View source for Big O notation

{{Order-of-approx}}
[[File:Big-O-notation.png|300px|thumb|Example of Big O notation: {{color|#ff0000|''f''(''x'')}} ∈
O({{color|#0000ff|''g''(''x'')}}) as there exists ''c''&nbsp;>&nbsp;0 (e.g., ''c''&nbsp;=&nbsp;1) and ''x''<sub>0</sub> (e.g., ''x''<sub>0</sub>&nbsp;=&nbsp;5) such that {{color|#ff0000|''f''(''x'')}}&nbsp;≤&nbsp;''c''{{color|#0000ff|''g''(''x'')}} whenever ''x''&nbsp;≥&nbsp;''x''<sub>0</sub>.]]

'''Big O notation''' is a mathematical notation that describes the [[asymptotic analysis|limiting behavior]] of a [[function (mathematics)|function]] when the [[Argument of a function|argument]] tends towards a particular value or infinity.  It is a member of a family of notations invented by [[Paul Gustav Heinrich Bachmann|Paul Bachmann]],<ref name=Bachmann /> [[Edmund Landau]],<ref name=Landau /> and others, collectively called '''Bachmann–Landau notation''' or '''asymptotic notation'''.

In [[computer science]], big O notation is used to [[Computational complexity theory|classify algorithms]] according to how their running time or space requirements grow as the input size grows.<ref name=quantumcomplexity>{{cite web|last1=Mohr|first1=Austin|title=Quantum Computing in Complexity Theory and Theory of Computation|url=http://www.austinmohr.com/Work_files/complexity.pdf|accessdate=7 June 2014|page=2}}</ref>  In [[analytic number theory]], big O notation is often used to express a bound on the difference between an [[arithmetic function|arithmetical function]] and a better understood approximation; a famous example of such a difference is the remainder term in the [[prime number theorem]].

Big O notation characterizes functions according to their growth rates: different functions with the same 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 usually only provides an [[upper bound]] on the growth rate of the function. Associated with big O notation are several related notations, using the symbols {{math|''o'', Ω, ω, and Θ}}, to describe other kinds of bounds on asymptotic growth rates.

Big O notation is also used in many other fields to provide similar estimates.

== Formal definition ==
Let ''f'' be a real or complex valued function and ''g'' a real valued function, both defined on some unbounded subset of the [[real number|real positive numbers]], such that ''g''(''x'') is strictly positive for all large enough values of ''x''.<ref name=LandauO>{{cite book |first=Edmund |last=Landau |authorlink=Edmund Landau |title=Handbuch der Lehre von der Verteilung der Primzahlen |publisher=B. G. Teubner |date=1909 |location=Leipzig |trans-title=Handbook on the theory of the distribution of the primes |language=de |page=31 | url=https://archive.org/stream/handbuchderlehre01landuoft#page/31/mode/2up}}</ref> One writes

:<math>f(x)=O(g(x))\text{ as }x\to\infty</math>

[[if and only if]] for all sufficiently large values of ''x'', the absolute value of ''f''(''x'') is at most a positive constant multiple of ''g''(''x''). That is, ''f''(''x'')&nbsp;=&nbsp;''O''(''g''(''x'')) if and only if there exists a positive real number ''M'' and a real number ''x''<sub>0</sub> such that

:<math>|f(x)| \le \; M g(x)\text{ for all }x \ge x_0.</math>

In many contexts, the assumption that we are interested in the growth rate as the variable ''x'' goes to infinity is left unstated, and one writes more simply that

:<math>f(x) = O(g(x)).</math>

The notation can also be used to describe the behavior of ''f'' near some real number ''a'' (often, ''a''&nbsp;=&nbsp;0): we say

:<math>f(x)=O(g(x))\text{ as }x\to a</math>

if and only if there exist positive numbers ''δ'' and ''M'' such that

:<math>|f(x)| \le \; M g(x)\text{ when } 0 < |x - a| < \delta.</math>

As ''g''(''x'') is chosen to be non-zero for values of ''x'' [[Mathematical jargon#sufficiently large|sufficiently close]] to ''a'', both of these definitions can be unified using the [[limit superior]]:

:<math>f(x)=O(g(x))\text{ as }x \to a</math>

if and only if

:<math>\limsup_{x\to a} \left|\frac{f(x)}{g(x)}\right| < \infty.</math> <!-- These absolute value bars may be left out. -->

== Example ==
In typical usage the ''O'' notation is asymptotical, that is, it refers to very large ''x''.  In this setting, the contribution of the terms that grow "most quickly" will eventually make the other ones irrelevant. As a result, the following simplification rules can be applied:
*If ''f''(''x'') is a sum of several terms, if there is one with largest growth rate, it can be kept, and all others omitted.
*If ''f''(''x'') is a product of several factors, any constants (terms in the product that do not depend on ''x'') can be omitted.
For example, let ''f''(''x'') = 6''x''<sup>4</sup>&nbsp;−&nbsp;2''x''<sup>3</sup>&nbsp;+&nbsp;5, and suppose we wish to simplify this function, using ''O'' notation, to describe its growth rate as ''x'' approaches infinity. This function is the sum of three terms: 6''x''<sup>4</sup>, −2''x''<sup>3</sup>, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of ''x'', namely 6''x''<sup>4</sup>. Now one may apply the second rule: 6''x''<sup>4</sup> is a product of 6 and ''x''<sup>4</sup> in which the first factor does not depend on ''x''. Omitting this factor results in the simplified form ''x''<sup>4</sup>. Thus, we say that ''f''(''x'') is a "big-oh" of (''x''<sup>4</sup>). Mathematically, we can write ''f''(''x'')&nbsp;=&nbsp;''O''(''x''<sup>4</sup>).
One may confirm this calculation using the formal definition: let ''f''(''x'')&nbsp;=&nbsp;6''x''<sup>4</sup>&nbsp;−&nbsp;2''x''<sup>3</sup>&nbsp;+&nbsp;5 and ''g''(''x'')&nbsp;=&nbsp;''x''<sup>4</sup>. Applying the [[#Formal definition|formal definition]] from above, the statement that ''f''(''x'')&nbsp;=&nbsp;''O''(''x''<sup>4</sup>) is equivalent to its expansion,
:<math>|f(x)| \le \; M x^4</math>
for some suitable choice of ''x''<sub>0</sub> and ''M'' and for all ''x''&nbsp;&gt;&nbsp;''x''<sub>0</sub>. To prove this, let ''x''<sub>0</sub>&nbsp;=&nbsp;1 and ''M''&nbsp;=&nbsp;13. Then, for all ''x''&nbsp;&gt;&nbsp;''x''<sub>0</sub>:
:<math>\begin{align}|6x^4 - 2x^3 + 5| &\le 6x^4 + |2x^3| + 5\\
                                      &\le 6x^4 + 2x^4 + 5x^4\\
                                      &= 13x^4\end{align}</math>
so
:<math> |6x^4 - 2x^3 + 5| \le 13 \,x^4 .</math>

== Usage ==
Big O notation has two main areas of application:
* in [[mathematics]], it is commonly used to describe [[Big O notation#Infinitesimal asymptotics|how closely a finite series approximates a given function]], especially in the case of a truncated [[Taylor series]] or [[asymptotic expansion]]
* in [[computer science]], it is useful in the [[Big O notation#Infinite asymptotics|analysis of algorithms]]

In both applications, the function ''g''(''x'') appearing within the ''O''(...) is typically chosen to be as simple as possible, omitting constant factors and lower order terms.

