Little O
Big O notation is a mathematical notation that describes the approximate size of a function on a domain. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann and Edmund Landau
and expanded by others, collectively called Bachmann–Landau notation. The letter O was chosen by Bachmann to stand for Ordnung, meaning the order of approximation.
In computer science, big O notation is used to classify algorithms according to how their run time or space requirements
grow as the input size grows.
In analytic number theory, big O notation is often used to express bounds on the growth of an arithmetical function; one well-known example is the remainder term in the prime number theorem.
In mathematical analysis, including calculus,
Big O notation is used to bound the error when truncating a power series and to express the quality
of approximation of a real or complex valued function
by a simpler function.
Often, big O notation characterizes functions according to their growth rates as the variable becomes large: different functions with the same asymptotic growth rate may be represented using the same O notation. The letter O is used because the growth rate of a function is also referred to as the order of the function. A description of a function in terms of big O notation only provides an upper bound on the growth rate of the function.
Associated with big O notation are several related notations, using the symbols
o
{\displaystyle o}
,
∼
{\displaystyle \sim }
,
Ω
{\displaystyle \Omega }
,
≪
{\displaystyle \ll }
,
≫
{\displaystyle \gg }
,
≍
{\displaystyle \asymp }
,
ω
{\displaystyle \omega }
, and
Θ
{\displaystyle \Theta }
to describe other kinds of bounds on growth rates.
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