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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.
Applying the Banach fixed-point theorem shows that the fixed point π is the unique fixed point on the interval, allowing for fixed-point iteration to be used. For example, the value 3 may be chosen to start the fixed-point iteration, as 3 π / 4 ≤ 3 ≤ 5 π / 4 {\displaystyle 3\pi /4\leq 3\leq 5\pi /4} .
3 Example of Picard iteration. 4 Example of non-uniqueness. 5 Detailed proof. ... In this context, this fixed-point iteration method is known as Picard iteration. Set
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...
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
X is a fixed-point of if and only if x is a root of , and x is an ε-residual fixed-point of if and only if x is an ε-root of . Chen and Deng [ 18 ] show that the discrete variants of these problems are computationally equivalent: both problems require Θ ( n d − 1 ) {\displaystyle \Theta (n^{d-1})} function evaluations.