Lagrange Interpolation Formula/Formulation 2

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Theorem

Let $f : \R \to \R$ be a real function.

Let $f$ have known values $y_i = \map f {x_i}$ for $n \in \set {0, 1, \ldots, n}$.

Let a value $y' = \map f {x'}$ be required to be estimated at some $x'$.

Then:

\(\ds y'\) \(\approx\) \(\ds \dfrac {y_1 \paren {x' - x_2} \paren {x' - x_3} \cdots \paren {x' - x_n} } {\paren {x_1 - x_2} \paren {x_1 - x_3} \cdots \paren {x_1 - x_n} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {y_2 \paren {x' - x_1} \paren {x' - x_3} \cdots \paren {x' - x_n} } {\paren {x_2 - x_1} \paren {x_2 - x_3} \cdots \paren {x_2 - x_n} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \cdots\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \dfrac {y_n \paren {x' - x_1} \paren {x' - x_2} \cdots \paren {x' - x_{n - 1} } } {\paren {x_n - x_1} \paren {x_n - x_3} \cdots \paren {x_n - x_{n - 1} } }\)


Proof


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Also known as

The Lagrange interpolation formula can also be styled as Lagrange's interpolation formula.


Also see


Source of Name

This entry was named for Joseph Louis Lagrange.


Sources

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