Newton-Raphson

The Newton-Raphson method (also called the Newton method) is an iterative technique for finding a root of a nonlinear equation f(x) = 0. Starting from an initial guess x_0, the method replaces the function locally by its tangent line at the current estimate and uses the x-intercept of that tangent as the next estimate. This gives the update x_(n+1) = x_n - f(x_n) / f'(x_n), where f'(x_n) is the derivative of the function at x_n. Geometrically, each iteration moves the estimate toward the point where the tangent line crosses the x-axis. When the initial guess is sufficiently close to a simple root and f'(x) is not near zero, convergence is quadratic: the number of correct digits roughly doubles each iteration, so the error at step n+1 is proportional to the square of the error at step n. This makes Newton-Raphson much faster than the bisection method near the root, but the method is not globally convergent. It can diverge, oscillate, or jump far from the root if the initial guess is poor, if f'(x_n) is close to zero (flat tangent), or if the function has inflection points or multiple roots nearby. A derivative that changes sign or a root with multiplicity greater than one slows convergence to linear. Practical implementations combine a bracketing method such as bisection with Newton-Raphson: bisection is used to get close to the root, then Newton-Raphson is used for fast final convergence. Stopping criteria include |f(x_n)| < tolerance, |x_(n+1) - x_n| < tolerance, or a maximum iteration count. In engineering, Newton-Raphson is widely used to solve nonlinear equations of state, power flow equations in electrical networks, nonlinear finite element equilibrium equations, and implicit time integration in dynamics. A modified Newton-Raphson method reuses the same Jacobian or derivative over several iterations to reduce computational cost at the expense of slower convergence.