Editing Linearization (section)

==Uses of linearization==

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\mathbf{x}}{dt} = \mathbf{F}(\mathbf{x},t)</math>,

the linearized system can be written as

:<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===
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>

===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===
In [[mathematical 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>