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Linearization

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In mathematics, linearization is finding the linear approximation to a function at a given point. 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. ^ G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron effects, International Journal of Bifurcation and Chaos, Vol. 17, No. 4, 2007, pp. 1079-1107
  4. ^ a b c Moffatt, Mike. (2008) About.com State-Space Approach Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.
  5. ^ S. Bagwell, P.D. Ledger, A.J. Gil, M. Mallett, M. Kruip, A linearised hp–finite element framework for acousto-magneto-mechanical coupling in axisymmetric MRI scanners, DOI: 10.1002/nme.5559

External linksEdit