Linearization

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 ${\displaystyle L_{a}(a)=f(a)}$, where ${\displaystyle L_{a}(x)}$ is the linearization of ${\displaystyle f(x)}$ at ${\displaystyle x=a}$. 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:

${\displaystyle y=(f(a)+f'(a)(x-a))}$

For ${\displaystyle x=a}$, ${\displaystyle f(a)=f(x)}$. The derivative of ${\displaystyle f(x)}$ is ${\displaystyle f'(x)}$, and the slope of ${\displaystyle f(x)}$ at ${\displaystyle a}$ is ${\displaystyle f'(a)}$.

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.

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

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

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

${\displaystyle f({\mathbf {x} })\approx f({\mathbf {p} })+\left.{\nabla f}\right|_{\mathbf {p} }\cdot ({\mathbf {x} }-{\mathbf {p} })}$

where ${\displaystyle \mathbf {x} }$ is the vector of variables, and ${\displaystyle \mathbf {p} }$ is the linearization point of interest .[2]

Linearization makes it possible to use tools for studying linear 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

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

where ${\displaystyle \mathbf {x_{0}} }$ is the point of interest and ${\displaystyle D\mathbf {F} (\mathbf {x_{0}} )}$ is the Jacobian of ${\displaystyle \mathbf {F} (\mathbf {x} )}$ evaluated at ${\displaystyle \mathbf {x_{0}} }$.

• Linear stability
• Tangent stiffness matrix
• Stability derivatives
• Linearization theorem
• Taylor approximation
• Functional equation (L-function)