Lagrange Interpolation Formula/Formulation 2

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

Equivalence of Formulations of Lagrange Interpolation Formula

Source of Name

This entry was named for Joseph Louis Lagrange.

Sources

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Lagrange's interpolation formula
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Lagrange's interpolation formula
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Lagrange interpolation formula