In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and its applications,

**linearization** refers to finding the

linear approximationIn mathematics, a linear approximation is an approximation of a general function using a linear function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.-Definition:Given a twice continuously...

to a

functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

at a given point. In the study of

dynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s, linearization is a method for assessing the local

stabilityIn mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

of an

equilibrium point of a

systemSystem is a set of interacting or interdependent components forming an integrated whole....

of nonlinear

differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s or discrete

dynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s. This method is used in fields such as

engineeringEngineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

,

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

,

economicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, and

ecologyEcology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...

.

## Linearization of a function

Linearizations of a

functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

are

linesIn mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....

— ones that are usually used for purposes of calculation. Linearization is an effective method for approximating the output of a function

at any

based on the value and

slopeIn mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

of the function at

, given that f(x) is continuous on

(or

) and that

is close to

. In, short, linearization approximates the output of a function near

.

For example, you might know that

. However, without a calculator, what would be a good approximation of

?

For any given function

,

can be approximated if it is near a known continuous point. The most basic requisite is that, where

is the linearization of f(x) at x = a,

. The point-slope form of an equation forms an equation of a line, given a point

and slope

. The general form of this equation is:

.

Using the point

,

becomes

. Because continuous functions are

locally linearLocal linearity is a property of functions that says — roughly — that the more you zoom in on a point on the graph of the function , the more the graph will look like a straight line. More precisely, a function is locally linear at a point if and only if a tangent line exists at that point...

, the best slope to substitute in would be the slope of the line

tangentIn geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

to

at

.

While the concept of local linearity applies the most to points arbitrarily close to

, those relatively close work relatively well for linear approximations. The slope

should be, most accurately, the slope of the tangent line at

.

Visually, the accompanying diagram shows the tangent line of

at x. At

, where

is any small positive or negative value, f(x+h) is very nearly the value of the tangent line at the point

.

The final equation for the linearization of a function at

is:

For

,

. The

derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

of

is

, and the slope of

at

is

.

## Example

To find

, we can use the fact that

. The linearization of

at

is

, because the function

defines the slope of the function

at

. Substituting in

, the linearization at 4 is

. In this case

, so

is approximately

. 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

at a point

is:

The general equation for the linearization of a multivariable function

at a point

is:

where

is the vector of variables, and

is the linearization point of interest

.

## Uses of linearization

Linearization makes it possible to use tools for studying

linear systemA linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case....

s 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

,

the linearized system can be written as

where

is the point of interest and

is the

JacobianIn vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space...

of

evaluated at

.

### Stability analysis

In

stabilityIn mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

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 pointIn mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...

. Any

complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

eigenvalues will appear in

complex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

pairs and indicate a

spiralIn mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...

.

### Microeconomics

In

microeconomicsMicroeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...

,

decision rulesA set of decision rules is the verbal equivalent of a graphical decision tree, which specifies class membership based on a hierarchical sequence of decisions...

may be approximated under the state-space approach to linearization. Under this approach, the Euler equations of the

utility maximization problemIn microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?" It is a type of optimal decision problem.-Basic setup:...

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
In computational mechanics, a tangent stiffness matrix is a matrix that describes the stiffness of a system in response to small changes in configuration...

- Stability derivatives
Stability Derivatives, and also Control Derivatives, are measures of how particular forces and moments on an aircraft change as other parameters related to stability change . For a defined "trim" flight condition, changes and oscillations occur in these parameters...

- Linearization theorem
- Taylor approximation

### Linearization tutorials