Linearization: Difference between revisions

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{{Short description|Finding linear approximation of function at given point}}

{{About||the linearization of a partial order|Linear extension|the linearization in concurrent computing|Linearizability}}

In [[mathematics]], '''linearization''' ([[British English]]: '''linearisation''') is finding the [[linear approximation]] to a [[function (mathematics)|function]] at a given point. The linear approximation of a function is the first order [[Taylor expansion]] around the point of interest. In the study of [[dynamical system]]s, linearization is a method for assessing the local [[stability theory|stability]] of an [[equilibrium point]] of a [[system]] of [[nonlinear]] [[differential equation]]s or discrete [[dynamical system]]s.<ref>[http://www.scholarpedia.org/article/Siegel_disks/Linearization The linearization problem in complex dimension one dynamical systems at Scholarpedia]</ref> This method is used in fields such as [[engineering]], [[physics]], [[economics]], and [[ecology]].

==Linearization of a function==

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While the concept of local linearity applies the most to points [[Limit of a function#Limit of a function at a point|arbitrarily close]] to <math>x = a</math>, those relatively close work relatively well for linear approximations. The slope <math>M</math> should be, most accurately, the slope of the tangent line at <math>x = a</math>.

[[Image:Tangent-calculus.svg|thumb|300px|An approximation of ''f''(''x'') = ''x^''² at (''x'', ''f''(''x''))]]

Visually, the accompanying diagram shows the tangent line of <math>f(x)</math> at <math>x</math>. At <math>f(x+h)</math>, where <math>h</math> is any small positive or negative value, <math>f(x+h)</math> is very nearly the value of the tangent line at the point <math>(x+h, L(x+h))</math>.

The final equation for the linearization of a function at <math>x = a</math> is:

<math display="block">y = (f(a) + f'(a)(x - a))</math>

<math>y = (f(a) + f'(a)(x - a))</math>

For <math>x = a</math>, <math>f(a) = f(x)</math>. The [[derivative]] of <math>f(x)</math> is <math>f'(x)</math>, and the slope of <math>f(x)</math> at <math>a</math> is <math>f'(a)</math>.

===Example===

To find <math>\sqrt{4.001}</math>, we can use the fact that <math>\sqrt{4} = 2</math>. The linearization of <math>f(x) = \sqrt{x}</math> at <math>x = a</math> is <math>y = \sqrt{a} + \frac{1}{2 \sqrt{a}}(x - a)</math>, because the function <math>f'(x) = \frac{1}{2 \sqrt{x}}</math> defines the slope of the function <math>f(x) = \sqrt{x}</math> at <math>x</math>. Substituting in <math>a = 4</math>, the linearization at 4 is <math>y = 2 + \frac{x-4}{4}</math>. In this case <math>x = 4.001</math>, so <math>\sqrt{4.001}</math> is approximately <math>2 + \frac{4.001-4}{4} = 2.00025</math>. The true value is close to 2.00024998, so the linearization approximation has a relative error of less than 1 millionth of a percent.

==Linearization of a multivariable function{{anchor|Multivariable functions}}==

{{See also|Taylor series#In several variables}}

The equation for the linearization of a function <math>f(x,y)</math> at a point <math>p(a,b)</math> is:

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:<math>f({\mathbf{x}}) \approx f({\mathbf{p}}) + \left. {\nabla f} \right|_{\mathbf{p}} \cdot ({\mathbf{x}} - {\mathbf{p}})</math>

where <math>\mathbf{x}</math> is the vector of variables, <math>{\nabla f}</math> is the [[gradient]], and <math>\mathbf{p}</math> is the linearization point of interest

.<ref>[http://www.ece.jhu.edu/~pi/Courses/454/linear.pdf Linearization. The Johns Hopkins University. Department of Electrical and Computer Engineering] {{webarchive|url=https://web.archive.org/web/20100607120539/http://www.ece.jhu.edu/~pi/Courses/454/linear.pdf |date=2010-06-07 }}</ref>

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:<math>\frac{d\mathbf{x}}{dt} \approx \mathbf{F}(\mathbf{x_0},t) + D\mathbf{F}(\mathbf{x_0},t) \cdot (\mathbf{x} - \mathbf{x_0})</math>

where <math>\mathbf{x_0}</math> is the point of interest and <math>D\mathbf{F}(\mathbf{x_0},t)</math> is the <math>\mathbf{x}</math>-[[Jacobian matrix and determinant|Jacobian]] of <math>\mathbf{F}(\mathbf{x},t)</math> evaluated at <math>\mathbf{x_0}</math>.

===Stability analysis===

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===Microeconomics===

In [[microeconomics]], [[decision rule]]s may be approximated under the state-space approach to linearization.<ref name="statespace">Moffatt, Mike. (2008) [[About.com]] ''[http://economics.about.com/od/economicsglossary/g/statespace.htm State-Space Approach] {{Webarchive|url=https://web.archive.org/web/20160304055023/http://economics.about.com/od/economicsglossary/g/statespace.htm |date=2016-03-04 }}'' Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.</ref> Under this approach, the [[Euler equations (fluid dynamics)#Conservation form|Euler equations]] of the [[utility maximization problem]] are linearized around the stationary steady state.<ref name="statespace"/> A unique solution to the resulting system of dynamic equations then is found.<ref name="statespace"/>

===Optimization===

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* [[Taylor approximation]]

* [[Functional equation (L-function)]]

* [[Quasilinearization]]

==References==