Fixed point Definition and 103 Threads

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




f
(
x
)
=

x

2



3
x
+
4
,


{\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 to generalize the fixed point iteration

    If we want to solve $$f(x)=0$$ we can re-write the equation as $$g(x)=x$$ and use the fixed point method, i.e, $$x_{n+1}=g(x_n)$$ starting with a guess $$x_0.$$ I was wondering if something similar can be done with $$\Lambda(x,y)=h(x,y).$$
  2. P

    How can I convert from Q3.5 format to decimal using an 8-bit microcontroller?

    Hello, I am currently trying to figure out if its possible to go from a number in (unsigned) Q3.5 format -> 8 bits given. And print it on a display in decimal. I only have a 8 bit microcontroller to work with. For example 011.11010 should have 3.8125 Getting to 3 is easy but I...
  3. H

    Oscillation of two masses connected to springs and a fixed point

    Q: Two masses m are connected by identical springs of constants k and they lie on a perfectly smooth surface. The extremity of one spring is fixed on the wall, the other one is loose. Find the equations for the motion of the system. Find the frequencies of oscillations. 1. Relevant equations...
  4. C

    Iteration functions for Fixed Point method

    Hi there. I need to find some iteration functions for x - 2\frac{sin(x)}{cos(x)}=0, as g(x)=2\frac{sin(x)}{cos(x)} does not converge. I can't find any others, maybe I didn't quite undertood how they're built. Any help will be appreciated Thanks
  5. A

    MHB Prove that a function does not have a fixed point

    it is a question in my book said Prove that the function f(x) = 2 + x - \tan ^{-1} x has the property \mid f'(x)\mid < 1 Prove that f dose not have a fixed point but i found that this function has a fixed point y = 2 + y - \tan ^{-1} y y = \tan 2 is it right that the question is...
  6. A

    MHB Prove that a function has a fixed point

    Let F be a continuous function from [a,b] onto [a,b] prove that F has a fixed point in the interval [a,b] it is clear for me by drawing the product of [a,b]x[a,b] any line which pass through all the image should intersect with the diagonal but i can't make a mathematical proof. I tried by...
  7. C

    Banach Fixed Point and Differential Equations

    Homework Statement Find the value of x, correct to three decimal places for which: \int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\frac{1}{2}. Homework Equations Banach's Fixed Point Theorem Picard's Theorem? The Attempt at a Solution I'm not sure where to start with this type of problem...
  8. D

    Convergence and stability in multivariate fixed point iteration

    Hi, I'm new to posting questions on forums, so I apologise if the problem is poorly described. My problem is solving a simulation of the state of a mineral processing froth flotation plant. In the form x@i+1 = f(x@i), f represents the flotation plant. f is a computationally intensive solution...
  9. S

    What is the relationship between stability and the derivative of a fixed point?

    Dear friends, I want to find the conditions of stability of a fixed point. consider the function "f" iterates to obtain fixed point "a": x_{n+1}= f(x_n) for this dynamic system, the fixed point "a" is stable if we have: |f ^{\prime}(a)| < 1 Currently I'm working on a bit different...
  10. B

    Generating Fixed-Point-Free Permutations in Sn

    Homework Statement What subgroup is generated by the fixed-point-free permutations? Homework Equations The Attempt at a Solution I know that the elements that have no fixed points are the ones whose cycle type adds up to n (i.e. all the numbers in {1,...,n} have to be used). I don't know what...
  11. S

    Fixed Point Iteration Convergence

    Homework Statement Consider the system x = \frac{1}{\sqrt{2}} * \sqrt{1+(x+y)^2} - 2/3 y = x = \frac{1}{\sqrt{2}} * \sqrt{1+(x-y)^2} - 2/3 Find a region D in the x,y-plane for which a fixed point iteration xn+1 = \frac{1}{\sqrt{2}} * \sqrt{1+(x_n + y_n)^2} - 2/3 yn+1 =...
  12. P

