Little o notation: Difference between revisions

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The little o notation is a mathematical notation which indicates that the decay (respectively, growth) rate of a certain function or sequence is faster (respectively, slower) than that of another function or sequence. It is often used in particular applications in physics, computer science, engineering and other applied sciences.

More formally, if f and g are real valued functions of the real numbers then the notation $f(t)=o(g(t))$ indicates that for every real number $\epsilon >0$ there exists a positive real number $T(\epsilon )$ (note the dependence of T on $\epsilon$ ) such that $|f(t)|\leq \epsilon |g(t)|$ for all $t>T(\epsilon ).$

When the function g does not vanish this may be rewritten simply as

\lim _{t\to \infty }{\frac {f(t)}{g(t)}}=0.

Similarly, if $a_{n}$ and $b_{n}$ are two numerical sequences then $a_{n}=O(b_{n})$ means that for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon>0} and n big enough one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n|\leq \epsilon |b_n| } (in case when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n} is not zero, this means the limit of the fraction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n/b_n} vanishes in the limit).

The little o notation is also often used to indicate that the absolute value of a real valued function goes to zero around some point at a rate faster than at which the absolute value of another function goes to zero at the same point. For example, suppose that f is a function with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t_0)=0} for some real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_0} . Then the notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=o(g(t-t_0))} , where g(t) is a function which is continuous at t=0 and with g(0)=0, denotes that for every real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon>0} there exists a neighbourhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(\epsilon)} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_0} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |f(t)|\leq \epsilon |g(t-t_0)|} holds on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(\epsilon)} .

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+{{subpages}}
 The '''little o notation''' is a mathematical notation which indicates that the decay (respectively, growth) rate of a certain function or sequence is faster (respectively, slower) than that of another function or sequence. It is often used in particular applications in [[physics]], [[computer science]], [[engineering]] and other [[applied sciences]].
-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 for every real number <math>\epsilon>0</math> there exists a positive real number (respectively, integer) <math>T(\epsilon)</math> (note the dependence of ''T'' on <math>\epsilon</math>) such that <math>|f(t)|\leq \epsilon |g(t)|</math> for all <math>t>T(\epsilon)</math> (respectively, <math>|f_n| \leq \epsilon |g_n|</math> for all <math>n>T(\epsilon)</math>).
+More formally, if ''f'' and ''g''  are real valued functions of the real numbers then the notation <math>f(t)=o(g(t))</math> indicates that for every real number <math>\epsilon>0</math> there exists a positive real number <math>T(\epsilon)</math> (note the dependence of ''T'' on <math>\epsilon</math>) such that <math>|f(t)|\leq \epsilon |g(t)|</math> for all <math>t>T(\epsilon).</math>
-The little O notation is also often used to indicate that the absolute value of a real valued function goes to zero around some point at a rate faster than at which the absolute value of another function goes to zero at the same point. For example, suppose that ''f'' is a function with <math>f(t_0)=0</math> for some real number <math>t_0</math>. Then the notation  <math>f(t)=o(g(t-t_0))</math>, where ''g(t)'' is a function which is [[continuity|continuous]] at ''t=0'' and with ''g(0)=0'',  denotes that for every real number <math>\epsilon>0</math> there exists a [[topological space#Some topological notions|neighbourhood]] <math>N(\epsilon)</math> of <math>t_0</math> such that <math>|f(t)|\leq \epsilon |g(t-t_0)|</math> holds on <math>N(\epsilon)</math>.
+When the function ''g'' does not vanish this may be rewritten simply as
+:<math> \lim_{t\to\infty} \frac{f(t)}{g(t)} = 0.</math>
-==See also==
+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 for any <math>\varepsilon>0</math> and ''n'' big enough one has <math>|a_n|\leq \epsilon |b_n| </math> (in case when <math>b_n</math> is not zero, this means the limit of the fraction <math>a_n/b_n</math> vanishes in the limit).
-[[Big O notation]]
+The little o notation is also often used to indicate that the absolute value of a real valued function goes to zero around some point at a rate faster than at which the absolute value of another function goes to zero at the same point. For example, suppose that ''f'' is a function with <math>f(t_0)=0</math> for some real number <math>t_0</math>. Then the notation  <math>f(t)=o(g(t-t_0))</math>, where ''g(t)'' is a function which is [[continuity|continuous]] at ''t=0'' and with ''g(0)=0'',  denotes that for every real number <math>\epsilon>0</math> there exists a [[topological space#Some topological notions|neighbourhood]] <math>N(\epsilon)</math> of <math>t_0</math> such that <math>|f(t)|\leq \epsilon |g(t-t_0)|</math> holds on <math>N(\epsilon)</math>.
-[[Category:Mathematics Workgroup]]
+==See also==
-[[Category: CZ Live]]
+[[Big O notation]][[Category:Suggestion Bot Tag]]

Little o notation: Difference between revisions

Latest revision as of 16:00, 12 September 2024

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