Linearization

Last updated

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, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. [1] This method is used in fields such as engineering, physics, economics, and ecology.

Contents

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 at any based on the value and slope of the function at , given that is differentiable on (or ) and that is close to . In short, linearization approximates the output of a function near .

For example, . However, what would be a good approximation of ?

For any given function , can be approximated if it is near a known differentiable point. The most basic requisite is that , where is the linearization of at . 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 differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent 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 .

An approximation of f(x) = x at (x, f(x)) Tangent-calculus.svg
An approximation of f(x) = x at (x, f(x))

Visually, the accompanying diagram shows the tangent line of at . At , where is any small positive or negative value, 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 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, is the gradient, and is the linearization point of interest . [2]

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

,

the linearized system can be written as

where is the point of interest and is the -Jacobian of evaluated at .

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. [3]

Microeconomics

In microeconomics, decision rules may be approximated under the state-space approach to linearization. [4] Under this approach, the Euler equations of the utility maximization problem are linearized around the stationary steady state. [4] A unique solution to the resulting system of dynamic equations then is found. [4]

Optimization

In mathematical optimization, 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. [5]

See also

Related Research Articles

In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

The derivative is a fundamental tool of calculus that quantifies the sensitivity of 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 the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation.

<span class="mw-page-title-main">Ellipse</span> Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

<span class="mw-page-title-main">Gradient</span> Multivariate derivative (mathematics)

In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field whose value at a point gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of . If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function may be defined by:

<span class="mw-page-title-main">Hyperbola</span> Plane curve: conic section

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

<span class="mw-page-title-main">Tangent</span> In mathematics, straight line touching a plane curve without crossing it

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, 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 precisely, a straight line is tangent to the curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

<span class="mw-page-title-main">Differential calculus</span> Area of mathematics; subarea of calculus

In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

<span class="mw-page-title-main">Ellipsoid</span> Quadric surface that looks like a deformed sphere

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints. It is named after the mathematician Joseph-Louis Lagrange.

<span class="mw-page-title-main">Heat equation</span> Partial differential equation describing the evolution of temperature in a region

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.

In the mathematical field of differential geometry, a metric tensor is an additional structure on a manifold M that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at p, and a metric field on M consists of a metric tensor at each point p of M that varies smoothly with p.

<span class="mw-page-title-main">Spherical harmonics</span> Special mathematical functions defined on the surface of a sphere

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics contains a list of common spherical harmonics.

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form

In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the amplitude or the phase is taken to be changing slowly.

<span class="mw-page-title-main">Linear approximation</span> Approximation of a function by its tangent line at a point

In 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.

<span class="mw-page-title-main">Arc length</span> Distance along a curve

Arc length is the distance between two points along a section of a curve.

The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz.

<span class="mw-page-title-main">Pendulum (mechanics)</span> Free swinging suspended body

A pendulum is a body suspended from a fixed support such that it freely swings back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back towards the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.

In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In a few important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

References

  1. The linearization problem in complex dimension one dynamical systems at Scholarpedia
  2. Linearization. The Johns Hopkins University. Department of Electrical and Computer Engineering Archived 2010-06-07 at the Wayback Machine
  3. Leonov, G. A.; Kuznetsov, N. V. (2007). "Time-Varying Linearization and the Perron effects". International Journal of Bifurcation and Chaos . 17 (4): 1079–1107. Bibcode:2007IJBC...17.1079L. doi:10.1142/S0218127407017732.
  4. 1 2 3 Moffatt, Mike. (2008) About.com State-Space Approach Archived 2016-03-04 at the Wayback Machine Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.
  5. Bagwell, S.; Ledger, P. D.; Gil, A. J.; Mallett, M.; Kruip, M. (2017). "A linearised hp–finite element framework for acousto-magneto-mechanical coupling in axisymmetric MRI scanners". International Journal for Numerical Methods in Engineering. 112 (10): 1323–1352. Bibcode:2017IJNME.112.1323B. doi: 10.1002/nme.5559 .

Linearization tutorials