There are two formally close, but noticeably different, usages of this notation:
* [[Infinity|infinite]] asymptotics
* [[infinitesimal]] asymptotics.

This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument.

=== Infinite asymptotics ===
[[File:comparison computational complexity.svg|thumb|Graphs of functions commonly used in the analysis of algorithms, showing the number of operations ''N'' versus input size ''n'' for each function]]
Big O notation is useful when [[analysis of algorithms|analyzing algorithms]] for efficiency.  For example, the time (or the number of steps) it takes to complete a problem of size ''n'' might be found to be ''T''(''n'') = 4''n''<sup>2</sup> − 2''n'' + 2.
As ''n'' grows large, the ''n''<sup>2</sup> [[term (mathematics)|term]] will come to dominate, so that all other terms can be neglected—for instance when ''n'' = 500, the term 4''n''<sup>2</sup> is 1000 times as large as the 2''n'' term. Ignoring the latter would have negligible effect on the expression's value for most purposes.
Further, the [[coefficient]]s become irrelevant if we compare to any other [[Orders of approximation|order]] of expression, such as an expression containing a term ''n''<sup>3</sup> or ''n''<sup>4</sup>.  Even if ''T''(''n'') = 1,000,000''n''<sup>2</sup>, if ''U''(''n'') = ''n''<sup>3</sup>, the latter will always exceed the former once ''n'' grows larger than 1,000,000 (''T''(1,000,000) = 1,000,000<sup>3</sup>= ''U''(1,000,000)). Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm.
So the big O notation captures what remains: we write either
:<math>\ T(n)= O(n^2) </math>
or
:<math>T(n)\in O(n^2) </math>
and say that the algorithm has ''order of n<sup>2</sup>'' time complexity.
The sign "=" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is sometimes considered more accurate (see the "[[#Equals sign|Equals sign]]" discussion below) while the first is considered by some as an [[abuse of notation]].<ref name="clrs3" />

=== Infinitesimal asymptotics ===
Big O can also be used to describe the [[Taylor series#Approximation error and convergence|error term]] in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term.  Consider, for example, the [[Exponential function#Formal definition|exponential series]] and two expressions of it that are valid when ''x'' is small:
:<math>\begin{align}
e^x &=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dotsb &\text{for all } x\\
    &=1+x+\frac{x^2}{2}+O(x^3)                             &\text{as } x\to 0\\
    &=1+x+O(x^2)                                           &\text{as } x\to 0\\
\end{align}</math>
The second expression (the one with ''O''(''x''<sup>3</sup>)) means the absolute-value of the error ''e''<sup>''x''</sup> − (1 + ''x'' + ''x''<sup>2</sup>/2) is at most some constant times {{!}}''x''<sup>3</sup>{{!}} when ''x'' is close enough to&nbsp;0.

== Properties ==
If the function ''f'' can be written as a finite sum of other functions, then the fastest growing one determines the order of ''f''(''n''). For example,
:<math>f(n) = 9 \log n + 5 (\log n)^4 + 3n^2 + 2n^3 = O(n^3) \qquad\text{as } n\to\infty .</math>
In particular, if a function may be bounded by a polynomial in ''n'', then as ''n'' tends to ''infinity'', one may disregard ''lower-order'' terms of the polynomial.
The sets ''O''(''n''<sup>''c''</sup>) and ''O''(''c''<sup>''n''</sup>) are very different. If ''c'' is greater than one, then the latter grows much faster. A function that grows faster than ''n''<sup>''c''</sup> for any ''c''  is called ''superpolynomial''.  One that grows more slowly than any exponential function of the form ''c''<sup>''n''</sup> is called ''subexponential''. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms for [[integer factorization]] and the function ''n''<sup>log ''n''</sup>.

We may ignore any powers of ''n'' inside of the logarithms. The set ''O''(log ''n'') is exactly the same as ''O''(log(''n''<sup>''c''</sup>)). The logarithms differ only by a constant factor (since
log(''n''<sup>''c''</sup>) = ''c'' log ''n'') and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. On the other hand, exponentials with different bases are not of the same order. For example, 2<sup>''n''</sup> and 3<sup>''n''</sup> are not of the same order.

Changing units may or may not affect the order of the resulting algorithm. Changing units is equivalent to multiplying the appropriate variable by a constant wherever it appears. For example, if an algorithm runs in the order of ''n''<sup>2</sup>, replacing ''n'' by ''cn'' means the algorithm runs in the order of ''c''<sup>2</sup>''n''<sup>2</sup>, and the big O notation ignores the constant ''c''<sup>2</sup>. This can be written as ''c''<sup>2</sup>''n''<sup>2</sup> = O(''n''<sup>2</sup>). If, however, an algorithm runs in the order of 2<sup>''n''</sup>, replacing ''n'' with ''cn'' gives 2<sup>''cn''</sup> = (2<sup>''c''</sup>)<sup>n</sup>. This is not equivalent to 2<sup>''n''</sup> in general.
Changing variables may also affect the order of the resulting algorithm. For example, if an algorithm's run time is ''O''(''n'') when measured in terms of the number ''n'' of ''digits'' of an input number ''x'', then its run time is ''O''(log ''x'') when measured as a function of the input number ''x'' itself, because ''n'' = ''O''(log ''x'').

=== Product ===
:<math> f_1 = O(g_1) \text{ and } f_2 = O(g_2)\, \Rightarrow f_1  f_2 = O(g_1  g_2)</math>
:<math>f\cdot O(g) = O(f g)</math>

=== Sum ===
:<math> f_1 = O(g_1) \text{ and }
  f_2= O(g_2)\, \Rightarrow f_1 + f_2 = O(\max(g_1, g_2))</math>
This implies <math> f_1 = O(g) \text{ and } f_2 = O(g) \Rightarrow f_1+f_2 \in O(g) </math>, which means that <math>O(g)</math> is a [[convex cone]].

