Linearize

Linearize means to approximate a nonlinear object by a linear one in a neighborhood of a reference point. In calculus, the linear approximation is the first-order term of the Taylor expansion. For a differentiable scalar function f of one variable, f(x) ≈ f(x0) + f'(x0)(x - x0). For a vector-valued function f: R^n -> R^m, the approximation near x0 uses the Jacobian matrix J_f(x0), giving f(x) ≈ f(x0) + J_f(x0)(x - x0). The linear map J_f(x0) is the tangent map to f at x0.

This local linear model is used to analyze and design nonlinear systems at an operating point. In dynamical systems, linearization around an equilibrium x* uses the Jacobian to assess local stability via eigenvalues. In control theory, a nonlinear system is often approximated by a linear state-space model dot x = A x + B u, y = C x + D u, obtained from A = ∂f/∂x and related partials evaluated at a nominal point. In numerical methods, linearization underpins Newton-Raphson and Gauss-Newton updates, solving F(x) ≈ F(x_k) + J_F(x_k)(x - x_k) to progress toward a solution.

Limitations: the approximation is valid only near the reference point and may fail farther away. Piecewise or more global nonlinear methods may be necessary. In statistics and data analysis, linearization can also refer to transforming a nonlinear relationship into a linear form to facilitate linear regression, for example through logarithmic or other monotone transformations.