Example of Big O notation: ${\displaystyle {\color {red}f(x)}=O{\color {blue}(g(x))}}$ as ${\displaystyle x\to \infty }$ since there exists ${\displaystyle M>0}$ (e.g., ${\displaystyle M=1}$) and ${\displaystyle x_{0}}$ (e.g.,${\displaystyle x_{0}=5}$) such that ${\displaystyle 0\leq {\color {red}f(x)}\leq M{\color {blue}g(x)}}$ whenever ${\displaystyle x\geq x_{0}}$.

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Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann,^[1] Edmund Landau,^[2] and others, collectively called Bachmann–Landau notation or asymptotic 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.^[3] In analytic number theory, big O notation is often used to express a bound on the difference between an 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 is also used in many other fields to provide similar estimates.

Big O notation characterizes functions according to their growth rates: 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 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 o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates.

Formal definition

Let ${\displaystyle f}$, the function to be estimated, be a real or complex valued function and let ${\displaystyle g}$, the comparison function, be a real valued function. Let both functions be defined on some unbounded subset of the positive real numbers, and ${\displaystyle g(x)}$ be strictly positive for all large enough values of ${\displaystyle x}$.^[4] One writes

$${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}x\to \infty }$$

and it is read "${\displaystyle f(x)}$ is big O of ${\displaystyle g(x)}$" if the absolute value of ${\displaystyle f(x)}$ is at most a positive constant multiple of ${\displaystyle g(x)}$ for all sufficiently large values of ${\displaystyle x}$. That is, ${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}$ if there exists a positive real number ${\displaystyle M}$ and a real number ${\displaystyle x_{0}}$ such that

$${\displaystyle |f(x)|\leq Mg(x)\quad {\text{ for all }}x\geq x_{0}.}$$

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

$${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}.}$$

The notation can also be used to describe the behavior of ${\displaystyle f}$ near some real number ${\displaystyle a}$ (often, ${\displaystyle a=0}$): we say

$${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}x\to a}$$

if there exist positive numbers ${\displaystyle \delta }$ and ${\displaystyle M}$ such that for all defined ${\displaystyle x}$ with ${\displaystyle 0<|x-a|<\delta }$,

$${\displaystyle |f(x)|\leq Mg(x).}$$

As ${\displaystyle g(x)}$ is chosen to be strictly positive for such values of ${\displaystyle x}$, both of these definitions can be unified using the limit superior:

$${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}x\to a}$$

if

$${\displaystyle \limsup _{x\to a}{\frac {\left|f(x)\right|}{g(x)}}<\infty .}$$

And in both of these definitions the limit point ${\displaystyle a}$ (whether ${\displaystyle \infty }$ or not) is a cluster point of the domains of ${\displaystyle f}$ and ${\displaystyle g}$, i. e., in every neighbourhood of ${\displaystyle a}$ there have to be infinitely many points in common. Moreover, as pointed out in the article about the limit inferior and limit superior, the ${\displaystyle \textstyle \limsup _{x\to a}}$ (at least on the extended real number line) always exists.

In computer science, a slightly more restrictive definition is common: ${\displaystyle f}$ and ${\displaystyle g}$ are both required to be functions from some unbounded subset of the positive integers to the nonnegative real numbers; then ${\displaystyle f(x)=O{\bigl (}g(x){\bigr )}}$ if there exist positive integer numbers ${\displaystyle M}$ and ${\displaystyle n_{0}}$ such that ${\displaystyle f(n)\leq Mg(n)}$ for all ${\displaystyle n\geq n_{0}}$.^[5]

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 (factors in the product that do not depend on x) can be omitted.