=== Multiplication by a constant ===
:Let ''k'' be constant. Then:
:<math>\ O(|k| g) = O(g)</math> if ''k'' is nonzero.
:<math>f= O(g) \Rightarrow kf = O(g). </math>

== Multiple variables ==
Big ''O'' (and little o, Ω, etc.) can also be used with multiple variables.
To define big ''O'' formally for multiple variables, suppose <math>f</math> and <math>g</math> are two functions defined on some subset of <math>\mathbb{R}^n</math>. We say
:<math>f(\vec{x})\text{ is }O(g(\vec{x}))\text{ as }\vec{x}\to\infty</math>
if and only if<ref>{{cite book|last1=Cormen|first1=Thomas|last2=Leiserson|first2=Charles|last3=Rivest|first3=Ronald|last4=Stein|first4=Clifford|title=Introduction to Algorithms|date=2009|publisher=MIT|page=53|edition=Third}}</ref>
:<math>\exists M\,\exists C>0\text{ such that for all }\vec{x} \text{ with } x_i \ge M \text{ for some }i, |f(\vec{x})| \le C |g(\vec{x})|.</math>
Equivalently, the condition that <math>x_i \geq M</math> for some <math>i</math> can be replaced with the condition that <math>\|\vec{x}\|_{\infty} \ge M</math>, where <math>\|\vec{x}\|_{\infty}</math> denotes the [[Chebyshev norm]]. For example, the statement
:<math>f(n,m) = n^2 + m^3 + O(n+m) \text{ as } n,m\to\infty</math>
asserts that there exist constants ''C'' and ''M'' such that
:<math>\forall \|(n, m)\|_{\infty} \geq M:  |g(n,m)| \le C|n+m|,</math>
where ''g''(''n'',''m'') is defined by
:<math>f(n,m) = n^2 + m^3 + g(n,m).</math>
This definition allows all of the coordinates of <math>\vec{x}</math> to increase to infinity. In particular, the statement
:<math>f(n,m) = O(n^m) \text{ as } n,m\to\infty</math>
(i.e., <math>\exists C\,\exists M\,\forall n\,\forall m\dots</math>) is quite different from
:<math>\forall m\colon f(n,m) = O(n^m) \text{ as } n\to\infty</math>
(i.e., <math>\forall m\,\exists C\,\exists M\,\forall n\dots</math>).

Under this definition, the subset on which a function is defined is significant when generalizing statements from the univariate setting to the multivariate setting. For example, if <math>f(n,m)=1</math> and <math>g(n,m)=n</math>, then <math>f(n,m)=O(g(n,m))</math> if we restrict <math>f</math> and <math>g</math> to <math>[1,\infty)^2</math>, but not if they are defined on <math>[0,\infty)^2</math>.

This is not the only generalization of big O to multivariate functions, and in practice, there is some inconsistency in the choice of definition.<ref>{{cite web|last1=Howell|first1=Rodney|title=On Asymptotic Notation with Multiple Variables|url=http://people.cis.ksu.edu/~rhowell/asymptotic.pdf|accessdate=2015-04-23}}</ref>

== Matters of notation ==

=== Equals sign ===
The statement "''f''(''x'') is ''O''(''g''(''x''))" as defined above is usually written as ''f''(''x'')&nbsp;=&nbsp;''O''(''g''(''x'')). Some consider this to be an [[abuse of notation]], since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As [[Nicolaas Govert de Bruijn|de Bruijn]] says, ''O''(''x'')&nbsp;=&nbsp;''O''(''x''<sup>2</sup>) is true but ''O''(''x''<sup>2</sup>)&nbsp;=&nbsp;''O''(''x'') is not.<ref>{{Cite book| author = [[N. G. de Bruijn]] | title=Asymptotic Methods in Analysis | place=Amsterdam |publisher=North-Holland | year=1958 | pages=5–7 | url=https://books.google.com/books?id=_tnwmvHmVwMC&pg=PA5&vq=%22The+trouble+is%22 | isbn=978-0-486-64221-5}}</ref> [[Donald Knuth|Knuth]] describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like ''n''&nbsp;=&nbsp;''n''<sup>2</sup> from the identities ''n''&nbsp;=&nbsp;''O''(''n''<sup>2</sup>) and ''n''<sup>2</sup>&nbsp;=&nbsp;''O''(''n''<sup>2</sup>)."<ref name="Concrete Mathematics">{{Cite book |last1=Graham |first1=Ronald |author1link=Ronald Graham |first2=Donald |last2=Knuth |author2link=Donald Knuth |last3=Patashnik |first3=Oren |author3link=Oren Patashnik |title=Concrete Mathematics |location=Reading, Massachusetts |publisher=Addison–Wesley |edition=2 |date=1994 |page=446 |url=https://books.google.com/books?id=pntQAAAAMAAJ |isbn=978-0-201-55802-9}}</ref>

For these reasons, it would be more precise to use [[set notation]] and write ''f''(''x'')&nbsp;∈&nbsp;''O''(''g''(''x'')), thinking of ''O''(''g''(''x'')) as the class of all functions ''h''(''x'') such that |''h''(''x'')|&nbsp;≤&nbsp;''C''|''g''(''x'')| for some constant ''C''.<ref name="Concrete Mathematics" /> However, the use of the equals sign is customary.  Knuth pointed out that "mathematicians customarily use the = sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle."<ref>{{Cite journal| author=Donald Knuth | title=Teach Calculus with Big O | date=June–July 1998 | journal=[[Notices of the American Mathematical Society]] | volume=45 | issue=6 | page=687 | url=http://www.ams.org/notices/199806/commentary.pdf}} ([http://www-cs-staff.stanford.edu/~knuth/ocalc.tex Unabridged version])</ref>

=== Other arithmetic operators ===
Big O notation can also be used in conjunction with other arithmetic operators in more complicated equations.   For example, ''h''(''x'') + ''O''(''f''(''x'')) denotes the collection of functions having the growth of ''h''(''x'') plus a part whose growth is limited to that of ''f''(''x''). Thus,
:<math>g(x) = h(x) + O(f(x))</math>
expresses the same as
:<math>g(x) - h(x) = O(f(x))\,.</math>

==== Example {{anchor|Example (Matters of notation)}} ====
Suppose an [[algorithm]] is being developed to operate on a set of ''n'' elements. Its developers are interested in finding a function ''T''(''n'')  that will express how long the algorithm will take to run (in some arbitrary measurement of time) in terms of the number of elements in the input set. The algorithm works by first calling a subroutine to sort the elements in the set and then perform its own operations. The sort has a known time complexity of ''O''(''n''<sup>2</sup>), and after the subroutine runs the algorithm must take an additional  55''n''<sup>3</sup>&nbsp;+&nbsp;2''n''&nbsp;+&nbsp;10 steps before it terminates.  Thus the overall time complexity of the algorithm can be expressed as ''T''(''n'') = 55''n''<sup>3</sup>&nbsp;+&nbsp;''O''(''n''<sup>2</sup>).
Here the terms 2''n''+10 are subsumed within the faster-growing ''O''(''n''<sup>2</sup>).  Again, this usage disregards some of the formal meaning of the "=" symbol, but it does allow one to use the big O notation as a kind of convenient placeholder.

=== Multiple uses ===
In more complicated usage, ''O''(...) can appear in different places in an equation, even several times on each side. For example, the following are true for <math>n\to\infty</math>
:<math>(n+1)^2 = n^2 + O(n)</math>
:<math>(n+O(n^{1/2}))(n + O(\log n))^2 = n^3 + O(n^{5/2})</math>
:<math>n^{O(1)} = O(e^n).</math>
The meaning of such statements is as follows: for ''any'' functions which satisfy each ''O''(...) on the left side, there are ''some'' functions satisfying each ''O''(...) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function ''f''(''n'') = ''O''(1), there is some function ''g''(''n'')  = ''O''(''e''<sup>''n''</sup>) such that ''n''<sup>''f''(''n'')</sup> = ''g''(''n'')." In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side. In this use the "=" is a formal symbol that unlike the usual use of "=" is not a [[symmetric relation]]. Thus for example ''n''<sup>''O''(1)</sup> = ''O''(''e''<sup>''n''</sup>) does not imply the false statement ''O''(''e''<sup>''n''</sup>) = ''n''<sup>''O''(1)</sup>

=== Typesetting ===
Big O consists of just an uppercase "O".  Unlike Greek-named Bachmann–Landau notations, it needs no special symbol.  Yet, commonly used calligraphic variants, like <math>\mathcal{O}</math>, are available, in [[LaTeX]] and derived typesetting systems.<ref>{{cite web |url=https://texblog.org/2014/06/24/big-o-and-related-notations-in-latex/ |title=Big O and related notations in LaTeX |date=24 June 2014 |author=Tom |work=texblog}}</ref>

== Orders of common functions ==
{{Further|Time complexity#Table of common time complexities}}

Here is a list of classes of functions that are commonly encountered when analyzing the running time of an algorithm.  In each case, ''c'' is a positive constant and ''n'' increases without bound. The slower-growing functions are generally listed first.

