- Linearization
In
mathematics and its applications, linearization refers to finding thelinear approximation to a function at a given point. In the study ofdynamical system s, linearization is a method for assessing the localstability of anequilibrium point of asystem ofnonlinear differential equation s. This method is used in fields such asengineering ,physics ,economics , andecology .Linearization of a function
Linearizations of a function are lines — ones that are usually 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 f(x) is continuous on (or )and that 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 .Example
To find , we can use the fact that . The linearization of at is , because the function defines the slope of the function at . Plugging in , the linearization at 4 is . In this case , so is approximately . 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 system s to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of itsTaylor 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 .tability analysis
In stability analysis, one can use the
eigenvalue s of theJacobian matrix evaluated at anequilibrium 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 asaddle point . Any complex eigenvalues will appear incomplex conjugate pairs and indicatespiral (or circular if the real components are zero around the equilibrium.Microeconomics
In
microeconomics , decision rules may be approximated under the state-space approach to linearization.Moffatt, Mike. (2008)About.com " [http://economics.about.com/od/economicsglossary/g/statespace.htm State-Space Approach] " Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.] Under this approach, the Euler equations of theutility maximization problem are linearized around the stationary steady state. A unique solution to the resulting system of dynamic equations then is found.ee also
*
Tangent stiffness matrix
*Stability derivatives
*Linearization theorem
*Taylor approximation References
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