- 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:

$y\; =\; f(a)\; +\; f\text{'}(a)(x\; -\; a),$

For $x\; =\; a$, $f(a)$ is $f(x)$ at $a$. The

derivative of $f(x)$ is $f\text{'}(x)$, and the slope of $f(x)$ at $a$ is $f\text{'}(a)$.**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 $f\text{'}(x)\; =\; frac\{1\}\{2\; sqrt\{x$ 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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