{| class="wikitable"
|-
!Notation !! Name !! Example
|-
|<math>O(1)</math> || [[Constant time|constant]] || Determining if a binary number is even or odd; Calculating <math>(-1)^n</math>; Using a constant-size [[lookup table]]
|-
|<math>O(\log \log n)</math> || double logarithmic || Number of comparisons spent finding an item using [[interpolation search]] in a sorted array of uniformly distributed values
|-
|<math>O(\log n)</math> || [[Logarithmic time|logarithmic]] || Finding an item in a sorted array with a [[Binary search algorithm|binary search]] or a balanced search [[Tree data structure|tree]] as well as all operations in a [[Binomial heap]]
|-
|<math>O((\log n)^c)</math><br /><math>\scriptstyle c>1</math> || [[Polylogarithmic time|polylogarithmic]] || Matrix chain ordering can be solved in polylogarithmic time on a [[parallel random-access machine]].
|-
|<math>O(n^c)</math><br /><math>\scriptstyle 0<c<1</math> || fractional power || Searching in a [[k-d tree]]
|-
|<math>O(n)</math> || [[linear time|linear]] || Finding an item in an unsorted list or in an unsorted array; adding two ''n''-bit integers by [[Ripple carry adder|ripple carry]]
|-
|<math>O(n\log^* n)</math> || n [[log-star]] n || Performing [[Polygon triangulation|triangulation]] of a simple polygon using [[Kirkpatrick–Seidel algorithm|Seidel's algorithm]], or the [[Proof of O(log*n) time complexity of union–find|union–find algorithm]]. Note that <math>\log^*(n) =
\begin{cases}
 0, & \text{if }n \leq 1 \\
 1 + \log^*(\log n), & \text{if }n>1
\end{cases}</math>
|-
|<math>O(n\log n)=O(\log n!)</math> || [[Linearithmic time|linearithmic]], loglinear, quasilinear, or "n log n" || Performing a [[fast Fourier transform]]; Fastest possible [[comparison sort]]; [[heapsort]] and [[merge sort]]
|-
|<math>O(n^2)</math> || [[quadratic time|quadratic]] || Multiplying two ''n''-digit numbers by a simple algorithm; simple sorting algorithms, such as [[bubble sort]], [[selection sort]] and [[insertion sort]]; (worst case) bound on some usually faster sorting algorithms such as [[quicksort]], [[Shellsort]], and [[tree sort]]
|-
|<math>O(n^c)</math> || [[polynomial time|polynomial]] or algebraic || [[Tree-adjoining grammar]] parsing; maximum [[Matching (graph theory)|matching]] for [[bipartite graph]]s; finding the [[determinant]] with [[LU decomposition]]
|-
|<math>L_n[\alpha,c] = e^{(c+o(1)) (\ln n)^\alpha (\ln \ln n)^{1-\alpha}}</math><br /><math>\scriptstyle 0 < \alpha < 1</math> || [[L-notation]] or [[sub-exponential time|sub-exponential]] || Factoring a number using the [[quadratic sieve]] or [[number field sieve]]
|-
|<math>O(c^n)</math><br /><math>\scriptstyle c>1</math> || [[exponential time|exponential]] || Finding the (exact) solution to the [[travelling salesman problem]] using [[dynamic programming]]; determining if two logical statements are equivalent using [[brute-force search]]
|-
|<math>O(n!)</math> || [[factorial]] || Solving the [[travelling salesman problem]] via brute-force search; generating all unrestricted permutations of a [[Partially ordered set|poset]]; finding the [[determinant]] with [[Laplace expansion]]; enumerating [[Bell number|all partitions of a set]]
|}
The statement <math>f(n)=O(n!)</math> is sometimes weakened to <math>f(n)=O\left(n^n\right)</math> to derive simpler formulas for asymptotic complexity.
For any <math>k>0</math> and <math>c>0</math>, <math>O(n^c(\log n)^k)</math> is a subset of <math>O(n^{c+\varepsilon })</math> for any <math> \varepsilon >0</math>, so may be considered as a polynomial with some bigger order.

== Related asymptotic notations ==
Big ''O'' is the most commonly used asymptotic notation for comparing functions.{{cn|date=December 2017}} Together with some  other related notations it forms the family of Bachmann–Landau notations.

===Little-o notation=== <!-- [[Little-o notation]] redirects here -->
{{redirect|Little o|the baseball player|Omar Vizquel}}
Intuitively, the assertion "{{math|''f''(''x'')}} is {{math|''o''(''g''(''x''))}}" (read "{{math|''f''(''x'')}} is little-o of {{math|''g''(''x'')}}") means that {{math|''g''(''x'')}} grows much faster than {{math|''f''(''x'')}}. Let as before ''f'' be a real or complex valued function and ''g'' a real valued function, both defined on some unbounded subset of the [[real number|real positive numbers]], such that ''g(x)'' is strictly positive for all large enough values of ''x''. One writes
:<math>f(x)=o(g(x))\text{ as }x\to\infty</math>
if for every positive constant {{math|''ε''}} there exists a constant {{math|''N''}} such that
:<math>|f(x)|\leq\varepsilon g(x)\qquad\text{ for all }x\geq N~.</math><ref name=Landausmallo>{{cite book |first=Edmund |last=Landau |authorlink=Edmund Landau |title=Handbuch der Lehre von der Verteilung der Primzahlen |publisher=B. G. Teubner |date=1909 |location=Leipzig |trans-title=Handbook on the theory of the distribution of the primes |language=de |page=61 | url=https://archive.org/stream/handbuchderlehre01landuoft#page/61/mode/2up}}</ref>
For example, one has
: <math>2x  = o(x^2) </math> and <math>1/x = o(1).</math>

The difference between the earlier [[#Formal definition|definition]] for the big-O notation and the present definition of little-o, is that while the former has to be true for ''at least one'' constant ''M'', the latter must hold for ''every'' positive constant {{math|''ε''}}, however small.<ref name="Introduction to Algorithms">Thomas H. Cormen et al., 2001, [http://highered.mcgraw-hill.com/sites/0070131511/ Introduction to Algorithms, Second Edition]{{page needed|date=February 2016}}</ref> In this way, little-o notation makes a ''stronger statement'' than the corresponding big-O notation: every function that is little-o of ''g'' is also big-O of ''g'', but not every function that is big-O of ''g'' is also little-o of ''g''.  For example, <math>2x^2 = O(x^2) </math> but <math>2x^2 \neq o(x^2). </math>

As ''g''(''x'') is nonzero, or at least becomes nonzero beyond a certain point, the relation {{math|''f''(''x'')&nbsp;{{=}}&nbsp;''o''(''g''(''x''))}} is equivalent to
:<math>\lim_{x \to \infty}\frac{f(x)}{g(x)}=0</math> (and this is in fact how Landau<ref name=Landausmallo /> originally defined the little-o notation).

