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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 .
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.
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.
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 ...
Attractive fixed points [ edit ] If an equation can be put into the form f ( x ) = x , and a solution x is an attractive fixed point of the function f , then one may begin with a point x 1 in the basin of attraction of x , and let x n +1 = f ( x n ) for n ≥ 1, and the sequence { x n } n ≥ 1 will converge to the solution x .
Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an attractive fixed point. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an unstable fixed point. [11]
% The fixed point iteration function is assumed to be input as an % inline function. % This function will calculate and return the fixed point, p, % that makes the expression f(x) = p true to within the desired % tolerance, tol. format compact % This shortens the output. format long % This prints more decimal places. for i = 1: 1000 % get ready ...
In his thesis, Boyce identified a pair of functions that commute under composition, but do not have a common fixed point, proving the fixed point conjecture to be false. [ 14 ] In 1963, Glenn Baxter and Joichi published a paper about the fixed points of the composite function h ( x ) = f ( g ( x ) ) = g ( f ( x ) ) {\displaystyle h(x)=f(g(x))=g ...