# Linearization

{{#invoke:Hatnote|hatnote}} 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 ${\displaystyle y=f(x)}$ at any ${\displaystyle x=a}$ based on the value and slope of the function at ${\displaystyle x=b}$, given that ${\displaystyle f(x)}$ is differentiable on ${\displaystyle [a,b]}$ (or ${\displaystyle [b,a]}$) and that ${\displaystyle a}$ is close to ${\displaystyle b}$. In short, linearization approximates the output of a function near ${\displaystyle x=a}$.

For example, ${\displaystyle {\sqrt {4}}=2}$. However, what would be a good approximation of ${\displaystyle {\sqrt {4.001}}={\sqrt {4+.001}}}$?

For any given function ${\displaystyle y=f(x)}$, ${\displaystyle f(x)}$ can be approximated if it is near a known differentiable point. The most basic requisite is that, where ${\displaystyle L_{a}(x)}$ is the linearization of ${\displaystyle f(x)}$ at ${\displaystyle x=a}$, ${\displaystyle L_{a}(a)=f(a)}$. The point-slope form of an equation forms an equation of a line, given a point ${\displaystyle (H,K)}$ and slope ${\displaystyle M}$. The general form of this equation is: ${\displaystyle y-K=M(x-H)}$.

Using the point ${\displaystyle (a,f(a))}$, ${\displaystyle L_{a}(x)}$ becomes ${\displaystyle y=f(a)+M(x-a)}$. Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to ${\displaystyle f(x)}$ at ${\displaystyle x=a}$.

While the concept of local linearity applies the most to points arbitrarily close to ${\displaystyle x=a}$, those relatively close work relatively well for linear approximations. The slope ${\displaystyle M}$ should be, most accurately, the slope of the tangent line at ${\displaystyle x=a}$.

An approximation of f(x)=x^2 at (x, f(x))

Visually, the accompanying diagram shows the tangent line of ${\displaystyle f(x)}$ at ${\displaystyle x}$. At ${\displaystyle f(x+h)}$, where ${\displaystyle h}$ is any small positive or negative value, ${\displaystyle f(x+h)}$ is very nearly the value of the tangent line at the point ${\displaystyle (x+h,L(x+h))}$.

The final equation for the linearization of a function at ${\displaystyle x=a}$ is:

## Example

To find ${\displaystyle {\sqrt {4.001}}}$, we can use the fact that ${\displaystyle {\sqrt {4}}=2}$. The linearization of ${\displaystyle f(x)={\sqrt {x}}}$ at ${\displaystyle x=a}$ is ${\displaystyle y={\sqrt {a}}+{\frac {1}{2{\sqrt {a}}}}(x-a)}$, because the function ${\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}}$ defines the slope of the function ${\displaystyle f(x)={\sqrt {x}}}$ at ${\displaystyle x}$. Substituting in ${\displaystyle a=4}$, the linearization at 4 is ${\displaystyle y=2+{\frac {x-4}{4}}}$. In this case ${\displaystyle x=4.001}$, so ${\displaystyle {\sqrt {4.001}}}$ is approximately ${\displaystyle 2+{\frac {4.001-4}{4}}=2.00025}$. 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 ${\displaystyle f(x,y)}$ at a point ${\displaystyle p(a,b)}$ is:

The general equation for the linearization of a multivariable function ${\displaystyle f(\mathbf {x} )}$ at a point ${\displaystyle \mathbf {p} }$ is:

where ${\displaystyle \mathbf {x} }$ is the vector of variables, and ${\displaystyle \mathbf {p} }$ 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

${\displaystyle {\frac {d{\mathbf {x} }}{dt}}={\mathbf {F} }({\mathbf {x} },t)}$,

the linearized system can be written as

${\displaystyle {\frac {d{\mathbf {x}}}{dt}}\approx {\mathbf {F}}({\mathbf {x_{0}}},t)+D{\mathbf {F}}({\mathbf {x_{0}}},t)\cdot ({\mathbf {x}}-{\mathbf {x_{0}}})}$

### 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]