Little-o respects a number of arithmetic operations.  For example,
: if {{mvar|c}} is a nonzero constant and <math>f = o(g)</math> then <math>c \cdot f = o(g)</math>, and
: if <math>f = o(F)</math> and <math>g = o(G)</math> then <math> f \cdot g = o(F \cdot G).</math>
It also satisfies a transitivity relation:
: if <math> f = o(g)</math> and <math> g = o(h)</math> then <math> f = o(h).</math>

=== Big Omega notation ===

There are two very widespread and incompatible definitions of the statement

:<math>f(x)=\Omega(g(x))\ (x\rightarrow a),</math>

where ''a'' is some real number, ∞,  or −∞, where ''f'' and ''g'' are real functions defined in a neighbourhood of ''a'', and where ''g'' is positive in this neighbourhood.

The first one (chronologically) is used in [[analytic number theory]], and the other one in [[computational complexity theory]]. When the two subjects meet, this situation is bound to generate confusion.

==== The Hardy–Littlewood definition ====

In 1914 [[Godfrey Harold Hardy]] and [[John Edensor Littlewood]] introduced the new symbol <math>\Omega</math>,<ref name="HL">{{cite journal|last1=Hardy|first1=G. H.|last2=Littlewood|first2=J. E.|title=Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic ϑ-functions|journal=Acta Mathematica|date=1914|volume=37|page=225|doi=10.1007/BF02401834|url=http://projecteuclid.org/download/pdf_1/euclid.acta/1485887376}}</ref> which is defined as follows:

:<math>f(x)=\Omega(g(x))\ (x\rightarrow\infty)\;\Leftrightarrow\;\limsup_{x \to \infty} \left|\frac{f(x)}{g(x)}\right|> 0.</math>

Thus <math>f(x)=\Omega(g(x))</math> is the negation of <math>f(x)=o(g(x))</math>.

In 1916 the same authors introduced the two new symbols <math>\Omega_R</math> and <math>\Omega_L</math>, defined as:<ref name="HL2">G. H. Hardy and J. E. Littlewood, « Contribution to the theory of the Riemann zeta-function and the theory of the distribution of primes », ''[[Acta Mathematica]]'', vol. 41, 1916.</ref>

:<math>f(x)=\Omega_R(g(x))\ (x\rightarrow\infty)\;\Leftrightarrow\;\limsup_{x \to \infty} \frac{f(x)}{g(x)}> 0</math>;

:<math>f(x)=\Omega_L(g(x))\ (x\rightarrow\infty)\;\Leftrightarrow\;\liminf_{x \to \infty} \frac{f(x)}{g(x)}< 0. </math>

These symbols were used by [[Edmund Landau]], with the same meanings, in 1924.<ref name="landau">E. Landau, "Über die Anzahl der Gitterpunkte in gewissen Bereichen. IV." Nachr. Gesell. Wiss. Gött. Math-phys. Kl. 1924, 137–150.</ref>  After Landau, the notations were never used again exactly thus; <math>\Omega_R</math> became <math>\Omega_+</math> and <math>\Omega_L</math> became <math>\Omega_-</math>.{{cn|date=December 2018}}

These three symbols <math>\Omega, \Omega_+, \Omega_-</math>, as well as <math>f(x)=\Omega_\pm(g(x))</math> (meaning that <math>f(x)=\Omega_+(g(x))</math> and <math>f(x)=\Omega_-(g(x))</math> are both satisfied), are now currently used in [[analytic number theory]].<ref name=Ivic>Aleksandar Ivić. The Riemann zeta-function, chapter 9. John Wiley & Sons 1985.</ref><ref>Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Chapter I.5. American Mathematical Society, Providence RI, 2015.</ref>

===== Simple examples =====

We have

:<math>\sin x=\Omega(1)\ (x\rightarrow\infty),</math>

and more precisely

:<math>\sin x=\Omega_\pm(1)\ (x\rightarrow\infty).</math>

We have

:<math>\sin x+1=\Omega(1)\ (x\rightarrow\infty),</math>

and more precisely

:<math>\sin x+1=\Omega_+(1)\ (x\rightarrow\infty);</math>

however

:<math>\sin x+1\not=\Omega_-(1)\ (x\rightarrow\infty).</math>

==== The Knuth definition ====

In 1976 [[Donald Knuth]] published a paper to justify his use of the <math>\Omega</math>-symbol to describe a stronger property. Knuth wrote: "For all the applications I have seen so far in computer science, a stronger requirement […] is much more appropriate". He defined

:<math>f(x)=\Omega(g(x))\Leftrightarrow g(x)=O(f(x))</math>

with the comment: "Although I have changed Hardy and Littlewood's definition of <math>\Omega</math>, I feel justified in doing so because their definition is by no means in wide use, and because there are other ways to say what they want to say in the comparatively rare cases when their definition applies."<ref name="knuth">Donald Knuth. "[http://www.phil.uu.nl/datastructuren/10-11/knuth_big_omicron.pdf Big Omicron and big Omega and big Theta]", SIGACT News, Apr.-June 1976, 18–24.</ref>