    Solving Fixed Point Problems: x=4-x^2, f(x)=7+sqrt(x-1), f(x)=sqrt(10+3x)-4

    Homework Statement Find all real values x that are fixed by the function y=4-x^2 f(x)=4-x^2 Homework Equations x=y The Attempt at a Solution x=4-x62 0=-x^2-x+4 0=-(x^2+x+(1/4))+(17/4) This is where i get stuck. I also have two other problems which iIdo not understand how to...
  13. D

    Unique Fixed Point: Proving F^3 is a Contraction

    Homework Statement Suppose F is mapping of a nonempty complete metric space into itself, and that F^3 = F o F o F is a contraction (o's denote composition). Show that f has a unique fixed point.The Attempt at a Solution Isn't this kind of a trick question? Suppose f does not have a unique...
  14. K

    Forcing a Least squares Polynomial through a fixed point

    Hi guys, Thanks for taking the time to read the post. I have a question related to curve fitting and polynomials that i was hoping someone might be able to help me with. I have a set of x and y data points, all on a graph. I have then calculated the 4th order least squares polynomial...
  15. H

    Angular momentum for combined rotation and translationa about a fixed point

    One more serious doubt... In feynman's tips on physics' problems... there is one regarding marble rolling... "An amusing trick is to press a finger down on the marble, on a horizontal table, in such a way that he marble is projected along the table with initial linear speed v0 (v-naught) and...
  16. ArcanaNoir

    Elementary proof of fixed point theorem.

    I'd like someone to check this proof out for violations of math law. It seems like hackery to me, but then so does a lot of what my professor says, so maybe it's not. If it's flawed, just tell me why. Please don't suggest other ways to prove this, that's my job this weekend. Thanks :) p.s...
  17. fluidistic

    Proof that a contractive function has a fixed point

    Homework Statement I must understand the proof that if F:[a,b] \to [a,b] and F is contractive then there exist a unique x \in [a,b] such that F(x)=x.Homework Equations Definition of a contractive function: F is contractive over [a,b] if and only if there exist \lambda such that 0<\lambda <1 and...
  18. radou

    Proof of Banach's fixed point theorem

    Homework Statement I was a bit surprised to find out that one of the exercises in Munkres is actually a proof to the Banach fixed point theorem, unless I'm mistaken. The exercise follows: If (X, d) is a complete metric space, and f : X --> X a contraction mapping, there is a unique point...
  19. A

    Linear approximations around a given fixed point.

    Saw this mentioned, didn't understand what it was or how it would be done. Given the continuous system given by x'1,x'2,x'3 Find the linear approximation for each x* (fixed point) Guessing first of course to find the fixed points. Then find the Jacobian Df for the solved system. What...
  20. J

    Proving a Continuous Function has a Fixed Point

    This is a question from the exam for the calculus class I took last semester: It looks like it might be able to be done with squeeze theorem, but I can't work it out. Please help me with this, before I descend into madness.
  21. W

    G must have an element with no fixed point when there is only one orbit

    I am studying for a modterm on Monday and asking for help on the homework questions I got WRONG on my problem sets (so I can hopefully improve my understanding and see my mistake). This is my reworked version of the incorrect HW problem and I would like to know if I am on the right track...
  22. S

    What are the Tensions in a System of Fixed Point Charges?

    A fixed point charge of +2q is connected by strings to point charges of +q and +4q, as shown below. Find the tensions T1 and T2. (Use the following as necessary: q, d and k.) For T1, I summed all the forces on each charge and got...
  23. P

    Consider the arrangement of two fixed point charges, equal in magnitude

    Consider the arrangement of two fixed point charges, equal in magnitude... Consider the arrangement of two fixed point charges, equal in magnitude, shown in the figure. Which of the following statements are correct for the initial motion of a third charge if it is released from rest in the...
  24. M

    How Does Fixed Point Iteration Converge with Nested Square Roots?

