- 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 y = f(x) at any x = a based on the value and
slope of the function at x = b, given that f(x) is continuous 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, you might know that sqrt{4} = 2. However, without a calculator, 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 continuous point. The most basic requisite is that, where L_a(x) is the linearization of f(x) at x = a, L(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 continuous 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. After all, a linearization is only an approximation. The slope M should be, most accurately, the slope of the tangent line at x = a.
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) is f(x) at a. The
derivative of f(x) is f'(x), and the slope of f(x) at a is f'(a).Example
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. Plugging 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 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:frac{dold{x{dt} = old{F}(old{x},t),
the linearized system can be written as
:frac{dold{x{dt} = Dold{F}(old{x_0},t) cdot (old{x} - old{x_0})
where old{x_0} is the point of interest and Dold{F}(old{x_0}) is the
Jacobian of old{F}(old{x}) evaluated at old{x_0}.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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