=== Family of Bachmann–Landau notations ===
{| class="wikitable"
|-
! Notation
! Name<ref name="knuth" />
! Description
! Formal Definition
! Limit Definition<ref name=Balcázar>{{cite journal|last1=Balcázar|first1=José L.|last2=Gabarró|first2=Joaquim|title=Nonuniform complexity classes specified by lower and upper bounds|journal=RAIRO – Theoretical Informatics and Applications – Informatique Théorique et Applications|volume=23|issue=2|page=180|url=http://archive.numdam.org/article/ITA_1989__23_2_177_0.pdf|accessdate=14 March 2017|language=en|issn=0988-3754}}</ref><ref name=Cucker>{{cite book|last1=Cucker|first1=Felipe|last2=Bürgisser|first2=Peter|title=Condition: The Geometry of Numerical Algorithms|date=2013|publisher=Springer|location=Berlin, Heidelberg|isbn=978-3-642-38896-5|pages=467–468|chapter=A.1  Big Oh, Little Oh, and Other Comparisons|doi=10.1007/978-3-642-38896-5}}</ref><ref name=Wild>{{cite journal | first1=Paul | last1=Vitányi | authorlink1=Paul Vitanyi | first2=Lambert | last2=Meertens | authorlink2=Lambert Meertens | title=Big Omega versus the wild functions | journal=ACM SIGACT News | volume=16 | issue=4 | date=April 1985 | pages=56–59 | doi=10.1145/382242.382835 | url=http://www.kestrel.edu/home/people/meertens/publications/papers/Big_Omega_contra_the_wild_functions.pdf| citeseerx=10.1.1.694.3072 }}</ref><ref name="knuth" /><ref name="HL" />
|-
| <math>f(n) = o(g(n))</math>
| Small O; Small Oh
| {{mvar|f}} is dominated by {{mvar|g}} asymptotically
| <math>\forall k>0 \; \exists n_0 \; \forall n>n_0 \; |f(n)| < k\cdot g(n)</math>
| <math>\lim_{n \to \infty} \frac{f(n)}{g(n)} = 0</math>
|-
| <math>f(n) = O(g(n))</math>
| Big O; Big Oh; Big Omicron
| <math>|f|</math> is bounded above by {{mvar|g}} (up to constant factor) asymptotically
| <math>\exists k>0 \; \exists n_0 \; \forall n>n_0 \; |f(n)| \leq k\cdot g(n)</math>
| <math>\limsup_{n \to \infty} \frac{\left|f(n)\right|}{g(n)} < \infty</math>
|-
| <math>f(n) = \Theta(g(n))</math>
| Big Theta
| {{mvar|f}} is bounded both above and below by {{mvar|g}} asymptotically
| <math>\exists k_1>0 \; \exists k_2>0 \; \exists n_0 \; \forall n>n_0</math><math>k_1\cdot g(n) \leq f(n) \leq k_2\cdot g(n)</math>
| <math>f(n) = O(g(n))</math> and <math>f(n) = \Omega(g(n))</math> (Knuth version)
|-
| <math>f(n)\sim g(n)</math>
| On the order of
| {{mvar|f}} is equal to {{mvar|g}} asymptotically
| <math>\forall \varepsilon>0\;\exists n_0\;\forall n>n_0\;\left|{f(n) \over g(n)}-1\right|<\varepsilon</math>
| <math>\lim_{n \to \infty} {f(n) \over g(n)} = 1</math>
|-
| <math>f(n) = \Omega(g(n))</math>
| Big Omega in number theory (Hardy–Littlewood)
| <math>|f|</math> is not dominated by {{mvar|g}} asymptotically
| <math>\exists k>0 \; \forall n_0 \; \exists n>n_0 \; |f(n)| \geq k\cdot g(n)</math>
| <math>\limsup_{n \to \infty} \left|\frac{f(n)}{g(n)}\right| > 0 </math>
|-
| <math>f(n) = \Omega(g(n))</math>
| Big Omega in complexity theory (Knuth)
| {{mvar|f}} is bounded below by {{mvar|g}} asymptotically
| <math>\exists k>0 \; \exists n_0 \; \forall n>n_0 \; f(n) \geq k\cdot g(n)</math>
| <math>\liminf_{n \to \infty} \frac{f(n)}{g(n)} > 0 </math>
|-
| <math>f(n) = \omega(g(n))</math>
| Small Omega
| {{mvar|f}} dominates {{mvar|g}} asymptotically
| <math>\forall k>0 \; \exists n_0 \; \forall n>n_0 \  |f(n)| > k\cdot |g(n)|</math>
| <math>\lim_{n \to \infty} \left|\frac{f(n)}{g(n)}\right| = \infty</math>
|}

The limit definitions assume <math>g(n) \neq 0</math> for sufficiently large {{mvar|n}}. The table is (partly) sorted from smallest to largest, in the sense that o, O, Θ, ∼, (Knuth's version of) Ω, ω on functions correspond to <, ≤, ≈, =, ≥, > on the real line<ref name=Wild /> (the Hardy-Littlewood version of Ω, however, doesn't correspond to any such description).

Computer science uses the big ''O'', big Theta Θ, little ''o'', little omega &omega; and Knuth's big Omega Ω notations.<ref>{{Introduction to Algorithms|edition=2|pages=41–50}}</ref> Analytic number theory often uses the big ''O'', small ''o'', Hardy–Littlewood's big Omega Ω (with or without the +, - or ± subscripts) and <math>\sim</math> notations.<ref name=Ivic /> The small omega ω notation is not used as often in analysis.<ref>for example it is omitted in: {{cite web|last1=Hildebrand|first1=A.J.|title=Asymptotic Notations|url=http://www.math.uiuc.edu/~ajh/595ama/ama-ch2.pdf|website=Asymptotic Methods in Analysis|publisher=Math 595, Fall 2009|accessdate=14 March 2017}}</ref>

=== Use in computer science ===
{{details|Analysis of algorithms}}
Informally, especially in computer science, the big ''O'' notation often can be used somewhat differently to describe an asymptotic [[Upper and lower bounds#Tight bounds|tight]] bound where using big Theta Θ notation might be more factually appropriate in a given context.{{Citation needed|reason=Wording (weasel-words) suggest a primarily opinion-based posit.|date=May 2015}} For example, when considering a function ''T''(''n'') = 73''n''<sup>3</sup> + 22''n''<sup>2</sup> + 58, all of the following are generally acceptable, but tighter bounds (such as numbers 2 and 3 below) are usually strongly preferred over looser bounds (such as number 1 below).
#''T''(''n'')&nbsp;=&nbsp;''O''(''n''<sup>100</sup>)
#''T''(''n'')&nbsp;=&nbsp;''O''(''n''<sup>3</sup>)
#''T''(''n'')&nbsp;=&nbsp;Θ(''n''<sup>3</sup>)
The equivalent English statements are respectively:
#''T''(''n'') grows asymptotically no faster than ''n''<sup>100</sup>
#''T''(''n'') grows asymptotically no faster than ''n''<sup>3</sup>
#''T''(''n'') grows asymptotically as fast as ''n''<sup>3</sup>.
So while all three statements are true, progressively more information is contained in each. In some fields, however, the big O notation (number 2 in the lists above) would be used more commonly than the big Theta notation (bullets number 3 in the lists above). For example, if ''T''(''n'') represents the running time of a newly developed algorithm for input size ''n'', the inventors and users of the algorithm might be more inclined to put an upper asymptotic bound on how long it will take to run without making an explicit statement about the lower asymptotic bound.

=== Other notation ===

In their book ''[[Introduction to Algorithms]]'', [[Thomas H. Cormen|Cormen]], [[Charles E. Leiserson|Leiserson]], [[Ronald L. Rivest|Rivest]] and [[Clifford Stein|Stein]] consider the set of functions ''f'' which satisfy

:<math> f(n) = O(g(n))\  (n\rightarrow\infty).</math>

In a correct notation this set can, for instance, be called ''O''(''g''), where

:<math>O(g) = \{ f :</math> there exist positive constants ''c'' and <math>n_0</math> such that <math>0 \le f(n) \le cg(n)</math> for all <math>n \ge n_0 \}</math>.<ref>{{cite book |  isbn=978-0-262-53305-8 | author1=Thomas H. Cormen |author2= Charles E. Leiserson |author3= Ronald L. Rivest | title=Introduction to Algorithms | location=Cambridge/MA | publisher=MIT Press | edition=3rd | year=2009 | page=47 | quote=When we have only an asymptotic upper bound, we use O-notation. For a given function ''g''(''n''), we denote by ''O''(''g''(''n'')) (pronounced “big-oh of ''g'' of ''n''” or sometimes just “oh of ''g'' of ''n''”) the set of functions ''O''(''g''(''n'')) = { ''f''(''n'') : there exist positive constants ''c'' and ''n''<sub>0</sub> such that 0 ≤ ''f''(''n'') ≤ ''cg''(''n'') for all ''n'' ≥ ''n''<sub>0</sub>} }}</ref>