    Let p>0 and x = \sqrt{p+\sqrt{p+\sqrt{p+ \cdots }}} , where all the square roots are positive. Design a fixed point iteration x_{n+1} = F (x_{n}) with some F which has x as a fixed point. We prove that the fixed point iteration converges for all choices of initial guesses greater than -p+1/4...
  25. P

    MATLAB: with My Fixed Point Iteration Program

    Homework Statement Fixed Point Iteration MATLAB program Homework Equations To test for convergence: abs(g'(x))<1 The Attempt at a Solution Hi all, I am trying to write a Fixed Point Iteration program but when I enter in the command line it kept giving me an error message. Can you...
  26. B

    Fixed point iteration to find the roots of 0=x-tan(x)

    Homework Statement The question wants me to first estimate the roots by drawing the graph and then by using a 'suitable' fixed point method to determine the first 4 positive roots. Homework Equations 0=x-tan (x) I rearranged to get x=arctan (x) so that the series x_n will converge...
  27. C

    Analysis: fixed point, contraction mapping

    Let p,q : \mathbb{C} \to \mathbb{C} be defined by \begin{align*} p(z) =& z^7 + z^3 - 9z - i, \\ q(z) =& \frac{z^7 + z^3 - i}{9} \end{align*} 1. Prove that p has a zero at z_0 if and only if z_0 is a fixed point for q. If z_0 is a fixed point for q then \begin{align*} q(z_0) =...
  28. M

    Fixed Point iteration using matlab, whats wrong with my code?

    Fixed Point iteration using matlab, what's wrong with my code?? Homework Statement We are suppose to use MatLab to make a program using the fixed point iteration to find the root of an equation. I just can't figure out what I'm doing wrong here... I'm pretty sure a while loop is the...
  29. D

    Brouwer's Fixed Point Theorem for Arbitrary Intervals

    Is it possible to prove Brouwer's Fixed Point Theorem (one-dimensional version) for intervals other than [-1,1]-->[-1,1], say [1,2]-->[0,3]? If so, how?
  30. K

    Cauchy sequence & Fixed point

    Cauchy sequence & "Fixed" point Homework Statement Suppose that f: Rd->Rd and there is a constant c E (0,1) such that ||f(x)-f(y)|| ≤ c||x-y|| for all x, y E Rd. Let xo E Rd be an arbitrary point in Rd, let xn+1=f(xn). Prove that a) f is continuous everywhere. b) (xn) is Cauchy. c) (xn)...
  31. C

    Convergence criteria for fixed point iteration

    Homework Statement Most functions can be rearranged in several ways to give x = g(x) with which to begin the fixed-point iteration method. For f(x) = e^x − 2x^2 , one g(x) is x = +- sqrt(e^x/2) a) Using the convergence criteria, show that this converges to the root near 1.5 if the positive...
  32. O

    Solving for a fixed point for a sine map

    Homework Statement Consider the sine map x{sub t+1} = f(x{sub t}) where f(x) = r*sin(x*pi). For r > 1/pi there are two fixed points, one at the origin that is unstable, and one elsewhere on the curve. The non-origin fixed point starts out, as you turn r just slightly above 1/pi, as stable...
  33. M

    Fixed Point Equations - Exam Revision Help

    hi I am working on my exam revision and need to know the fixed point equation.if you could help it would be apreciated. Homework Equations The Attempt at a Solution
  34. F

    Twice-differentiable, mean value theorem, fixed point

    Homework Statement Let g:[0,1] \to \mathbb{R} be twice-differentiable (i.e. both g and g' are differentiable functions) with g''(x) > 0 for all x \in [0,1]. If g(0) > 0 and g(1) = 1, show that g(d) = d for some d \in (0,1) if and only if g'(1) > 1. Homework Equations The Attempt...
  35. Z

    A tire sliding about a fixed point

    Homework Statement There are two tires separated by a few feet, with a weighted beam attached on top of them. The beam's weight isn't distributed evenly. One of the tires is a fixed point. The other tire slides (doesn't roll) 90 degrees. How do you determine the force required to slide the...
  36. A

    Finding Tensions of Fixed Point Charges

    A fixed point charge of +2q is connected by strings to point charges of +q and +4q (see attached diagram), Find the tensions T_1 and T_2. For T_2, I start summing the forces on the +4q point charge. F_{net,4q}=F_{q}+F_{2q}+T_2 0=F_{q}+F_{2q}+T_{2} T_2=-(F_{q}+F_{2q}) Is this the...
  37. L

    C/C++ What Is Fixed Point Notation in C++?

