Linearization Theorem
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, linearization is finding the
linear approximation In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving o ...
to a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
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 seri ...
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. Examples include the mathematical models that describe the swinging of a ...
s, linearization is a method for assessing the local
stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural sta ...
of an
equilibrium point In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. Formal definition The point \tilde\in \mathbb^n is an equilibrium point for the differential equation :\frac = \ ...
of a
system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
of
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s 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. Examples include the mathematical models that describe the swinging of a ...
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 r ...
,
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
, 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 overlaps wi ...
.


Linearization of a function

Linearizations of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
are
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
—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 use ...
of the function at x = b, given that f(x) is differentiable on , b/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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
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. F ...
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 system In systems theory, 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 nonlinear case. As a mathematical abstraction o ...
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 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 seri ...
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 Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural sta ...
analysis of
autonomous systems An autonomous robot is a robot that acts without recourse to human control. The first autonomous robots environment were known as Elmer and Elsie, which were constructed in the late 1940s by W. Grey Walter. They were the first robots in history t ...
, 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 denoted b ...
s of the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
evaluated at a
hyperbolic equilibrium point In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbo ...
to determine the nature of that equilibrium. This is the content of the
linearization theorem 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 around the point of interest. In the study of dynamical systems, lineari ...
. For time-varying systems, the linearization requires additional justification.


Microeconomics

In
microeconomics Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...
,
decision rule In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics, and are closely related to the concept of a strategy (game theory ...
s may be approximated under the state-space approach to linearization.Moffatt, Mike. (2008)
About.com Dotdash Meredith (formerly About.com) is an American digital media company based in New York City. The company publishes online articles and videos about various subjects across categories including health, home, food, finance, tech, beauty, ...

State-Space Approach
' Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.
Under this approach, the
Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are 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 ...
of the
utility maximization problem Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my u ...
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 In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are n ...
. The optimized result is reached much more efficiently and is deterministic as a
global optimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
.


Multiphysics

In
multiphysics In computational modelling, multiphysics simulation (often shortened to simply "multiphysics") is defined as the simultaneous simulation of different aspects of a physical system or systems and the interactions among them. For example, simultaneous ...
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 In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-va ...
. Examples of this include
MRI scanner The physics of magnetic resonance imaging (MRI) concerns fundamental physical considerations of MRI techniques and technological aspects of MRI devices. MRI is a medical imaging technique mostly used in radiology and nuclear medicine in order ...
systems which results in a system of electromagnetic, mechanical and acoustic fields.


See also

*
Linear stability In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the for ...
*
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 w ...
*
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 (parameters such as airspeed, altitude, angle of attack, etc.). For a def ...
*
Linearization theorem 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 around the point of interest. In the study of dynamical systems, lineari ...
*
Taylor approximation In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
*
Functional equation (L-function) In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which i ...


References


External links


Linearization tutorials


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