The authors state that the use of equality operator (=) to denote set membership rather than the set membership operator (∈) is an abuse of notation, but that doing so has advantages.<ref name="clrs3">{{cite book |  isbn=978-0-262-53305-8 | author1=Thomas H. Cormen |author2= Charles E. Leiserson |author3= Ronald L. Rivest | title=Introduction to Algorithms | location=Cambridge/MA | publisher=MIT Press | edition=3rd | year=2009 |page=45|quote=Because ''θ''(''g''(''n'')) is a set, we could write "''f''(''n'') ∈ ''θ''(''g''(''n''))" to indicate that ''f''(''n'') is a member of ''θ''(''g''(''n'')). Instead, we will usually write ''f''(''n'') = ''θ''(''g''(''n'')) to express the same notion. You might be confused because we abuse equality in this way, but we shall see later in this section that doing so has its advantages.}}</ref> Inside an equation or inequality, the use of asymptotic notation stands for an anonymous function in the set ''O''(''g''), which eliminates lower-order terms, and helps to reduce inessential clutter in equations, for example:<ref>{{cite book |  isbn=978-0-262-53305-8 | author1=Thomas H. Cormen |author2= Charles E. Leiserson |author3= Ronald L. Rivest | title=Introduction to Algorithms | location=Cambridge/MA | publisher=MIT Press | edition=3rd | year=2009 | page=49 | quote= When the asymptotic notation stands alone (that is, not within a larger formula) on the right-hand side of an equation (or inequality), as in n = O(n²), we have already defined the equal sign to mean set membership: n ∈ O(n²). In general, however, when asymptotic notation appears in a formula, we interpret it as standing for some anonymous function that we do not care to name. For example, the formula 2''n''<sup>2</sup> + 3''n'' + 1 = 2''n''<sup>2</sup> + ''θ''(''n'') means that 2''n''<sup>2</sup> + 3''n'' + 1 = 2''n''<sup>2</sup> + ''f''(''n''), where ''f''(''n'') is some function in the set ''θ''(''n''). In this case, we let ''f''(''n'') = 3''n'' + 1, which is indeed in ''θ''(''n''). Using asymptotic notation in this manner can help eliminate inessential detail and clutter in an equation.}}</ref>

:<math> 2n^2 + 3n + 1=2n^2 + O(n).</math>

=== Extensions to the Bachmann–Landau notations ===
Another notation sometimes used in computer science is Õ (read ''soft-O''): ''f''(''n'')&nbsp;=&nbsp;''Õ''(''g''(''n'')) is shorthand
for ''f''(''n'')&nbsp;=&nbsp;''O''(''g''(''n'')&nbsp;log<sup>''k''</sup>&nbsp;''g''(''n'')) for some ''k''. Essentially, it is big O notation, ignoring logarithmic factors because the growth-rate effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s). This notation is often used to obviate the "nitpicking" within growth-rates that are stated as too tightly bounded for the matters at hand (since log<sup>''k''</sup>&nbsp;''n'' is always ''o''(''n''<sup>ε</sup>) for any constant ''k'' and any ε&nbsp;>&nbsp;0).

Also the [[L-notation|L notation]], defined as
:<math>L_n[\alpha,c]=e^{(c+o(1))(\ln n)^\alpha(\ln\ln n)^{1-\alpha}}</math>
is convenient for functions that are between [[Time complexity#Polynomial time|polynomial]] and [[Time complexity#Exponential time|exponential]] in terms of <math>\ln n</math>.

== Generalizations and related usages ==
The generalization to functions taking values in any [[normed vector space]] is straightforward (replacing absolute values by norms), where ''f'' and ''g'' need not take their values in the same space. A generalization to functions ''g'' taking values in any [[topological group]] is also possible{{Citation needed|date=May 2017}}.
The "limiting process" ''x''&nbsp;→&nbsp;''x''<sub>o</sub> can also be generalized by introducing an arbitrary [[filter base]], i.e. to directed [[net (mathematics)|nets]] ''f'' and&nbsp;''g''.
The ''o'' notation can be used to define [[derivative]]s and [[differentiability]] in quite general spaces, and also (asymptotical) equivalence of functions,
:<math> f\sim g \iff (f-g) \in o(g) </math>
which is an [[equivalence relation]] and a more restrictive notion than the relationship "''f'' is Θ(''g'')" from above. (It reduces to lim ''f'' / ''g'' = 1 if ''f'' and ''g'' are positive real valued functions.)  For example, 2''x'' is Θ(''x''), but 2''x''&nbsp;−&nbsp;''x'' is not ''o''(''x'').

== History (Bachmann–Landau, Hardy, and Vinogradov notations) ==

The symbol O was first introduced by number theorist [[Paul Bachmann]] in 1894, in the second volume of his book ''Analytische Zahlentheorie'' ("[[analytic number theory]]"), the first volume of which (not yet containing big O notation) was published in 1892.<ref name=Bachmann>{{cite book |first=Paul |last=Bachmann |authorlink=Paul Bachmann |title=Analytische Zahlentheorie |trans-title=Analytic Number Theory |language=de |volume=2 |location=Leipzig |publisher=Teubner |date=1894 |url=https://archive.org/stream/dieanalytischeza00bachuoft#page/402/mode/2up}}</ref> The number theorist [[Edmund Landau]] adopted it, and was thus inspired to introduce in 1909 the notation o;<ref name=Landau>{{cite book |first=Edmund |last=Landau |authorlink=Edmund Landau |title=Handbuch der Lehre von der Verteilung der Primzahlen |publisher=B. G. Teubner |date=1909 |location=Leipzig |trans-title=Handbook on the theory of the distribution of the primes |language=de |page=883 | url=https://archive.org/details/handbuchderlehre01landuoft}}</ref> hence both are now called Landau symbols. These notations were used in applied mathematics during the 1950s for asymptotic analysis.<ref>{{cite book |title=Asymptotic Expansions |last=Erdelyi |first=A. |year=1956 |isbn=978-0-486-60318-6}}</ref>
The symbol <math>\Omega</math> (in the sense "is not an ''o'' of") was introduced in 1914 by Hardy and Littlewood.<ref name="HL" /> Hardy and Littlewood also introduced in 1918 the symbols <math>\Omega_R</math> ("right") and <math>\Omega_L</math> ("left"),<ref name="HL2" />  precursors of the modern symbols <math>\Omega_+</math> ("is not smaller than a small o of") and <math>\Omega_-</math> ("is not larger than a small o of"). Thus the Omega symbols (with their original meanings) are sometimes also referred to as "Landau symbols". This notation <math>\Omega</math> became commonly used in number theory at least since the 1950s.<ref name="titchmarsh">E. C. Titchmarsh, The Theory of the Riemann Zeta-Function (Oxford; Clarendon Press, 1951)</ref>
In the 1970s the big O was popularized in computer science by [[Donald Knuth]], who introduced the related Theta notation, and proposed a different definition for the Omega notation.<ref name="knuth" />

Landau never used the big Theta and small omega symbols.

