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Lagrange interpolation formula (Theorem)

Let $ (x_1,y_1), (x_2,y_2),\dotsc,(x_n,y_n)$ be $ n$ points in the plane ( $ x_i\neq x_j$ for $ i\neq j$). Then there exists a unique polynomial $ p(x)$ of degree at most $ n-1$ such that $ y_i=p(x_i)$ for $ i=1,\ldots,n$.

Such polynomial can be found using Lagrange's interpolation formula:

$\displaystyle p(x)=\frac{f(x)}{(x-x_1)f'(x_1)}y_1+\frac{f(x)}{(x-x_2)f'(x_2)}y_2+\cdots+\frac{f(x)}{(x-x_n)f'(x_n)}y_n $
where $ f(x)=(x-x_1)(x-x_2)\cdots(x-x_n)$.

To see this, notice that the above formula is the same as

$\displaystyle p(x)$ $\displaystyle = y_1 \frac{(x-x_2)(x-x_3)\dots(x-x_n)}{(x_1-x_2)(x_1-x_3)\dots(x_1-x_n)} + y_2 \frac{(x-x_1)(x-x_3)\dots(x-x_n)}{(x_2-x_1)(x_2-x_3)\dots(x_2-x_n)}$    
  $\displaystyle \phantom{=}\qquad+\dots+y_n \frac{(x-x_1)(x-x_2)\dots(x-x_{n-1})}{(x_n-x_1)(x_n-x_2)\dots(x_n-x_{n-1})}$    

and that for all $ x_i$, every numerator except one vanishes, and this numerator will be identical to the denominator, making the overall quotient equal to 1. Therefore, each $ p(x_i)$ equals $ y_i$.



"Lagrange interpolation formula" is owned by drini. [ full author list (4) | owner history (1) ]
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See Also: Simpson's rule, lecture notes on polynomial interpolation

Other names:  Lagrange's Interpolation formula
Also defines:  Lagrange polynomial
Keywords:  Interpolation, Extrapolation, Polynomial, Derivative

Attachments:
proof of uniqueness of Lagrange Interpolation formula (Proof) by rspuzio
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Cross-references: quotient, denominator, vanishes, numerator, degree, polynomial, plane, points
There are 6 references to this entry.

This is version 11 of Lagrange interpolation formula, born on 2001-10-15, modified 2007-10-28.
Object id is 229, canonical name is LagrangeInterpolationFormula.
Accessed 31611 times total.

Classification:
AMS MSC: 41A05 (Approximations and expansions :: Interpolation)
 65D05 (Numerical analysis :: Numerical approximation and computational geometry :: Interpolation)
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