Linearization

Linearization is a method of approximating a nonlinear function or system by a linear one in the vicinity of a chosen point. In one variable, the linearization of a differentiable function f at x0 is the first-order Taylor expansion: L(x) = f(x0) + f'(x0)(x − x0). In several variables, the linearization uses the Jacobian matrix Jf(x0): L(x) = f(x0) + Jf(x0)(x − x0). This yields the tangent-line or tangent-hyperplane approximation to f near x0.

The linearized model is accurate close to x0 because the remaining terms are of higher order in (x − x0).

Linearization has wide applications: in solving nonlinear equations, in stability analysis of dynamical systems, in control theory for design around an operating point, and in numerical methods such as Newton’s method, where the linearization provides update steps. In physics and biology, small-signal analysis and local behavior near equilibria rely on linear models.

Limitations: it describes local, not global behavior; it may misrepresent nonlinear phenomena like bifurcations or limit cycles; the linear model may alter stability properties; requires differentiability and a valid neighborhood.

Example: f(x) = sin(x) around x0 = 0 has linearization L(x) = x, since sin(0)=0 and cos(0)=1.