NewtonRaphson

The Newton-Raphson method is an efficient iterative technique for locating zeros of a real-valued function f. It requires a differentiable function near the desired root α and an initial guess x0 close to α. The iteration is x_{n+1} = x_n - f(x_n)/f'(x_n). Geometrically, it uses the tangent line to f at x_n, with the root approximated by the x-intercept of the tangent.

Convergence properties: near a simple root where f'(α) ≠ 0 and with a sufficiently good initial guess, the method exhibits quadratic convergence, doubling accuracy roughly each iteration. However, if f' is zero at the root or near singularities, convergence can be slow or fail. Poor initial guesses can lead to divergence or convergence to a different root. The method can be extended to systems by solving F: R^n -> R^n with x_{n+1} = x_n - J_F(x_n)^{-1} F(x_n), where J_F is the Jacobian matrix.

The method is named for Isaac Newton and Joseph Raphson, who developed early versions in the 17th century. It is widely used in numerical analysis, engineering, and applied sciences to find polynomial and transcendental roots, often with safeguards such as damping (using x_{n+1} = x_n - λ f/f'), or line search, to improve robustness. Practical stopping criteria include |f(x_n)| < tol or |x_{n+1}-x_n| < tol or a maximum number of iterations.