NewtonRaphsonbased

NewtonRaphsonbased is a term used to describe methods and algorithms that rely on the Newton-Raphson iteration as the core computational step for solving equations or optimizing functions.

In the univariate case, given a differentiable function f, the iteration is x_{n+1} = x_n - f(x_n)/f'(x_n). This simple update seeks a root of f by using the tangent line at each current estimate.

In multiple dimensions, the approach solves F(x) = 0 using x_{k+1} = x_k - J_F(x_k)^{-1} F(x_k), where F is a vector-valued function and J_F is its Jacobian. This extends the idea to systems of nonlinear equations and can be applied to unconstrained optimization by locating stationary points through solving ∇f(x)=0.

Convergence is quadratic near a simple root, provided the derivative (or Jacobian) is well-behaved and the initial guess is reasonably close to a solution. If the derivative vanishes or the problem is ill-conditioned, convergence can fail or be slow. To improve robustness, damped Newton, line-search, or trust-region variants are often used to control step size and enhance global convergence.

Applications in science and engineering include root finding, solving nonlinear systems, computing equilibria, and optimization problems where stationary points are determined by solving derivative equations. The method is sensitive to scaling and noise, so practical implementations employ preconditioning and clear stopping criteria. Historically, Newton and Raphson developed the method, which remains a fundamental tool in numerical analysis and is built into most scientific computing environments. Related approaches include the secant method and Gauss-Newton for specific problem classes.