Linearization: Difference between revisions

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

{{for|the linearization of a partial order|Linear extension}}

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

In [[mathematics]], '''linearization''' refers([[British toEnglish]]: '''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:

:<math> f(x,y) \approx f(a,b) + \left. {\frac{{\partial f(x,y)}}{{\partial x}}} \right|_{a,b} (x - a) + \left. {\frac{{\partial f(x,y)}}{{\partial y}}} \right|_{a,b} (y - b)</math>

The general equation for the linearization of a multivariable function <math>f(\mathbf{x})</math> at a point <math>\mathbf{p}</math> is:

:<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>

==Uses of linearization==

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Linearization makes it possible to use tools for studying [[linear system]]s to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its [[Taylor expansion]] around the point of interest. For a system defined by the equation

:<math>\frac{d\boldmathbf{x}}{dt} = \boldmathbf{F}(\boldmathbf{x},t)</math>,

the linearized system can be written as

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

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

===Stability analysis===

In [[stability theory|stability]] analysis of [[Autonomous_system_Autonomous system (mathematics)|autonomous systems]], one can use the [[eigenvalue]]s of the [[Jacobian matrix and determinant|Jacobian matrix]] evaluated at a [[hyperbolic equilibrium point]] to determine the nature of that equilibrium. This is the content of the [[linearization theorem]]. For time-varying systems, the linearization requires additional justification.<ref>{{cite journal |first=G. A. |last=Leonov, |first2=N. V. |last2=Kuznetsov, |title=Time-Varying Linearization and the Perron effects, |journal=[[International Journal of Bifurcation and Chaos, Vol.]] |volume=17, No. |issue=4, |year=2007, pp|pages=1079–1107 |doi=10.1142/S0218127407017732 1079-1107|bibcode=2007IJBC...17.1079L }}</ref>

===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_Euler equations (fluid_dynamicsfluid dynamics)#Linearized_formConservation 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===

In [[Mathematicalmathematical optimization]], cost functions and non-linear components within can be linearized in order to apply a linear solving method such as the [[Simplex algorithm]]. The optimized result is reached much more efficiently and is deterministic as a [[global optimum]].

===Multiphysics===

In [[multiphysics]] systems—systems involving multiple physical fields that interact with one another—linearization with respect to each of the physical fields may be performed. This linearization of the system with respect to each of the fields results in a linearized monolithic equation system that can be solved using monolithic iterative solution procedures such as the [[Newton–Raphson method]]. Examples of this include [[MRI scanner]] systems which results in a system of electromagnetic, mechanical and acoustic fields.<ref>{{cite journal |first=S. |last=Bagwell |first2=P. D. |last2=Ledger |first3=A. J. |last3=Gil |first4=M. |last4=Mallett |first5=M. |last5=Kruip |year=2017 |title=A linearised ''hp''–finite element framework for acousto-magneto-mechanical coupling in axisymmetric MRI scanners |journal=International Journal for Numerical Methods in Engineering |volume=112 |issue=10 |pages=1323–1352 |doi=10.1002/nme.5559 |bibcode=2017IJNME.112.1323B |doi-access=free }}</ref>

==See also==

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* [[Taylor approximation]]

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

* [[Quasilinearization]]

==References==

{{reflistReflist}}

==External links==

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===Linearization tutorials===

* [http://www.mathworks.com/discovery/linearization.html Linearization for Model Analysis and Control Design]

{{Authority control}}

[[Category:Differential calculus]]

[[Category:Dynamical systems]]

[[Category:Approximations]]