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Linearization

For the linearization in concurrent computing, see Linearizability.

In mathematics linearization refers to 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.

Linearization of a function

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 <math>y = f(x)</math> at any <math>x = a</math> based on the value and slope of the function at <math>x = b</math>, given that <math>f(x)</math> is differentiable on <math>[a, b]</math> (or <math>[b, a]</math>) and that <math>a</math> is close to <math>b</math>. In short, linearization approximates the output of a function near <math>x = a</math>.

For example, <math>\sqrt{4} = 2</math>. However, what would be a good approximation of <math>\sqrt{4.001} = \sqrt{4 + .001}</math>?

For any given function <math>y = f(x)</math>, <math>f(x)</math> can be approximated if it is near a known differentiable point. The most basic requisite is that, where <math>L_a(x)</math> is the linearization of <math>f(x)</math> at <math>x = a</math>, <math>L_a(a) = f(a)</math>. The point-slope form of an equation forms an equation of a line, given a point <math>(H, K)</math> and slope <math>M</math>. The general form of this equation is: <math>y - K = M(x - H)</math>.

Using the point <math>(a, f(a))</math>, <math>L_a(x)</math> becomes <math>y = f(a) + M(x - a)</math>. Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to <math>f(x)</math> at <math>x = a</math>.

While the concept of local linearity applies the most to points arbitrarily close to <math>x = a</math>, those relatively close work relatively well for linear approximations. The slope <math>M</math> should be, most accurately, the slope of the tangent line at <math>x = a</math>.

File:Tangent-calculus.svg
An approximation of f(x)=x^2 at (x, f(x))

Visually, the accompanying diagram shows the tangent line of <math>f(x)</math> at <math>x</math>. At <math>f(x+h)</math>, where <math>h</math> is any small positive or negative value, <math>f(x+h)</math> is very nearly the value of the tangent line at the point <math>(x+h, L(x+h))</math>.

The final equation for the linearization of a function at <math>x = a</math> is:

<math>y = f(a) + f'(a)(x - a)\,</math>

For <math>x = a</math>, <math>f(a) = f(x)</math>. The derivative of <math>f(x)</math> is <math>f'(x)</math>, and the slope of <math>f(x)</math> at <math>a</math> is <math>f'(a)</math>.

Example

To find <math>\sqrt{4.001}</math>, we can use the fact that <math>\sqrt{4} = 2</math>. The linearization of <math>f(x) = \sqrt{x}</math> at <math>x = a</math> is <math>y = \sqrt{a} + \frac{1}{2 \sqrt{a}}(x - a)</math>, because the function <math>f'(x) = \frac{1}{2 \sqrt{x}}</math> defines the slope of the function <math>f(x) = \sqrt{x}</math> at <math>x</math>. Substituting in <math>a = 4</math>, the linearization at 4 is <math>y = 2 + \frac{x-4}{4}</math>. In this case <math>x = 4.001</math>, so <math>\sqrt{4.001}</math> is approximately <math>2 + \frac{4.001-4}{4} = 2.00025</math>. 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 function

The equation for the linearization of a function <math>f(x,y)</math> at a point <math>p(a,b)</math> is:

<math> f(x,y) \approx f(a,b) + \left. {\fracTemplate:\partial f(x,y)Template:\partial x} \right|_{a,b} (x - a) + \left. {\fracTemplate:\partial f(x,y)Template:\partial y} \right|_{a,b} (y - b)</math>

The general equation for the linearization of a multivariable function <math>f(\mathbf{x})</math> at a point <math>\mathbf{p}</math> is:

<math>f({\mathbf{x}}) \approx f({\mathbf{p}}) + \left. {\nabla f} \right|_{\mathbf{p}} \cdot ({\mathbf{x}} - {\mathbf{p}})</math>

where <math>\mathbf{x}</math> is the vector of variables, and <math>\mathbf{p}</math> is the linearization point of interest .[2]

Uses of linearization

Linearization makes it possible to use tools for studying nonlinear 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

<math>\frac{d\bold{x}}{dt} = \bold{F}(\bold{x},t)</math>,

the linearized system can be written as

<math>\frac{d\bold{x}}{dt} \approx \bold{F}(\bold{x_0},t) + D\bold{F}(\bold{x_0},t) \cdot (\bold{x} - \bold{x_0})</math>

where <math>\bold{x_0}</math> is the point of interest and <math>D\bold{F}(\bold{x_0})</math> is the Jacobian of <math>\bold{F}(\bold{x})</math> evaluated at <math>\bold{x_0}</math>.

Stability analysis

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]

Microeconomics

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]

See also

References

  1. ^ The linearization problem in complex dimension one dynamical systems at Scholarpedia
  2. ^ Linearization. The Johns Hopkins University. Department of Electrical and Computer Engineering
  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.

External links

Linearization tutorials