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A fixed-point representation of a fractional number is essentially an integer that is to be implicitly multiplied by a fixed scaling factor. For example, the value 1.23 can be stored in a variable as the integer value 1230 with implicit scaling factor of 1/1000 (meaning that the last 3 decimal digits are implicitly assumed to be a decimal fraction), and the value 1 230 000 can be represented ...
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.
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 ...
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 ...
Fixed-point computation refers to the process of computing an exact or approximate fixed point of a given function. [1] In its most common form, the given function f {\displaystyle f} satisfies the condition to the Brouwer fixed-point theorem : that is, f {\displaystyle f} is continuous and maps the unit d -cube to itself.
The fixed point iteration x n+1 = cos x n with initial value x 1 = −1.. An attracting fixed point of a function f is a fixed point x fix of f with a neighborhood U of "close enough" points around x fix such that for any value of x in U, the fixed-point iteration sequence , (), (()), ((())), … is contained in U and converges to x fix.
Fixed point, a benchmark (surveying) used by geodesists; Fixed point join, also called a recursive join; Fixed point, in quantum field theory, a coupling where the beta function vanishes – see renormalization group § Conformal symmetry; Temperature reference point, usually defined by a phase change or triple point.
The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.