For example, let f(x) = 6x⁴ − 2x³ + 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: 6x⁴, −2x³, 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 6x⁴. Now one may apply the second rule: 6x⁴ is a product of 6 and x⁴ in which the first factor does not depend on x. Omitting this factor results in the simplified form x⁴. Thus, we say that f(x) is a "big O" of x⁴. Mathematically, we can write f(x) = O(x⁴). One may confirm this calculation using the formal definition: let f(x) = 6x⁴ − 2x³ + 5 and g(x) = x⁴. Applying the formal definition from above, the statement that f(x) = O(x⁴) is equivalent to its expansion,

$${\displaystyle |f(x)|\leq Mx^{4}}$$

for some suitable choice of a real number x₀ and a positive real number M and for all x > x₀. To prove this, let x₀ = 1 and M = 13. Then, for all x > x₀:

$${\displaystyle {\begin{aligned}|6x^{4}-2x^{3}+5|&\leq 6x^{4}+|2x^{3}|+5\\&\leq 6x^{4}+2x^{4}+5x^{4}\\&=13x^{4}\end{aligned}}}$$

so

$${\displaystyle |6x^{4}-2x^{3}+5|\leq 13x^{4}.}$$

Usage

Big O notation has two main areas of application:

• In mathematics, it is commonly used to describe 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 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:

• 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

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 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) = 4n² − 2n + 2. As n grows large, the n² term will come to dominate, so that all other terms can be neglected—for instance when n = 500, the term 4n² is 1000 times as large as the 2n term. Ignoring the latter would have negligible effect on the expression's value for most purposes. Further, the coefficients become irrelevant if we compare to any other order of expression, such as an expression containing a term n³ or n⁴. Even if T(n) = 1,000,000n², if U(n) = n³, the latter will always exceed the former once n grows larger than 1,000,000, viz. T(1,000,000) = 1,000,000³ = 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

${\displaystyle T(n)=O(n^{2})}$

or

${\displaystyle T(n)\in O(n^{2})}$

and say that the algorithm has order of n² 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" discussion below) while the first is considered by some as an abuse of notation.^[6]

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,

${\displaystyle f(n)=9\log n+5(\log n)^{4}+3n^{2}+2n^{3}=O(n^{3})\qquad {\text{as }}n\to \infty .}$

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^c) and O(cⁿ) are very different. If c is greater than one, then the latter grows much faster. A function that grows faster than n^c for any c is called superpolynomial. One that grows more slowly than any exponential function of the form cⁿ 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^{log n}.

We may ignore any powers of n inside of the logarithms. The set O(log n) is exactly the same as O(log(n^c)). The logarithms differ only by a constant factor (since log(n^c) = 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, 2ⁿ and 3ⁿ 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 n², replacing n by cn means the algorithm runs in the order of c²n², and the big O notation ignores the constant c². This can be written as c²n² = O(n²). If, however, an algorithm runs in the order of 2ⁿ, replacing n with cn gives 2^cn = (2^c)ⁿ. This is not equivalent to 2ⁿ 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).

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 ${\displaystyle f}$ and ${\displaystyle g}$ are two functions defined on some subset of ${\displaystyle \mathbb {R} ^{n}}$. We say

${\displaystyle f(\mathbf {x} ){\text{ is }}O(g(\mathbf {x} ))\quad {\text{ as }}\mathbf {x} \to \infty }$

if and only if there exist constants ${\displaystyle M}$ and ${\displaystyle C>0}$ such that ${\displaystyle |f(\mathbf {x} )|\leq C|g(\mathbf {x} )|}$ for all ${\displaystyle \mathbf {x} }$ with ${\displaystyle x_{i}\geq M}$ for some ${\displaystyle i.}$^[7] Equivalently, the condition that ${\displaystyle x_{i}\geq M}$ for some ${\displaystyle i}$ can be written ${\displaystyle \|\mathbf {x} \|_{\infty }\geq M}$, where ${\displaystyle \|\mathbf {x} \|_{\infty }}$ denotes the Chebyshev norm. For example, the statement

${\displaystyle f(n,m)=n^{2}+m^{3}+O(n+m)\quad {\text{ as }}n,m\to \infty }$

asserts that there exist constants C and M such that

${\displaystyle |f(n,m)-(n^{2}+m^{3})|\leq C|n+m|}$

whenever either ${\displaystyle m\geq M}$ or ${\displaystyle n\geq M}$ holds. This definition allows all of the coordinates of ${\displaystyle \mathbf {x} }$ to increase to infinity. In particular, the statement

${\displaystyle f(n,m)=O(n^{m})\quad {\text{ as }}n,m\to \infty }$

(i.e., ${\displaystyle \exists C\,\exists M\,\forall n\,\forall m\,\cdots }$) is quite different from

${\displaystyle \forall m\colon ~f(n,m)=O(n^{m})\quad {\text{ as }}n\to \infty }$

(i.e., ${\displaystyle \forall m\,\exists C\,\exists M\,\forall n\,\cdots }$).

