In mathematics, more specifically in order theory, the greatest fixed point (gfp or GFP) of a function is the fixed point which is greater than or equal to all other fixed points, according to some partial order. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... Partial plot of a function f. ... In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...
For example, the greatest fixed point of the real function
f(x) = –x2
is x = 0 with the usual order on the real numbers. Fixed-point theorems can yield algorithms for locating the greatest fixed point, although least fixed points are more commonly used. Greatest and least fixed points may have desirable properties that arbitrary fixed points do not have. In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. ... In mathematics, the least fixed point in order theory of a function is the fixed point which is less than or equal to all other fixed points, according to some partial order. ...
In mathematics, a fixedpoint (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function.
In graphical terms, a fixedpoint means the point (x, f(x)) is on the line y = x, or in other words the graph of f has a point in common with that line.
Attractive fixedpoints are a special case of a wider mathematical concept of attractors.
Fixedpoint numbers can be assigned, compared, added, subtracted, negated and shifted (for multiplying or dividing by powers of two) using the normal integer operators, but you should take care to use the appropriate conversion routines when mixing fixedpoint with integer or floating point values.
The fixedpoint square root, sin, cos, tan, inverse sin, and inverse cos functions are implemented using lookup tables, which are very fast but not particularly accurate.
The fixedpoint functions used to be named with an "f" prefix instead of "fix", eg.
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