Definition:Gregory-Newton Backward Difference Formula
(Redirected from Definition:Newton's Backward Difference Formula)
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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 $f$:
- $\forall i \in \set {0, 1, 2, \ldots, n}: y_i = \map f {x_i}$
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: $\nabla y_n = y_n - y_{n - 1}$
- $\nabla^i y_n := \paren {\nabla y_n}^i$
Also known as
The Gregory-Newton backward difference formula is also known as:
- the Newton-Gregory backward difference formula
- Newton's backward difference formula
- the backward difference formula
and so on.
Also see
- Gregory-Newton Backward Difference Formula for a proof that the technique is valid
- Results about Gregory-Newton interpolation can be found here.
Source of Name
This entry was named for James Gregory and Isaac Newton.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gregory-Newton interpolation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gregory-Newton interpolation