linearisation

Linearisation, or linearization, is the process of approximating a nonlinear system, function, or relationship by a linear one near a chosen reference point. It is typically achieved through a first-order Taylor expansion and is used to study local behavior or to simplify calculations.

In one dimension, if a function f is differentiable at a, the linearisation is the tangent line L(x) = f(a) + f'(a)(x − a). The approximation is accurate for x close to a; the error is of order (x − a)^2. In several variables, for a differentiable function f: R^n → R^m, the linearisation at a is L(x) = f(a) + J_f(a)(x − a), where J_f(a) is the Jacobian matrix. As x approaches a, f(x) ≈ L(x).

Applications of linearisation are widespread. It is used for small-signal analysis in electrical engineering and control theory, and for studying local behavior of dynamical systems near equilibria. Numerical methods such as the Newton–Raphson method rely on linear tangent approximations to find zeros. In differential equations, linearisation around an operating point or equilibrium can simplify the analysis of stability and response to perturbations.

Limitations should be noted. Linearisation provides a local approximation and may be inaccurate for points far from the reference or for strongly nonlinear systems. Neglected higher-order terms can lead to incorrect conclusions about global behavior, and in some cases piecewise linear models or higher-order methods are preferred.