x=x_{i}}
then all terms in the sum other than the i th contain a factor
x
−
x
i
{\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
x
i
−
x
j
{\displaystyle x_{i}-x_{j}}
both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return
y
i
{\displaystyle y_{i}}
as the function of
x
i
{\displaystyle x_{i}}
for any i in the set
{
1
,
.
.
.
,
n
}
{\displaystyle \{1,...,n\}}
.