Definition:Theta Notation/Also defined as

$\Theta$ Notation: Also defined as

Sources which utilise order notation so as to explore the behaviour of algorithms are concerned only with algorithm run times, necessarily positive.

Hence they may define the $\Theta$ notation on positive real sequences only, as follows:

Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.

Then $\map \Theta g$ is defined as:

$\map \Theta g = \set {f: \N \to \R: \exists c_1, c_2 \in \R_{>0}: \exists n_0 \in \N: \forall n \ge n_0: 0 \le c_1 \cdot \map g n \le \map f n \le c_2 \cdot \map g n}$

Some sources define some or all of the inequalities in this expression to be strict, that is:

$\map \Theta g = \set {f: \N \to \R: \exists c_1, c_2 \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le c_1 \cdot \map g n < \map f n < c_2 \cdot \map g n}$

Also see

• Results about $\Theta$ notation can be found here.