Linearization: Difference between revisions

Revision as of 18:41, 17 March 2021

Revision as of 17:37, 21 May 2021

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

{{for|the linearization in concurrent computing|Linearizability}}

In [[mathematics]], '''linearization''' 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]].

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===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]] |volume=17 |issue=4 |year=2007 |pages=1079–1107 |doi=10.1142/S0218127407017732 |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]'' Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.</ref> Under this approach, the [[Euler_equations_Euler equations (fluid_dynamicsfluid dynamics)#Linearized_formLinearized 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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==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]]