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 ${\displaystyle f(n,m)=1}$ and ${\displaystyle g(n,m)=n}$, then ${\displaystyle f(n,m)=O(g(n,m))}$ if we restrict ${\displaystyle f}$ and ${\displaystyle g}$ to ${\displaystyle [1,\infty )^{2}}$, but not if they are defined on ${\displaystyle [0,\infty )^{2}}$.

This is not the only generalization of big O to multivariate functions, and in practice, there is some inconsistency in the choice of definition.^[8]

Matters of notation

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 ${\displaystyle n\to \infty }$:

$${\displaystyle {\begin{aligned}(n+1)^{2}&=n^{2}+O(n),\\(n+O(n^{1/2}))\cdot (n+O(\log n))^{2}&=n^{3}+O(n^{5/2}),\\n^{O(1)}&=O(e^{n}).\end{aligned}}}$$

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(eⁿ) such that n^f(n) = 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^O(1) = O(eⁿ) does not imply the false statement O(eⁿ) = n^O(1).

Orders of common functions

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.

Notation	Name	Example
${\displaystyle O(1)}$	constant	Finding the median value for a sorted array of numbers; Calculating ${\displaystyle (-1)^{n}}$; Using a constant-size lookup table
${\displaystyle O(\log \log n)}$	double logarithmic	Average number of comparisons spent finding an item using interpolation search in a sorted array of uniformly distributed values
${\displaystyle O(\log n)}$	logarithmic	Finding an item in a sorted array with a binary search or a balanced search tree as well as all operations in a binomial heap
${\displaystyle O((\log n)^{c})}$ ${\textstyle c>1}$	polylogarithmic	Matrix chain ordering can be solved in polylogarithmic time on a parallel random-access machine.
${\displaystyle O(n^{c})}$ ${\textstyle 0<c<1}$	fractional power	Searching in a k-d tree
${\displaystyle O(n)}$	linear	Finding an item in an unsorted list or in an unsorted array; adding two n-bit integers by ripple carry
${\displaystyle O(n\log ^{*}n)}$	n log-star n	Performing triangulation of a simple polygon using Seidel's algorithm, where ${\displaystyle \log ^{*}(n)={\begin{cases}0,&{\text{if }}n\leq 1\\1+\log ^{*}(\log n),&{\text{if }}n>1\end{cases}}}$
${\displaystyle O(n\log n)=O(\log n!)}$	linearithmic, loglinear, quasilinear, or "n log n"	Performing a fast Fourier transform; fastest possible comparison sort; heapsort and merge sort
${\displaystyle O(n^{2})}$	quadratic	Multiplying two n-digit numbers by schoolbook multiplication; 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
${\displaystyle O(n^{c})}$	polynomial or algebraic	Tree-adjoining grammar parsing; maximum matching for bipartite graphs; finding the determinant with LU decomposition
${\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }}}$ ${\textstyle 0<\alpha <1}$	L-notation or sub-exponential	Factoring a number using the quadratic sieve or number field sieve
${\displaystyle O(c^{n})}$ ${\textstyle c>1}$	exponential	Finding the (exact) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using brute-force search
${\displaystyle O(n!)}$	factorial	Solving the travelling salesman problem via brute-force search; generating all unrestricted permutations of a poset; finding the determinant with Laplace expansion; enumerating all partitions of a set

The statement ${\displaystyle f(n)=O(n!)}$ is sometimes weakened to ${\displaystyle f(n)=O\left(n^{n}\right)}$ to derive simpler formulas for asymptotic complexity. For any ${\displaystyle k>0}$ and ${\displaystyle c>0}$, ${\displaystyle O(n^{c}(\log n)^{k})}$ is a subset of ${\displaystyle O(n^{c+\varepsilon })}$ for any ${\displaystyle \varepsilon >0}$, so may be considered as a polynomial with some bigger order.

