Linearization

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

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 $x = a$.

File:Tangent-calculus.svg
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

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

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

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

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

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

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

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]