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In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
In mathematics, Anderson acceleration, also called Anderson mixing, is a method for the acceleration of the convergence rate of fixed-point iterations.Introduced by Donald G. Anderson, [1] this technique can be used to find the solution to fixed point equations () = often arising in the field of computational science.
A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition. [1] For example, the Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, fixed-point iteration will always converge to a fixed point.
For example, the secant method, ... A practical method to calculate the order of convergence for a sequence generated by a fixed point iteration is ... Download as PDF;
3 Example of Picard iteration. ... In this context, this fixed-point iteration method is known as Picard iteration. Set ... (PDF). Ordinary ...
Many methods compute subsequent values by evaluating an auxiliary function on the preceding values. The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points and for converging rapidly to these fixed points.
Fixed-point iteration; 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
Mathematical methods relating to successive approximation include: Babylonian method, for finding square roots of numbers [3] Fixed-point iteration [4] Means of finding zeros of functions: Halley's method; Newton's method; Differential-equation matters: Picard–Lindelöf theorem, on existence of solutions of differential equations