and its applications, 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
. This method is used in fields such as engineering
, and ecology
Linearization of a function
Linearizations of a function
— ones that are usually used for purposes of calculation. Linearization is an effective method for approximating the output of a function
based on the value and slope
of the function at
, given that f(x) is continuous on
is close to
. In, short, linearization approximates the output of a function near
For example, you might know that . However, without a calculator, what would be a good approximation of ?
For any given function , can be approximated if it is near a known continuous point. The most basic requisite is that, where is the linearization of f(x) at x = a, . 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 continuous 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. After all, a linearization is only an approximation. The slope should be, most accurately, the slope of the tangent line at .
Visually, the accompanying diagram shows the tangent line of at x. At , where is any small positive or negative value, f(x+h) is very nearly the value of the tangent line at the point .
The final equation for the linearization of a function at is:
For , is at . The derivative of is , and the slope of at is .
, we can use the fact that
. The linearization of
, because the function
defines the slope of the function
. Plugging in
, the linearization at 4 is
. In this case
. The true value is close to 2.00024998, so the linearization approximation is amazingly accurate.
Uses of linearization
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 .
analysis, one can use the eigenvalues
of the Jacobian matrix
evaluated at an equilibrium point
to determine the nature of that equilibrium. If all the eigenvalues are positive
, the equilibrium is unstable; if they are all negative the equilibrium is stable; and if the values are of mixed signs, the equilibrium is a saddle point
. Any complex
eigenvalues will appear in complex conjugate
pairs and indicate spiral
(or circular if the real
components are zero around the equilibrium.
, decision rules
may be approximated under the state-space approach to linearization. Under this approach, the Euler equations
of the utility maximization problem
are linearized around the stationary steady state. A unique solution to the resulting system of dynamic equations then is found.