Category:Lagrange Interpolation Formula
This category contains pages concerning Lagrange Interpolation Formula:
Theorem
Formulation 1
Let $\tuple {x_0, \ldots, x_n}$ and $\tuple {a_0, \ldots, a_n}$ be ordered tuples of real numbers such that $x_i \ne x_j$ for $i \ne j$.
Then there exists a unique polynomial $P \in \R \sqbrk X$ of degree at most $n$ such that:
- $\map P {x_i} = a_i$ for all $i \in \set {0, 1, \ldots, n}$
Moreover $P$ is given by the formula:
- $\ds \map P X = \sum_{j \mathop = 0}^n a_i \map {L_j} X$
where $\map {L_j} X$ is the $j$th Lagrange basis polynomial associated to the $x_i$.
Formulation 2
Let $f : \R \to \R$ be a real function.
Let $f$ have known values $y_i = \map f {x_i}$ for $n \in \set {0, 1, \ldots, n}$.
Let a value $y' = \map f {x'}$ be required to be estimated at some $x'$.
Then:
\(\ds y'\) | \(\approx\) | \(\ds \dfrac {y_1 \paren {x' - x_2} \paren {x' - x_3} \cdots \paren {x' - x_n} } {\paren {x_1 - x_2} \paren {x_1 - x_3} \cdots \paren {x_1 - x_n} }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac {y_2 \paren {x' - x_1} \paren {x' - x_3} \cdots \paren {x' - x_n} } {\paren {x_2 - x_1} \paren {x_2 - x_3} \cdots \paren {x_2 - x_n} }\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \cdots\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac {y_n \paren {x' - x_1} \paren {x' - x_2} \cdots \paren {x' - x_{n - 1} } } {\paren {x_n - x_1} \paren {x_n - x_3} \cdots \paren {x_n - x_{n - 1} } }\) |
Also known as
The Lagrange interpolation formula can also be styled as Lagrange's interpolation formula.
Also see
Source of Name
This entry was named for Joseph Louis Lagrange.
Source of Name
This entry was named for Joseph Louis Lagrange.
Subcategories
This category has only the following subcategory.
L
Pages in category "Lagrange Interpolation Formula"
The following 6 pages are in this category, out of 6 total.