NewtonSchritten

NewtonSchritten, commonly referred to as Newton steps, denotes a class of iterative methods for solving nonlinear equations and optimization problems that build on Newton's method by applying successive corrections toward a solution.

In the standard root-finding form, a differentiable F: R^n -> R^n is given. The Newton step s_k solves J_F(x_k) s_k = -F(x_k) and the iterate is x_{k+1} = x_k + s_k. For optimization, with objective f and gradient g = ∇f, F is g and J_F is ∇^2 f, yielding x_{k+1} = x_k - ∇^2 f(x_k)^{-1} ∇f(x_k).

Globalization techniques such as line search or trust region modify the step by choosing a scalar α_k in (0,1] so that a decrease condition is satisfied, producing damped Newton steps.

Advantages and limitations: near a well-behaved solution NewtonSchritten converge quadratically; they require evaluating and inverting a Jacobian or Hessian, which can be expensive or problematic if it is singular or ill-conditioned. Variants include inexact Newton methods and quasi-Newton approaches to reduce cost.

Applications include solving systems of nonlinear equations, nonlinear data fitting, and various problems in engineering and physical sciences where reliable root finding or optimization is needed.

Origin and usage: The term reflects the German-language emphasis on the stepwise nature of the method and is encountered in mathematical texts and software documentation describing Newton-based solvers.