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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 =. However, to optimize a twice-differentiable f {\displaystyle f} , our goal is to find the roots of f ′ {\displaystyle f'} .
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 assumes the function f to have a continuous derivative. 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 ...
It has similarities with Quasi-Newton methods. Conditional gradient method (Frank–Wolfe) for approximate minimization of specially structured problems with linear constraints, especially with traffic networks. For general unconstrained problems, this method reduces to the gradient method, which is regarded as obsolete (for almost all problems).
Newton's method — based on linear approximation around the current iterate; quadratic convergence Kantorovich theorem — gives a region around solution such that Newton's method converges; Newton fractal — indicates which initial condition converges to which root under Newton iteration; Quasi-Newton method — uses an approximation of the ...
In a quasi-Newton method, such as that due to Davidon, Fletcher and Powell or Broyden–Fletcher–Goldfarb–Shanno (BFGS method) an estimate of the full Hessian is built up numerically using first derivatives only so that after n refinement cycles the method closely approximates to Newton's method in performance. Note that quasi-Newton ...
Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization which may be considered a quasi-Newton method.SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable, but not necessarily convex.
These minimization problems arise especially in least squares curve fitting. The LMA interpolates between the Gauss–Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. For well-behaved functions and ...