Polynomial interpolation

In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, polynomial interpolation is the <a href="/facts/Interpolation/8PyetX4f">interpolation</a> of a given <a href="/facts/Data_set/eDvegUTK">data set</a> by the <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> of lowest possible degree that passes through the <a href="/facts/Ordered_pair/EInLVI0r">ordered pair</a> of points in the dataset.
Given a set of n + 1 data points 
 
 
 
 (
 
 x
 
 0
 
 
 ,
 
 y
 
 0
 
 
 )
 ,
 …
 ,
 (
 
 x
 
 n
 
 
 ,
 
 y
 
 n
 
 
 )
 
 
 {\displaystyle (x_{0},y_{0}),\ldots ,(x_{n},y_{n})}
 
, with no two 
 
 
 
 
 x
 
 j
 
 
 
 
 {\displaystyle x_{j}}
 
 the same, a polynomial function 
 
 
 
 p
 (
 x
 )
 =
 
 a
 
 0
 
 
 +
 
 a
 
 1
 
 
 x
 +
 ⋯
 +
 
 a
 
 n
 
 
 
 x
 
 n
 
 
 
 
 {\displaystyle p(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}}
 
 is said to interpolate the data if 
 
 
 
 p
 (
 
 x
 
 j
 
 
 )
 =
 
 y
 
 j
 
 
 
 
 {\displaystyle p(x_{j})=y_{j}}
 
 for each 
 
 
 
 j
 ∈
 {
 0
 ,
 1
 ,
 …
 ,
 n
 }
 
 
 {\displaystyle j\in \{0,1,\dotsc ,n\}}
 
.
There is always a unique such polynomial, commonly given by two explicit formulas, the <a href="/facts/Lagrange_polynomial/MnpMmhtC">Lagrange polynomials</a> and <a href="/facts/Newton_polynomial/2vRsjpA2">Newton polynomials</a>.

Polynomial interpolation open-in-new

Polynomial interpolation