Definition:Big-O Notation/Sequence
Definition
Definition 1
Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.
Then $\map \OO g$ is defined as:
- $\map \OO g = \set {f: \N \to \R: \exists c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le \size {\map f n} \le c \cdot \size {\map g n} }$
Definition 2
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.
$a_n$ is big-$\OO$ of $b_n$ if and only if:
- $\exists c \in \R_{\ge 0}: \exists n_0 \in \N: \forall n \in \N: \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$
That is:
- $\size {a_n} \le c \cdot \size {b_n}$
for all sufficiently large $n$.
This is denoted:
- $a_n \in \map \OO {b_n}$
Notation
The expression $\map f n \in \map \OO {\map g n}$ is read as:
- $\map f n$ is big-O of $\map g n$
While it is correct and accurate to write:
- $\map f n \in \map \OO {\map g n}$
it is a common abuse of notation to write:
- $\map f n = \map \OO {\map g n}$
This notation offers some advantages.
Also defined as
Some authors require that $b_n$ be nonzero for $n$ sufficiently large.
Some authors require that the functions appearing in the $\OO$-estimate be positive or strictly positive.
Also denoted as
The big-$\OO$ notation, along with little-$\oo$ notation, are also referred to as Landau's symbols or the Landau symbols, for Edmund Georg Hermann Landau.
Also see
- Big-O Notation for Sequences Coincides with General Definition where it is shown that this definition coincides with the general definition if $\N$ is given the discrete topology.