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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 and its applications, linearization refers to finding the linear approximation
Linear approximation

In mathematics, a linear approximation is an approximation of a general function using a linear function ....
 to a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 at a given point. In the study of dynamical system
Dynamical system

The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
s, linearization is a method for assessing the local stability
Stability theory

In mathematics, stability theory deals with the stability of solutions for differential equations and dynamical systems....
 of an equilibrium point
Equilibrium point

In mathematics, the point is an equilibrium point for the differential equationif for all .Similarly, the point is an equilibrium point for the difference equation...
 of a system
System

System is a set of interacting or interdependent entities, real or abstract, forming an integrated whole.The concept of an "integrated whole" can also be stated in terms of a system embodying a set of relationships which are differentiated from relationships of the set to other elements, and from relationships between an element of the se...
 of nonlinear differential equation
Differential equation

A differential equation is a mathematics equation for an unknown function of one or several variable that relates the values of the function itself and its derivatives of various orders....
s or discrete dynamical system
Dynamical system

The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
s. This method is used in fields such as engineering
Engineering

Engineering is the discipline and profession of applying Technology and science knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and process that safely realize a desired objective and meet specified criteria....
, physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, economics
Economics

File:Ballard Farmers' Market - vegetables.jpgEconomics is the Social sciences that studies the Production theory basics, Distribution , and Consumption of Good and Service ....
, and ecology
Ecology

Ecology is the science study of the distribution and Abundance of life and the interactions between organisms and their nature environment ....
.

arizations of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 are lines
Linear function

In mathematics, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; or a map between two vector spaces that preserves vector addition and scalar multiplication....
 — ones that are usually used for purposes of calculation.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 and its applications, linearization refers to finding the linear approximation
Linear approximation

In mathematics, a linear approximation is an approximation of a general function using a linear function ....
 to a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 at a given point. In the study of dynamical system
Dynamical system

The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
s, linearization is a method for assessing the local stability
Stability theory

In mathematics, stability theory deals with the stability of solutions for differential equations and dynamical systems....
 of an equilibrium point
Equilibrium point

In mathematics, the point is an equilibrium point for the differential equationif for all .Similarly, the point is an equilibrium point for the difference equation...
 of a system
System

System is a set of interacting or interdependent entities, real or abstract, forming an integrated whole.The concept of an "integrated whole" can also be stated in terms of a system embodying a set of relationships which are differentiated from relationships of the set to other elements, and from relationships between an element of the se...
 of nonlinear differential equation
Differential equation

A differential equation is a mathematics equation for an unknown function of one or several variable that relates the values of the function itself and its derivatives of various orders....
s or discrete dynamical system
Dynamical system

The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
s. This method is used in fields such as engineering
Engineering

Engineering is the discipline and profession of applying Technology and science knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and process that safely realize a desired objective and meet specified criteria....
, physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, economics
Economics

File:Ballard Farmers' Market - vegetables.jpgEconomics is the Social sciences that studies the Production theory basics, Distribution , and Consumption of Good and Service ....
, and ecology
Ecology

Ecology is the science study of the distribution and Abundance of life and the interactions between organisms and their nature environment ....
.

Linearization of a function

Linearizations of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 are lines
Linear function

In mathematics, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; or 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 slope
Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
 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
Linear equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.Linear equations can have one or more variables....
 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 linear
Local linearity

Local linearity is a property of functions that says ? roughly ? that if you zoom in on a point on the graph of the function , the graph will eventually look like a line ....
, the best slope to substitute in would be the slope of the line tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 to at .

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

In mathematics, the limit of a function is a fundamental concept in calculus and mathematical analysis concerning the behavior of that Function near a particular independent variable....
 to , those relatively close work relatively well for linear approximations. After all, a 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 derivative
Derivative

In 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 a quantity is changing at a given point....
 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 . Plugging 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.

Uses of linearization


Linearization makes it possible to use tools for studying linear system
Linear system

A 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 Jacobian
Jacobian

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.In algebraic geometry the Jacobian of a algebraic curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded....
 of evaluated at .

Stability analysis

In stability
Stability theory

In mathematics, stability theory deals with the stability of solutions for differential equations and dynamical systems....
 analysis, one can use the eigenvalues of the Jacobian matrix evaluated at an equilibrium point
Equilibrium point

In mathematics, the point is an equilibrium point for the differential equationif for all .Similarly, the point is an equilibrium point for the difference equation...
 to determine the nature of that equilibrium. If all the eigenvalues are positive
Negative and non-negative numbers

A negative number is a real number that is inequality 0 , such as -3. A positive number is a real number that is greater than zero, such as 2....
, the equilibrium is unstable; if they are all negative the equilibrium is stable; and if the values are of mixed signs, the equilibrium is a saddle point
Saddle point

In mathematics, a saddle point is a point in the domain of a function of two variables which is a stationary point but not a local extremum....
. Any complex
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 eigenvalues will appear in complex conjugate
Complex conjugate

In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number...
 pairs and indicate a spiral
Spiral

In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point....
.

Microeconomics

In microeconomics
Microeconomics

Microeconomics is a branch of economics that studies how individuals, households and firms and some states make decisions to allocate limited resources, typically in markets where goods or services are being bought and sold....
, decision rules
Decision rules

A 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
Euler equations

In fluid dynamics, the Euler equations govern inviscid flow. They correspond to the Navier-Stokes equations with zero viscosity and heat conduction terms....
 of the utility maximization problem
Utility maximization problem

In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?"...
 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
    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

    Stability derivatives are a means of linearising the equations of motion of an atmospheric flight vehicle so that conventional control engineering methods may be applied to assess their stability....
  • Linearization theorem
  • Taylor approximation