Definition:Newton Divided Difference Interpolation Formula

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Definition

Let a real function $y = \map f x$ have the known values:

$y_0, y_1, \ldots, y_n$

at the real numbers:

$x_0, x_1, \ldots, x_n$

respectively.


Let a value $y'$ be required to be estimated at the real number $x'$.


Then:

\(\ds y'\) \(=\) \(\ds a_0\) \(\ds {} + a_1 \paren {x' - x_0}\)
\(\ds \) \(\) \(\ds \) \(\ds {} + a_2 \paren {x' - x_0} \paren {x' - x_1}\)
\(\ds \) \(\) \(\ds \) \(\ds {} + \cdots\)
\(\ds \) \(\) \(\ds \) \(\ds {} + a_n \paren {x' - x_0} \paren {x' - x_1} \cdots \paren {x' - x_{n - 1} }\)

where:

$a_k = f \sqbrk {x_0, x_1, \ldots, x_k}$

where $f$ is defined recursively as:

$ \begin {cases} f \sqbrk {x_k} & = & \map f {x_k} \\ f \sqbrk {x_0, x_1, \ldots, x_{k + 1} } & = & \dfrac {f \sqbrk {x_1, x_2, \ldots, x_{k + 1} } - f \sqbrk {x_0, x_1, \ldots, x_k} } {x_{k + 1} - x_0} \end {cases}$

This is called the Newton divided difference interpolation formula.



This article is complete as far as it goes, but it could do with expansion.
In particular: There are very probably limitations on $f$ in the area of continuity which would have bearing on how accurate $y'$ is expected to be, but such matters can be investigated in due course.
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Examples

Example: $f \sqbrk {0, 1}$

$f \sqbrk {0, 1} = \dfrac {\map f {x_1} - \map f {x_0} } {x_1 - x_0}$


Also known as

The Newton divided difference interpolation formula is also known just as the divided difference interpolation formula.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ either form may be used, but when exploring the historical context the former is preferred.


Also see

  • Results about the Newton divided difference interpolation formula can be found here.


Source of Name

This entry was named for Isaac Newton.


Historical Note


This needs considerable tedious hard slog to complete it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
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Sources

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