Category:Newton Divided Difference Interpolation Formula
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This category contains results about Newton Divided Difference Interpolation Formula.
Definitions specific to this category can be found in Definitions/Newton Divided Difference Interpolation Formula.
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}$
Subcategories
This category has only the following subcategory.