Linearization
In mathematics, linearization is finding the linear approximation to a function at a given point. 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 .
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, 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 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
- Linear stability
- Tangent stiffness matrix
- Stability derivatives
- Linearization theorem
- Taylor approximation
- Functional equation (L-function)
References
- ^ The linearization problem in complex dimension one dynamical systems at Scholarpedia
- ^ Linearization. The Johns Hopkins University. Department of Electrical and Computer Engineering Archived 2010-06-07 at the Wayback Machine.
- ^ G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron effects, International Journal of Bifurcation and Chaos, Vol. 17, No. 4, 2007, pp. 1079-1107
- ^ ^{a} ^{b} ^{c} Moffatt, Mike. (2008) About.com State-Space Approach Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.
- ^ S. Bagwell, P.D. Ledger, A.J. Gil, M. Mallett, M. Kruip, A linearised hp–finite element framework for acousto-magneto-mechanical coupling in axisymmetric MRI scanners, DOI: 10.1002/nme.5559
External links
Linearization tutorials
- Linearization for Model Analysis and Control Design