What is Fixed point: Definition and 103 Discussions

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f if f(c) = c. This means f(f(...f(c)...)) = f n(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.
For example, if f is defined on the real numbers by





{\displaystyle f(x)=x^{2}-3x+4,}
then 2 is a fixed point of f, because f(2) = 2.
Not all functions have fixed points: for example, f(x) = x + 1, has no fixed points, since x is never equal to x + 1 for any real number. In graphical terms, a fixed point x 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.
Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a projectivity has been called a double point.In Galois theory, the set of the fixed points of a set of field automorphisms is a field called the fixed field of the set of automorphisms.

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  1. C

    How does one convert a decimal float to binary using the multiply by 2 method?

    I've started reading A concise introduction to numerical analysis, A.C. Foul and on the first page there's the following paragraph about how a floating point in fixed point precision can be represented: I don't understand the example where it says " ##\beta=2## and ##p=20##, the decimal number...
  2. Infrared

    Challenge Math Challenge - July 2023

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  3. PragmaticYak

    Fixed point free automorphism of order 2

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  4. drooble122

    Algebra: distance from a fixed point

    The first image is the question and the second is the answer. My thinking is let's say North is positive, and South is negative. Fixed point O is the starting point. Then the question becomes +(2a-b)-(3a+2b). The answer should be -a-3b. I cannot fathom why the book gives the answer as a+b. Any...
  5. S

    I Is this statement an aspect of the Hairy Ball or Fixed Point Theorem?

    “Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes…” The reason this statement sounds like one of those theora is that I recall reading a Time-Life book on Mathematics, and there...
  6. wrobel

    I Rigid body with fixed point

    Today I read a book in mechanics and encountered a funny proposition about rigid body with fixed point. Perhaps somebody will be interested to propose it to students as a task. This proposition is almost correct:) Consider a rigid body with a fixed point ##O##. Let ##Oxyz## be a coordinate...
  7. L

    MHB Fixed point iteration convergence

    Question: For the following functions, does the fixed point iteration for finding the fixed point in $\left [ 0,1 \right ]$ converge for all first points $ p_{0} \in \left [ 0,1 \right ]$? Justify your answer. a.$ g(x) = e^{\frac{-x}{2}}$ b.$ g(x) = 3x - 1$ Let me attempt for part a first...
  8. M

    MHB Fixed point,, Jacobi- & Newton Method, Linear Systems

    Hey! :giggle: Question 1 : Let $g(x)-=x-x^3$. The point $x=0$ is a fixed point for $g$. Show that if $x^{\star}$ is a fixed point of $g$, $g(x^{\star})=x^{\star}$, then $x^{\star}=0$. If $(x_k)$ the sequence $x_{k+1}=g(x_k)$, $k=0,1,2,\ldots$ show that if $0>x_0>-1$ then $(x_k)$ is...
  9. facenian

    I Proving a Fixed Point Theorem for Shrinking Maps on Compact Spaces

    Show that if ##f## is a shrinking map ##d(f(x),f(y)) < d(x,y)## and ##X## is compact, then ##f## has a unique fixed point. Hint. Let ##A_n=f^n(X)## and ##A=\cap A_n##. Given ##x\in A##, choose ##x_n## so that ##x=f^{n+1}(x_n)##. If ##a## is the limit of some subsequence of the sequence...
  10. nomadreid

    I Kripke's fixed point for truth predicate: justification?

    If I understand correctly (dubious), given a consistent theory C (collection of sentences), Kripke proposes to add a predicate T so that, if K = the collection of all sentences T("S") for every sentence S in C, ("." being some appropriate coding) then the closure of K∪C forms a new theory C*...
  11. A

    How close will body approach fixed point charge?

    Homework Statement Charged sphere with a mass of 15 mg and charge 2 nC moves with a speed of 15 cm/s towards a fixed point charge of 3 nC. How close will sphere approach charge? Homework Equations K=(1/2)*mv2 U=k*(Q1Q2/r) The Attempt at a Solution So I am not sure I approached correctly but...
  12. M

    MHB Fixed point iteration: g is a contraction mapping

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  13. peadar2211

    Determine the stability of a fixed point of oscillations

    Homework Statement I have a system of coupled differential equations representing chemical reactions and given certain initial conditions for the equations I can observe oscillation behaviour when I solved the equations numerically using Euler's Method. However, then it asks to investigate the...
  14. B

    Unique Fixed Point: Prove Contraction Map in Compact Space Has Unique Solution

    Homework Statement Let ##(X,d)## be some metric space, and let ##f : X \to X## be such that ##d(f(x),f(y)) \le a d(x,y)## for every ##x,y \in X## for some ##a \in (0,1)## (such a map is a called a contraction map) If ##f## is a contraction and ##X## is compact, show that ##f## has a unique...
  15. S

    Torque and rotation around a fixed point

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  16. Zafa Pi

    I Dense orbit and fixed point question

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  17. I

    Proving that three closed orbits must contain a fixed point

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  18. I

    A Critical exponents - experimental values

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  19. Hercuflea

    Moving point and it's distance relative to a fixed point

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  20. C

    Fixed Point Theorem: Necessary & Sufficient Conditions for Convergence

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  21. JasMath33

    Using the Intermediate Value Theorem to Find Fixed Points

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  22. F

    Find the force a distance from a fixed point

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  23. H

    Example of torque-free rotation with a fixed point

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  24. awholenumber

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  25. karush

    MHB A particle traveling in a strainght line passes a fixed point O

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  26. K

    Does Loop quantum gravity have an effective UV fixed point?

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  27. RicardoMP

    Fixed point method for nonlinear systems - complex roots

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  28. Samuel Williams

    Does T have a unique fixed point in X?

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  29. O

    MHB Cauchy sequence in fixed point theory

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  30. O

    MHB Continuous mapping and fixed point

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  31. O

    MHB Fixed Point Theorem & Contractive Maps

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  32. O

    MHB What does priori estimate do in fixed point theory

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  33. O

    MHB Fixed Point Theory: Lipschitz or Contraction?

    I see that if a mapping is contraction then it is contractive then it is nonexpensive and then it is lipschtiz...so, which class of mapping is general ? lipschitz or contraction ? which one ? thank you for your attention :)
  34. M

    How Does a Fixed Point Theorem Explain Convergence in Iterative Methods?

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  35. J

    Fixed point iteration, locally convergent

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  36. Einj

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  37. Math Amateur

    MHB Metric Spaces - Fixed Point Theorem (Apostol, Theorem 4.48)

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  38. D

    MHB Error in fixed point arithmetic library

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  39. K

    MHB Fixed point, interval of existence, & stability

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  40. evinda

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  41. M

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  42. M

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  43. U

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  44. A

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  45. F

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  46. O

    MHB Fixed point for a complex mapping.

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  47. C

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  48. H

    Fixed Point Iteration for Solving Equations

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  49. S

    Fixed Point Theorem: Unit Square Injection to Bigger Square

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  50. N

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