Linearization

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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[edit]

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

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

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

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

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

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

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

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

y = f(a) + f'(a)(x - a)\,

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

Example[edit]

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

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

 f(x,y) \approx f(a,b) + \left. {\frac{{\partial f(x,y)}}{{\partial x}}} \right|_{a,b} (x - a) + \left. {\frac{{\partial f(x,y)}}{{\partial y}}} \right|_{a,b} (y - b)

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

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

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

Uses of linearization[edit]

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

\frac{d\bold{x}}{dt} = \bold{F}(\bold{x},t),

the linearized system can be written as

\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})

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

Stability analysis[edit]

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[edit]

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[edit]

References[edit]

  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[edit]

Linearization tutorials[edit]