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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, linearization (
British English British English is the set of Variety (linguistics), varieties of the English language native to the United Kingdom, especially Great Britain. More narrowly, it can refer specifically to the English language in England, or, more broadly, to ...
: linearisation) is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s, linearization is a method for assessing the local stability of an equilibrium point of a
system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...
of nonlinear differential equations or discrete
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s. This method is used in fields such as
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
,
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
,
economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...
, and
ecology Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ...
.


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 of the function at x = b, given that f(x) is differentiable on , b/math> (or , a/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 is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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, is the gradient, 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 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 of \mathbf(\mathbf,t) evaluated at \mathbf.


Stability analysis

In stability analysis of autonomous systems, one can use the eigenvalues 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 Microeconomics is a branch of economics that studies the behavior of individuals and Theory of the firm, firms in making decisions regarding the allocation of scarcity, scarce resources and the interactions among these individuals and firms. M ...
, 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 In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
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 criteria, from some set of available alternatives. It is generally divided into two subfiel ...
, 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 * Stability derivatives * Linearization theorem * Taylor approximation * Functional equation (L-function) * Quasilinearization


References


External links


Linearization tutorials


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