Definition:Gregory-Newton Interpolation

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Definition

Let $f$ be a real function.

Let $x_0, x_1, x_2, \ldots, x_n \in \R$ be equally spaced:

$\forall i \in \set {1, 2, \ldots, n}: x_i - x_{i - 1} = d$

where $d$ is constant.

Let $y_0, y_1, y_2, \ldots, y_n$ be values of $x_0, x_1, x_2, \ldots, x_n$ under $f$:

$\forall i \in \set {0, 1, 2, \ldots, n}: y_i = \map f {x_i}$


Gregory-Newton Forward Difference Formula

Let $x_0 < x' < x_1$.

Let $k = \dfrac {x' - x_0} d$.

Then $y' = \map f {x'}$ can be approximated by the formula:

$y' = y_0 + \dbinom k 1 \Delta y_0 + \dbinom k 2 \Delta^2 y_0 + \ldots + \dbinom k n \Delta^n y_0$

where:

$\Delta y_0$ is the forward difference operator: $\Delta y_0 = y_1 - y_0$
$\Delta^i y_0 := \paren {\Delta y_0}^i$


Gregory-Newton Backward Difference Formula

Let $x_{n - 1} < x' < x_n$.

Let $k = \dfrac {x' - x_{n - 1} } d$.

Then $y' = \map f {x'}$ can be approximated by the formula:

$y' = y_n + \dbinom k 1 \nabla y_n + \dbinom k 2 \nabla^2 y_n + \ldots + \dbinom k n \nabla^n y_n$

where:

$\nabla y_n$ is the backward difference operator: $\nabla y_n = y_n - y_{n - 1}$
$\nabla^i y_n := \paren {\nabla y_n}^i$


Gauss Interpolation Formula


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Also see

  • Results about Gregory-Newton interpolation can be found here.


Source of Name

This entry was named for James Gregory and Isaac Newton.


Sources

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