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Linearization

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In mathematics, linearization is finding the linear approximation to a 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 systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems.[1] This method is used in fields such as engineering, physics, economics, and ecology.

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Linearization of a functionEdit

Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function   at any   based on the value and slope of the function at  , given that   is differentiable on   (or  ) and that   is close to  . In short, linearization approximates the output of a function near  .

For example,  . However, what would be a good approximation of  ?

For any given function  ,   can be approximated if it is near a known differentiable point. The most basic requisite is that  , where   is the linearization of   at  . The point-slope form of an equation forms an equation of a line, given a point   and slope  . The general form of this equation is:  .

Using the point  ,   becomes  . Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to   at  .

While the concept of local linearity applies the most to points arbitrarily close to  , those relatively close work relatively well for linear approximations. The slope   should be, most accurately, the slope of the tangent line at  .

 
An approximation of f(x)=x^2 at (x, f(x))

Visually, the accompanying diagram shows the tangent line of   at  . At  , where   is any small positive or negative value,   is very nearly the value of the tangent line at the point  .

The final equation for the linearization of a function at   is:

 

For  ,  . The derivative of   is  , and the slope of   at   is  .

ExampleEdit

To find  , we can use the fact that  . The linearization of   at   is  , because the function   defines the slope of the function   at  . Substituting in  , the linearization at 4 is  . In this case  , so   is approximately  . 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 functionEdit

The equation for the linearization of a function   at a point   is:

 

The general equation for the linearization of a multivariable function   at a point   is:

 

where   is the vector of variables, and   is the linearization point of interest .[2]

Uses of linearizationEdit

Linearization makes it possible to use tools for studying linear systems 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

 ,

the linearized system can be written as

 

where   is the point of interest and   is the Jacobian of   evaluated at  .

Stability analysisEdit

In stability analysis of autonomous systems, one can use the eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium. This is the content of linearization theorem. For time-varying systems, the linearization requires additional justification.[3]

MicroeconomicsEdit

In microeconomics, decision rules may be approximated under the state-space approach to linearization.[4] Under this approach, the Euler equations of the utility maximization problem are linearized around the stationary steady state.[4] A unique solution to the resulting system of dynamic equations then is found.[4]

OptimizationEdit

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.

MultiphysicsEdit

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.[5]

See alsoEdit

ReferencesEdit

  1. ^ The linearization problem in complex dimension one dynamical systems at Scholarpedia
  2. ^ Linearization. The Johns Hopkins University. Department of Electrical and Computer Engineering Archived 2010-06-07 at the Wayback Machine.
  3. ^ Leonov, G. A.; Kuznetsov, N. V. (2007). "Time-Varying Linearization and the Perron effects". International Journal of Bifurcation and Chaos. 17 (4): 1079–1107. doi:10.1142/S0218127407017732.
  4. ^ a b c Moffatt, Mike. (2008) About.com State-Space Approach Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.
  5. ^ Bagwell, S.; Ledger, P. D.; Gil, A. J.; Mallett, M.; Kruip, M. (2017). "A linearised hp–finite element framework for acousto-magneto-mechanical coupling in axisymmetric MRI scanners". International Journal for Numerical Methods in Engineering. 112 (10): 1323–1352. doi:10.1002/nme.5559.

External linksEdit