Hardy's symbols were (in terms of the modern ''O'' notation)
:<math> f \preccurlyeq g\iff f \in O(g) </math> &nbsp; and &nbsp; <math> f\prec g\iff f\in o(g); </math>

(Hardy however never defined or used the notation <math>\prec\!\!\prec</math>, nor <math>\ll</math>, as it has been sometimes reported).
Hardy introduced the symbols <math>\preccurlyeq </math> and <math>\prec </math> (as well as some other symbols) in his 1910 tract "Orders of Infinity", and made use of them only in three papers (1910–1913). In his nearly 400 remaining papers and books he consistently used the Landau symbols O and o.

Hardy's notation is not used anymore. On the other hand, in the 1930s,<ref>See for instance "A new estimate for ''G''(''n'') in Waring's problem" (Russian). Doklady Akademii Nauk SSSR 5, No 5-6 (1934), 249–253. Translated in English in: Selected works / Ivan Matveevič Vinogradov; prepared by the Steklov Mathematical Institute of the Academy of Sciences of the USSR on the occasion of his 90th birthday. Springer-Verlag, 1985.</ref> the Russian number theorist  [[Ivan Matveyevich Vinogradov]]  introduced his notation	<math>\ll</math>, which  has been increasingly used in number theory instead of  the <math>O</math> notation. We have
:<math> f\ll g \iff f \in O(g), </math>
and frequently both notations are used in the same paper.

The big-O originally stands for "order of" ("Ordnung", Bachmann 1894), and is thus a Latin letter. Neither Bachmann nor Landau ever call it "Omicron". The symbol was much later on (1976) viewed by Knuth as a capital [[omicron]],<ref name="knuth" /> probably in reference to his definition of the symbol [[Omega]]. The digit [[0 (number)|zero]] should not be used.

== See also ==
* [[Asymptotic expansion]]: Approximation of functions generalizing Taylor's formula
* [[Asymptotically optimal algorithm]]: A phrase frequently used to describe an algorithm that has an upper bound asymptotically within a constant of a lower bound for the problem
* [[Big O in probability notation]]: ''O<sub>p</sub>'',''o<sub>p</sub>''
* [[Limit superior and limit inferior]]: An explanation of some of the limit notation used in this article
* [[Nachbin's theorem]]: A precise method of bounding [[complex analytic]] functions so that the domain of convergence of [[integral transform]]s can be stated
* [[Orders of approximation]]
* [[Computational complexity of mathematical operations]]

== References and notes ==
{{Reflist|30em}}

== Further reading ==
* {{cite book | first=G. H. | last=Hardy | authorlink=G. H. Hardy | title=Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-Reymond | date=1910 | url=https://archive.org/details/ordersofinfinity00harduoft | publisher=[[Cambridge University Press]]}}
* {{cite book | first=Donald | last=Knuth | authorlink=Donald Knuth | title=Fundamental Algorithms | volume=1 | series=The Art of Computer Programming | edition=3rd | publisher=Addison–Wesley | date=1997 | isbn=978-0-201-89683-1 | section=1.2.11: Asymptotic Representations}}
* {{cite book | first1=Thomas H. | last1=Cormen | authorlink1=Thomas H. Cormen | first2=Charles E. | last2=Leiserson | authorlink2=Charles E. Leiserson | first3=Ronald L. | last3=Rivest | authorlink3=Ronald L. Rivest | first4=Clifford | last4=Stein | authorlink4=Clifford Stein | title=Introduction to Algorithms | edition=2nd | publisher=MIT Press and McGraw–Hill | date=2001 | isbn=978-0-262-03293-3 | section=3.1: Asymptotic notation| title-link=Introduction to Algorithms }}
* {{cite book | first=Michael | last=Sipser | authorlink=Michael Sipser | year = 1997 | title = Introduction to the Theory of Computation | publisher = PWS Publishing | isbn = 978-0-534-94728-6 | pages=226–228}}
* {{cite conference | first1=Jeremy | last1=Avigad | first2=Kevin | last2=Donnelly | url=http://www.andrew.cmu.edu/~avigad/Papers/bigo.pdf | format=PDF | title=Formalizing O notation in Isabelle/HOL | doi=10.1007/978-3-540-25984-8_27 | conference=International Joint Conference on Automated Reasoning | date=2004}}
* {{cite web | first=Paul E. | last=Black | url=https://xlinux.nist.gov/dads/HTML/bigOnotation.html | title=big-O notation | work=Dictionary of Algorithms and Data Structures | editor-first=Paul E. | editor-last=Black | publisher=U.S. National Institute of Standards and Technology | date=11 March 2005 | access-date=December 16, 2006}}
* {{cite web | first=Paul E. | last=Black | url=https://xlinux.nist.gov/dads/HTML/littleOnotation.html | title=little-o notation | work=Dictionary of Algorithms and Data Structures | editor-first=Paul E. | editor-last=Black | publisher=U.S. National Institute of Standards and Technology | date=17 December 2004 | access-date=December 16, 2006}}
* {{cite web | first=Paul E. | last=Black | url=https://xlinux.nist.gov/dads/HTML/omegaCapital.html | title=Ω | work=Dictionary of Algorithms and Data Structures | editor-first=Paul E. | editor-last=Black | publisher=U.S. National Institute of Standards and Technology | date=17 December 2004 | access-date=December 16, 2006}}
* {{cite web | first=Paul E. | last=Black | url=https://xlinux.nist.gov/dads/HTML/omega.html | title=ω | work=Dictionary of Algorithms and Data Structures | editor-first=Paul E. | editor-last=Black | publisher=U.S. National Institute of Standards and Technology | date=17 December 2004 | access-date=December 16, 2006}}
* {{cite web | first=Paul E. | last=Black | url=https://xlinux.nist.gov/dads/HTML/theta.html | title=Θ | work=Dictionary of Algorithms and Data Structures | editor-first=Paul E. | editor-last=Black | publisher=U.S. National Institute of Standards and Technology | date=17 December 2004 | access-date=December 16, 2006}}

== External links ==
{{Wikibooks|Data Structures|Asymptotic Notation#Big-O Notation|Big-O Notation}}
* [http://oeis.org/wiki/Growth_of_sequences Growth of sequences — OEIS (Online Encyclopedia of Integer Sequences) Wiki]
* [https://classes.soe.ucsc.edu/classes/cmps102/Spring04/TantaloAsymp.pdf Introduction to Asymptotic Notations]
* [http://mathworld.wolfram.com/LandauSymbols.html Landau Symbols]
* [http://www.perlmonks.org/?node_id=573138 Big-O Notation – What is it good for]
* [https://stackoverflow.com/questions/487258/what-is-a-plain-english-explanation-of-big-o-notation/50288253#50288253 Big O Notation explained in plain english]
*[https://autarkaw.org/2013/01/30/making-sense-of-the-big-oh/ An example of Big O in accuracy of central divided difference scheme for first derivative]

[[Category:Mathematical notation]]
[[Category:Asymptotic analysis]]
[[Category:Analysis of algorithms]]