Search results
Results from the WOW.Com Content Network
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 ...
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.
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 .
The function () = + (shown in red) has the fixed points 0, 1, and 2. In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically, for functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed ...
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 f ( x ) = x {\displaystyle f(x)=x} often arising in the field of computational ...
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 ...
The fixed-point theorem shows that no total computable function is fixed-point free, but there are many non-computable fixed-point-free functions. Arslanov's completeness criterion states that the only recursively enumerable Turing degree that computes a fixed-point-free function is 0′ , the degree of the halting problem .
In the finite-dimensional case, the Lefschetz fixed-point theorem provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to Banach spaces. [41] This generalization is known as Schauder's fixed-point theorem, a result generalized further by S. Kakutani to set-valued functions. [42]