Definition:Order of Convergence
Definition
Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence of numbers or vectors in a normed vector space.
Let $\alpha \in \R$.
Let $p \in \R_{\ge 1}$.
Then $\sequence {x_n}$ converges to $\alpha$ with order $p$ if and only if there exists a sequence $\sequence {\epsilon_n}_{n \mathop \in \N}$ such that:
- $(1): \quad \size {x_n - \alpha} \le \epsilon_n$ for every $n \in \N$
- $(2): \quad \ds \lim_{n \mathop \to \infty} \frac {\epsilon_{n + 1} } { {\epsilon_n}^p} = c$ where $c > 0$
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If $p = 1$, the constant $c$ is additionally required to be less than $1$.
First-Order Convergence
$\sequence {x_n}$ converges to $\alpha$ with order $1$ if and only if there exists a sequence $\sequence {\epsilon_n}_{n \mathop \in \N}$ such that:
- $(1): \quad \size {x_n - \alpha} \le \epsilon_n$ for every $n \in \N$
- $(2): \quad \ds \lim_{n \mathop \to \infty} \frac {\epsilon_{n + 1} } {\epsilon_n} = c$
where $0 < c < 1$.
Second-Order Convergence
$\sequence {x_n}$ converges to $\alpha$ with order $2$ if and only if there exists a sequence $\sequence {\epsilon_n}_{n \mathop \in \N}$ such that:
- $\size {x_n - \alpha} \le \epsilon_n$ for every $n \in \N$
- $(2): \quad \ds \lim_{n \mathop \to \infty} \frac {\epsilon_{n + 1} } { {\epsilon_n}^2} = c$
where $0 < c < 1$.
Also known as
Order of convergence is also known as rate of convergence.
Also see
- Results about order of convergence can be found here.
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order: 12. (of convergence of a sequence)
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