Everything about Fixed Point Mathematics totally explained
In
mathematics, a
fixed point (sometimes shortened to
fixpoint) of a
function is a point that's mapped to itself by the function. That is to say,
x is a fixed point of the function
f if and only if f(
x) =
x.
For example, if
f is defined on the
real numbers by
»
then 2 is a fixed point of
f, because
f(2) = 2.
Not all functions have fixed points: for example, if
f is a function defined on the real numbers as
f(
x) =
x + 1, then it has no fixed points, since
x is never equal to
x + 1 for any real number. In graphical terms, a fixed point 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. The example is a case where the graph and the line are a pair of
parallel lines.
Points which come back to the same value after a finite number of
iterations of the function are known as
periodic points; a fixed point is a periodic point with period equal to one.
Attractive fixed points
An
attractive fixed point of a function
f is a fixed point
x0 of
f such that for any value of
x in the domain that's close enough to
x0, the
iterated function sequence
»
converges to
x0. How close is "close enough" is sometimes a subtle question.
The natural
cosine function ("natural" means in
radians, not degrees or other units) has exactly one fixed point, which is attractive. In this case, "close enough" isn't a stringent criterion at all -- to demonstrate this, start with
any real number and repeatedly press the
cos key on a calculator. It quickly converges to about 0.73908513, which is a fixed point. That is where the graph of the cosine function intersects the line
.
Not all fixed points are attractive: for example,
x = 0 is a fixed point of the function
f(
x) = 2
x, but iteration of this function for any value other than zero rapidly diverges. However, if the function
f is continuously differentiable in an open neighbourhood of a fixed point
x0, and |
f'(
x0)| < 1, attraction is guaranteed.
Attractive fixed points are a special case of a wider mathematical concept of
attractors.
An attractive fixed point is said to be a
stable fixed point if it's also
Lyapunov stable.
A fixed point is said to be a
neutrally stable fixed point if it's
Lyapunov stable but not attracting. The center of a
second order homogeneous linear differential equation is an example of a neutrally stable fixed point.
Theorems guaranteeing fixed points
There are numerous theorems in different parts of mathematics that guarantee that functions, if they satisfy certain conditions, have at least one fixed point. These are amongst the most basic qualitative results available: such
fixed-point theorems that apply in generality provide valuable insights.
Applications
In many fields, equilibria or
stability are fundamental concepts that can be described in terms of fixed points. For example, in
economics, a
Nash equilibrium of a
game is a fixed point of the game's
best response correspondence.
In
compilers, fixed point computations are used for whole program analysis, which are often required to do code
optimization. The vector of
PageRank values of all web pages is the fixed point of a
linear transformation derived from the
World Wide Web's link structure.
Logician
Saul Kripke makes use of fixed points in his influential theory of truth. He shows how one can generate a partially defined truth predicate (one which remains undefined for problematic sentences like "This sentence isn't true"), by recursively defining "truth" starting from the segment of a language which contains no occurrences of the word, and continuing until the process ceases to yield any newly well-defined sentences. (This will take a denumerable infinity of steps.) That is, for a language L, let L-prime be the language generated by adding to L, for each sentence S in L, the sentence "
S is true." A fixed point is reached when L-prime is L; at this point sentences like "This sentence isn't true" remain undefined, so, according to Kripke, the theory is suitable for a natural language which contains its
own truth predicate.
Further Information
Get more info on 'Fixed Point Mathematics'.
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