NewtonRaphsonIteration

Newton-Raphson iteration, commonly called Newton's method, is an iterative procedure for finding roots of a real-valued function f. Starting from an initial guess x0, it generates a sequence by x_{n+1} = x_n - f(x_n)/f'(x_n), i.e., the x-intercept of the tangent to the graph at x_n.

Convergence: If f has a simple root r with f(r) = 0 and f'(r) ≠ 0, and the starting value x0 is sufficiently close to r, the method converges quadratically, meaning the error decreases roughly as the square of the previous error with each iteration. However, there is no general guarantee of global convergence; if f'(x_n) is zero near the current estimate or the initial guess lies far from any root, the sequence may fail to converge or diverge.

Algorithm: Start with x0. Then repeat: compute f(x_n) and f'(x_n). If f'(x_n) = 0, stop or modify. Update x_{n+1} = x_n - f(x_n)/f'(x_n). Check convergence criteria, such as |f(x_n)| < tol or |x_{n+1} - x_n| < tol.

Extensions and variants: The method extends to systems of equations f: R^n → R^n using the Jacobian J_f(x_n), with x_{n+1} = x_n - J_f(x_n)^{-1} f(x_n). For multiple roots, convergence can degrade; remedies include multiplicity-adjusted Newton methods or line-search or damping strategies to improve global behavior.

History and usage: The method is named after Isaac Newton and Joseph Raphson; Newton introduced the idea, and Raphson popularized it in the 17th century. It remains a foundational tool in numerical analysis and is widely used in science and engineering to solve nonlinear equations, often as a component of larger algorithms.