Related asymptotic notations

Big O is widely used in computer science. Together with some other related notations, it forms the family of Bachmann–Landau notations.

Family of Bachmann–Landau notations

Notation	Name[24]	Description	Formal definition	Limit definition[25][26][27][24][19]
${\displaystyle f(n)=o(g(n))}$	Small O; Small Oh; Little O; Little Oh	f is dominated by g asymptotically	${\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|<k\,g(n)}$	${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=0}$
${\displaystyle f(n)=O(g(n))}$	Big O; Big Oh; Big Omicron	${\displaystyle |f|}$ is bounded above by g (up to constant factor) asymptotically	${\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon |f(n)|\leq k\,g(n)}$	${\displaystyle \limsup _{n\to \infty }{\frac {f(n)}{g(n)}}<\infty }$
${\displaystyle f(n)=\Theta (g(n))}$	Big Theta	f is bounded both above and below by g asymptotically	${\displaystyle \exists k_{1}>0\,\exists k_{2}>0\,\exists n_{0}\,\forall n>n_{0}\colon }$ ${\displaystyle k_{1}\,g(n)\leq f(n)\leq k_{2}\,g(n)}$	${\displaystyle f(n)=O(g(n))}$ and ${\displaystyle f(n)=\Omega (g(n))}$ (Knuth version)
${\displaystyle f(n)\sim g(n)}$	On the order of	f is equal to g asymptotically	${\displaystyle \forall \varepsilon >0\,\exists n_{0}\,\forall n>n_{0}\colon \left|{\frac {f(n)}{g(n)}}-1\right|<\varepsilon }$	${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1}$
${\displaystyle f(n)=\Omega (g(n))}$	Big Omega in complexity theory (Knuth)	f is bounded below by g asymptotically	${\displaystyle \exists k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)\geq k\,g(n)}$	${\displaystyle \liminf _{n\to \infty }{\frac {f(n)}{g(n)}}>0}$
${\displaystyle f(n)=\omega (g(n))}$	Small Omega; Little Omega	f dominates g asymptotically	${\displaystyle \forall k>0\,\exists n_{0}\,\forall n>n_{0}\colon f(n)>k\,g(n)}$	${\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=\infty }$
${\displaystyle f(n)=\Omega (g(n))}$	Big Omega in number theory (Hardy–Littlewood)	${\displaystyle |f|}$ is not dominated by g asymptotically	${\displaystyle \exists k>0\,\forall n_{0}\,\exists n>n_{0}\colon |f(n)|\geq k\,g(n)}$	${\displaystyle \limsup _{n\to \infty }{\frac {\left|f(n)\right|}{g(n)}}>0}$

The limit definitions assume ${\displaystyle g(n)>0}$ for sufficiently large ${\displaystyle n}$. The table is (partly) sorted from smallest to largest, in the sense that ${\displaystyle o,O,\Theta ,\sim ,}$ (Knuth's version of) ${\displaystyle \Omega ,\omega }$ on functions correspond to ${\displaystyle <,\leq ,\approx ,=,}$${\displaystyle \geq ,>}$ on the real line^[27] (the Hardy–Littlewood version of ${\displaystyle \Omega }$, however, doesn't correspond to any such description).

Computer science uses the big ${\displaystyle O}$, big Theta ${\displaystyle \Theta }$, little ${\displaystyle o}$, little omega ${\displaystyle \omega }$ and Knuth's big Omega ${\displaystyle \Omega }$ notations.^[28] Analytic number theory often uses the big ${\displaystyle O}$, small ${\displaystyle o}$, Hardy–Littlewood's big Omega ${\displaystyle \Omega }$ (with or without the +, − or ± subscripts) and ${\displaystyle \sim }$ notations.^[22] The small omega ${\displaystyle \omega }$ notation is not used as often in analysis.^[29]

Other notation

In their book Introduction to Algorithms, Cormen, Leiserson, Rivest and Stein consider the set of functions f which satisfy

