Newtonlike

Newtonlike refers to a broad family of iterative methods for solving nonlinear equations f(x)=0 or for unconstrained optimization that generalize Newton's method. In the classic Newton method, one forms the linearization f(x_k) + J_f(x_k)(x - x_k) and solves for the Newton step s_k = -J_f(x_k)^{-1} f(x_k), obtaining x_{k+1} = x_k + s_k. Newtonlike methods retain the core idea of using local linear models to drive iterates but allow variations that enhance robustness, reduce computational cost, or adapt to large-scale problems. These variations include using approximate Jacobians or Hessians, solving the linear system inexactly, adding damping, or employing line searches or trust-region strategies to control step size.

Common variants include inexact Newton methods, which solve the linear system only approximately; damped Newton or line-search Newton methods, which scale the update to improve global convergence; quasi-Newton methods like BFGS and L-BFGS that build an approximate Hessian without exact second derivatives; Gauss-Newton methods tailored to nonlinear least squares; and Levenberg-Marquardt methods that interpolate between Newton steps and gradient descent.

Convergence properties depend on the method and problem. Under standard regularity assumptions and near a simple root, Newtonlike methods often exhibit quadratic convergence when the Jacobian or Hessian is non-singular. Global convergence generally requires globalization techniques such as line search or trust regions, and the use of inexact solves or imperfect Hessians may affect convergence speed. They are widely used in engineering, physics, computer vision, and machine learning for solving systems of nonlinear equations and for optimizing high-dimensional objectives, especially when exact second-order information is expensive to obtain.

See also: Newton's method; quasi-Newton methods; Gauss-Newton; Levenberg-Marquardt; inexact Newton method; optimization.