Category:Newton Divided Difference Interpolation Formula
Jump to navigation
Jump to search
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}$
This is called the Newton divided difference interpolation formula.
Subcategories
This category has only the following subcategory.