Big O notation: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Subpagination Bot
m (Add {{subpages}} and remove any categories (details))
mNo edit summary
 
(5 intermediate revisions by 3 users not shown)
Line 3: Line 3:
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'' (respectively, <math>f_n</math>) and ''g'' (respectively, <math>g_n</math>) are real valued functions of the real numbers (respectively, sequences)  then the notation <math>f(t)=O(g(t))</math> (respectively, <math>f_n=O(g_n)</math>) denotes that there exist a positive real number (respectively, integer) ''T'' and a positive constant ''C'' such that <math>|f(t)|\leq C |g(t)|</math> for all <math>t>T</math>(respectively, <math>|f_n| \leq C |g_n|</math> for all n>T).   
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> -->


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>.  
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.-->


==See also==
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]]
 
[[Little o notation]]

Latest revision as of 12:01, 18 July 2024

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

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 .