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In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
around the point of interest. In the study of
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s. This method is used in fields such as
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analy ...
, and
ecology Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...
.


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 y = f(x) at any x = a based on the value and
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used ...
of the function at x = b, given that f(x) is differentiable on
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> (or
, a The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
/math>) and that a is close to b. In short, linearization approximates the output of a function near x = a. For example, \sqrt = 2. However, what would be a good approximation of \sqrt = \sqrt? For any given function y = f(x), f(x) can be approximated if it is near a known differentiable point. The most basic requisite is that L_a(a) = f(a), where L_a(x) 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 (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 differentiable 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. 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 In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of f(x) is f'(x), and the slope of f(x) at a is f'(a).


Example

To find \sqrt, we can use the fact that \sqrt = 2. The linearization of f(x) = \sqrt at x = a is y = \sqrt + \frac(x - a), because the function f'(x) = \frac defines the slope of the function f(x) = \sqrt at x. Substituting in a = 4, the linearization at 4 is y = 2 + \frac. In this case x = 4.001, so \sqrt is approximately 2 + \frac = 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. \_ (x - a) + \left. \_ (y - b) The general equation for the linearization of a multivariable function f(\mathbf) at a point \mathbf is: :f() \approx f() + \left. \_ \cdot ( - ) where \mathbf is the vector of variables, and \mathbf 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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
around the point of interest. For a system defined by the equation :\frac = \mathbf(\mathbf,t), the linearized system can be written as :\frac \approx \mathbf(\mathbf,t) + D\mathbf(\mathbf,t) \cdot (\mathbf - \mathbf) where \mathbf is the point of interest and D\mathbf(\mathbf,t) is the \mathbf-
Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: *Jacobian matrix and determinant *Jacobian elliptic functions *Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähler m ...
of \mathbf(\mathbf,t) evaluated at \mathbf.


Stability analysis

In stability analysis of autonomous systems, one can use the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium. This is the content of the linearization theorem. For time-varying systems, the linearization requires additional justification.


Microeconomics

In microeconomics, decision rules may be approximated under the state-space approach to linearization.Moffatt, Mike. (2008) About.com
State-Space Approach
' Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.
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.


Optimization

In
mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
, cost functions and non-linear components within can be linearized in order to apply a linear solving method such as the Simplex algorithm. The optimized result is reached much more efficiently and is deterministic as a global optimum.


Multiphysics

In multiphysics systems—systems involving multiple physical fields that interact with one another—linearization with respect to each of the physical fields may be performed. This linearization of the system with respect to each of the fields results in a linearized monolithic equation system that can be solved using monolithic iterative solution procedures such as the Newton–Raphson method. Examples of this include MRI scanner systems which results in a system of electromagnetic, mechanical and acoustic fields.


See also

* Linear stability *
Tangent stiffness matrix Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science (also called scientific computing) as a "third ...
* Stability derivatives * Linearization theorem * Taylor approximation * Functional equation (L-function)


References


External links


Linearization tutorials


Linearization for Model Analysis and Control Design
{{Authority control Differential calculus Dynamical systems Approximations