${\displaystyle f(n)=O(g(n))\quad (n\to \infty )~.}$

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

$${\displaystyle O(g)=\{f:{\text{there exist positive constants}}~c~{\text{and}}~n_{0}~{\text{such that}}~0\leq f(n)\leq cg(n){\text{ for all }}n\geq n_{0}\}.}$$

^[31]

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.^[6] 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:^[32]

${\displaystyle 2n^{2}+3n+1=2n^{2}+O(n).}$

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. The "limiting process" x → x_o can also be generalized by introducing an arbitrary filter base, i.e. to directed nets f and g. The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions,

${\displaystyle f\sim g\iff (f-g)\in o(g)}$

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, 2x is Θ(x), but 2x − 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").^[1] The number theorist Edmund Landau adopted it, and was thus inspired to introduce in 1909 the notation o;^[2] hence both are now called Landau symbols. These notations were used in applied mathematics during the 1950s for asymptotic analysis.^[35] The symbol ${\displaystyle \Omega }$ (in the sense "is not an o of") was introduced in 1914 by Hardy and Littlewood.^[19] Hardy and Littlewood also introduced in 1916 the symbols ${\displaystyle \Omega _{R}}$ ("right") and ${\displaystyle \Omega _{L}}$ ("left"),^[20] precursors of the modern symbols ${\displaystyle \Omega _{+}}$ ("is not smaller than a small o of") and ${\displaystyle \Omega _{-}}$ ("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 ${\displaystyle \Omega }$ became commonly used in number theory at least since the 1950s.^[36] 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.^[24]

Landau never used the big Theta and small omega symbols.

Hardy's symbols were (in terms of the modern O notation)

${\displaystyle f\preccurlyeq g\iff f\in O(g)}$   and   ${\displaystyle f\prec g\iff f\in o(g);}$

(Hardy however never defined or used the notation ${\displaystyle \prec \!\!\prec }$, nor ${\displaystyle \ll }$, as it has been sometimes reported). Hardy introduced the symbols ${\displaystyle \preccurlyeq }$ and ${\displaystyle \prec }$ (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,^[37] the Russian number theorist Ivan Matveyevich Vinogradov introduced his notation ${\displaystyle \ll }$, which has been increasingly used in number theory instead of the ${\displaystyle O}$ notation. We have

${\displaystyle f\ll g\iff f\in O(g),}$

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,^[24] probably in reference to his definition of the symbol Omega. The digit 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_p, o_p
• Limit inferior and limit superior: An explanation of some of the limit notation used in this article
• Master theorem (analysis of algorithms): For analyzing divide-and-conquer recursive algorithms using Big O notation
• Nachbin's theorem: A precise method of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated
• Order of approximation
• Computational complexity of mathematical operations

References and notes

Further reading

• Hardy, G. H. (1910). Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-Reymond. Cambridge University Press.
• Knuth, Donald (1997). "1.2.11: Asymptotic Representations". Fundamental Algorithms. The Art of Computer Programming. Vol. 1 (3rd ed.). Addison-Wesley. ISBN 978-0-201-89683-1.
• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "3.1: Asymptotic notation". Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. ISBN 978-0-262-03293-3.
• Sipser, Michael (1997). Introduction to the Theory of Computation. PWS Publishing. pp. 226–228. ISBN 978-0-534-94728-6.
• Avigad, Jeremy; Donnelly, Kevin (2004). Formalizing O notation in Isabelle/HOL (PDF). International Joint Conference on Automated Reasoning. doi:10.1007/978-3-540-25984-8_27.
• Black, Paul E. (11 March 2005). Black, Paul E. (ed.). "big-O notation". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.
• Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "little-o notation". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.
• Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "Ω". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.
• Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "ω". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.
• Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "Θ". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006.

External links

The Wikibook Data Structures has a page on the topic of: Big-O Notation

Wikiversity solved a MyOpenMath problem using Big-O Notation

• Growth of sequences — OEIS (Online Encyclopedia of Integer Sequences) Wiki
• Introduction to Asymptotic Notations
• Big-O Notation – What is it good for
• An example of Big O in accuracy of central divided difference scheme for first derivative Archived 2018-10-07 at the Wayback Machine
• A Gentle Introduction to Algorithm Complexity Analysis