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An illustration of Newton's method. In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.
Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function, which are solutions to the equation =.
Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method; its order of convergence is usually quadratic whereas the bisection method's is linear. Newton's method is also important because it readily generalizes to higher-dimensional problems.
Sequential quadratic programming (SQP) — replace problem by a quadratic programming problem, solve that, and repeat; Newton's method in optimization. See also under Newton algorithm in the section Finding roots of nonlinear equations; Nonlinear conjugate gradient method; Derivative-free methods
Anderson's iterative method, which uses a least squares approach to the Jacobian. [9] Schubert's or sparse Broyden algorithm – a modification for sparse Jacobian matrices. [10] The Pulay approach, often used in density functional theory. [11] [12] A limited memory method by Srivastava for the root finding problem which only uses a few recent ...
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
The main advantage of Steffensen's method is that it has quadratic convergence [1] like Newton's method – that is, both methods find roots to an equation just as 'quickly'. In this case quickly means that for both methods, the number of correct digits in the answer doubles with each step.
Quasi-Newton methods are a generalization of the secant method to find the root of the first derivative for multidimensional problems. In multiple dimensions the secant equation is under-determined , and quasi-Newton methods differ in how they constrain the solution, typically by adding a simple low-rank update to the current estimate of the ...