Linearization

For the linearization in concurrent computing, see Linearizability.

In mathematics and its applications, 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. This method is used in fields such as engineering, physics, economics, and ecology.

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(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. The slope M should be, most accurately, the slope of the tangent line at x = a.

An approximation of f(x) 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:

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

The general equation for the linearization of a multivariable function $f(\mathbf{x})$ 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 .

Uses of linearization

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

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

the linearized system can be written as

$\frac{d\bold{x}}{dt} = \bold{F}(\bold{x_0},t) + D\bold{F}(\bold{x_0},t) \cdot (\bold{x} - \bold{x_0})$

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, one can use the eigenvalues of the Jacobian matrix evaluated at an equilibrium point to determine the nature of that equilibrium. If all of 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 possibly a saddle point. Any complex eigenvalues will appear in complex conjugate pairs and indicate a spiral.

Microeconomics

In microeconomics, decision rules may be approximated under the state-space approach to linearization. Under this approach, the Euler equations of the utility maximization problem are linearized around the stationary steady state. A unique solution to the resulting system of dynamic equations then is found.

See also

• Tangent stiffness matrix
• Stability derivatives
• Linearization theorem
• Taylor approximation

External Links

Linearization tutorials

• Linearization for Model Analysis and Control Design