Linearization

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. This method is used in fields such as engineering, physics, economics, and ecology.
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 formula_1 at any formula_2 based on the value and slope of the function at formula_3, given that formula_4 is differentiable on formula_5 (or formula_6) and that formula_7 is close to formula_8. In short, linearization approximates the output of a function near formula_2.
For example, formula_10. However, what would be a good approximation of formula_11?
For any given function formula_1, formula_4 can be approximated if it is near a known differentiable point. The most basic requisite is that formula_14, where formula_15 is the linearization of formula_4 at formula_2. The point-slope form of an equation forms an equation of a line, given a point formula_18 and slope formula_19. The general form of this equation is: formula_20.
Using the point formula_21, formula_15 becomes formula_23. Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to formula_4 at formula_2.
While the concept of local linearity applies the most to points arbitrarily close to formula_2, those relatively close work relatively well for linear approximations. The slope formula_19 should be, most accurately, the slope of the tangent line at formula_2.
Visually, the accompanying diagram shows the tangent line of formula_4 at formula_30. At formula_31, where formula_32 is any small positive or negative value, formula_31 is very nearly the value of the tangent line at the point formula_34.
The final equation for the linearization of a function at formula_2 is:
formula_36
For formula_2, formula_38. The derivative of formula_4 is formula_40, and the slope of formula_4 at formula_7 is formula_43.
Example.
To find formula_44, we can use the fact that formula_10. The linearization of formula_46 at formula_2 is formula_48, because the function formula_49 defines the slope of the function formula_46 at formula_30. Substituting in formula_52, the linearization at 4 is formula_53. In this case formula_54, so formula_44 is approximately formula_56. 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 formula_57 at a point formula_58 is:
formula_59
The general equation for the linearization of a multivariable function formula_60 at a point formula_61 is:
formula_62
where formula_63 is the vector of variables, and formula_61 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
the linearized system can be written as
where formula_67 is the point of interest and formula_68 is the Jacobian of formula_69 evaluated at formula_67.
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, decision rules may be approximated under the state-space approach to linearization. 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, 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.