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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 =.
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.
The iteration stops when a fixed point of the auxiliary function is reached to the desired precision, i.e., when a new computed value is sufficiently close to the preceding ones. Newton's method (and similar derivative-based methods)
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 Jacobian: Broyden's method — uses a rank-one update for the Jacobian
The first problem solved by Newton with the Newton-Raphson-Simpson method was the polynomial equation =. He observed that there should be a solution close to 2. Replacing y = x + 2 transforms the equation into = = + + +.
Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the ...
In numerical analysis, a quasi-Newton method is an iterative numerical method used either to find zeroes or to find local maxima and minima of functions via an iterative recurrence formula much like the one for Newton's method, except using approximations of the derivatives of the functions in place of exact derivatives.
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the i-th approximation (called an "iterate") is derived from the previous ones.