Linearization: Difference between revisions

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{{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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===Stability analysis===

In [[stability theory|stability]] analysis of [[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> In the context of [[network dynamics]] the system's fixed-point behavior is captured by its Jacobian matrix, as derived from the [[dynamic Jacobian ensemble]].<ref>{{Cite journal|last=C. Meena, C. Hens, S. Acharyya, S. Haber, S. Boccaletti and B. Barzel|date=2023|title=Emergent stability in complex network dynamics|journal=Nature Physics|volume= |issue= |pages= |doi=10.1038/s41567-023-02020-8}}</ref> This family of Jacobians links the <math>i,j</math> terms of the Jacobian to the degrees of nodes <math>i</math> and <math>j</math> in the network.

===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==