Big O notation: Difference between revisions
imported>Aleksander Stos (alternative formulation; also deleted adjective positive -- e.g. T might be negative (well, it was OK but unnecessary restrictive)) |
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The '''big O notation''' is a mathematical notation to express various bounds concerning asymptotic behaviour of functions. It is often used in particular applications in [[physics]], [[computer science]], [[engineering]] and other [[applied sciences]]. For example, a typical context use in computer science is to express the [[complexity of algorithms]]. | The '''big O notation''' is a mathematical notation to express various bounds concerning asymptotic behaviour of functions. It is often used in particular applications in [[physics]], [[computer science]], [[engineering]] and other [[applied sciences]]. For example, a typical context use in computer science is to express the [[complexity of algorithms]]. | ||
More formally, if ''f'' and ''g'' are real valued functions of the real variable <math>t</math> then the notation <math>f(t)=O(g(t))</math> indicates that there exist a real number ''T'' and a constant ''C'' such that <math>|f(t)|\leq C |g(t)|</math> for all <math>t>T.</math> Equivalently, | More formally, if ''f'' and ''g'' are real valued functions of the real variable <math>t</math> then the notation <math>f(t)=O(g(t))</math> indicates that there exist a real number ''T'' and a constant ''C'' such that <math>|f(t)|\leq C |g(t)|</math> for all <math>t>T.</math> <!-- Equivalently, | ||
:<math>\displaystyle\limsup_{t\to\infty}\frac{|f(t)|}{|g(t)|} \le C.</math> | :<math>\displaystyle\limsup_{t\to\infty}\frac{|f(t)|}{|g(t)|} \le C.</math> --> | ||
Similarly, if <math>a_n</math> and <math>b_n</math> are two numerical sequences then <math>a_n=O(b_n)</math> means that <math>|a_n|\le C |b_n|</math> for all <math>n</math> big enough. | Similarly, if <math>a_n</math> and <math>b_n</math> are two numerical sequences then <math>a_n=O(b_n)</math> means that <math> |a_n|\le C|b_n|</math> for all ''n'' big enough. | ||
<!-- <math>|a_n|\le C |b_n|</math> for all <math>n</math> big enough.--> | |||
The big O notation is also often used to indicate that the absolute value of a real valued function around some [[topological space#Some topological notions|neighbourhood]] of a point is upper bounded by a constant multiple of the absolute value of another function, in that neigbourhood. For example, for a real number <math>t_0</math> the notation <math>f(t)=O(g(t-t_0))</math>, where ''g(t | The big O notation is also often used to indicate that the absolute value of a real valued function around some [[topological space#Some topological notions|neighbourhood]] of a point is upper bounded by a constant multiple of the absolute value of another function, in that neigbourhood. For example, for a real number <math>t_0</math> the notation <math>f(t)=O(g(t-t_0))</math>, where ''g''(''t'') is a function which is [[continuity|continuous]] at ''t'' = 0 with ''g''(0) = 0, denotes that there exists a real positive constant ''C'' such that <math>|f(t)|\leq C|g(t-t_0)|</math> on ''some'' neighbourhood ''N'' of <math>t_0</math>.[[Category:Suggestion Bot Tag]] | ||
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Latest revision as of 11:01, 18 July 2024
The big O notation is a mathematical notation to express various bounds concerning asymptotic behaviour of functions. It is often used in particular applications in physics, computer science, engineering and other applied sciences. For example, a typical context use in computer science is to express the complexity of algorithms.
More formally, if f and g are real valued functions of the real variable then the notation indicates that there exist a real number T and a constant C such that for all
Similarly, if and are two numerical sequences then means that for all n big enough.
The big O notation is also often used to indicate that the absolute value of a real valued function around some neighbourhood of a point is upper bounded by a constant multiple of the absolute value of another function, in that neigbourhood. For example, for a real number the notation , where g(t) is a function which is continuous at t = 0 with g(0) = 0, denotes that there exists a real positive constant C such that on some neighbourhood N of .