    Recently in C++ code I came across the notation 1 << 8. At first, I thought it was just a standard bitshift, till an explanation of the code told me it was fixed point notation, something I had not heard of till then. I have tried to read up about fixed point notation, but am still confused...
  38. S

    Proving there is a fixed point in a discrete group of rotations

    Homework Statement Let G be a discrete group in which every element is orientation-preserving. Prove that the point group G' is a cyclic group of rotations and that there is a point p in the plane such that the set of group elements which fix p is isomorphic to G' The Attempt at a...
  39. A

    Proving Existence of Fixed Points in Continuous Sets

    Homework Statement Suppose f:[a,b] \rightarrow [a,b] is continuous. Prove that there is at least one fixed point in [a,b] - that is, x such that f(x) = x. Homework Equations The Attempt at a Solution I was going to try something with the IVT, but then I realized I wasn't sure what...
  40. M

    Trajectory for a runner aiming to a fixed point, with initial velocity?

    Hello, I want to know how long time a running runner needs to reach a point, when he is initially running in the wrong direction. I'd like to do this with minimal calculation, so an approximation at 10% would be enough. To even complicate the problem more, the runner can decelerate...
  41. A

    Spiraling Inwards: Proving a Particle's Time to Reach a Fixed Point

    Homework Statement A particle P of mass m moves under the infulence of a central force of magnitude mkr^{-3} directed towards a fixed point O. Initially r=a and P has a velocity V perpendicular to OP, where V^2 < \frac{k}{a^2}. Prove that P spirals in towards O and reaches O in a time T =...
  42. S

    How do i find k in the banach fixed point theorem

    how do i find k in the banach fixed point theorem. so say i have a function f(x)=1+3x-x^2 in the interval [1,2] then how do i find k? thank you
  43. S

    What Does the Banach Fixed Point Theorem Mean?

    I really don't understand nothing from the Banach fixed point theorem, i know that it should satisfy: [g(x)-g(y)]<K(x-y) for all x and y in[a,b] but i don't even understand what that's supposed to mean? any help will be appreciated. thank you.
  44. K

    Fixed Point Iteration Requirements

    Hi I wrote a numerical analysis midterm earlier this week and there was one question I couldn't figure out. I was wondering if anyone had some insight. What I've been told and what I've read in many many places is that f(x) will converge to a fixed point on an interval I if 1. f(x) is...
  45. S

    About convex hull and fixed point

    First, I wonder whether I can put the post here... Given X=[0,1]^2 a(x)={y in X:||y-x||>=1/4} b(x)is the convex hull of a(x). Identify the set of fixed points. My answer is 3/4>=x>=1/4, 3/4>=y>=1/4, but I am not sure... What if we have a(x)={y in X:||y-x||>=1/2}? (My answer is...
  46. S

    About convex hull and fixed point

    X=[0,1]^2 a(x)={y in X:||y-x||>=1/4} b(x)is the convex hull of a(x). Identify the set of fixed points. My answer is 3/4>=x>=1/4, 3/4>=y>=1/4, but I am not sure... Thanks.
  47. B

    How Can I Integrate Fixed Point Iterations with Excel Using VBA?

    Hey guys...I'm not real smooth with programming, but I've got some of it done. I need to use the formulas I have in the code, and export the data back into my excel sheet...The first Do-Loop is for the Fixed Point iterations, and the second part is for the NR method. I just need to learn...
  48. N

    Simple Fixed Point Iteration for Root-solving

    I just want to know why in the world this works? I am speaking about the simple iteration of taking a function, f(x), setting it to 0, f(x) = 0, solving for x in the function and setting it equal to g(x)...and then iterating. For example the function :f(x) = x^2 +2x - 1 Setting it to 0 and...
  49. B

    Proving a fixed point on a function

    Hello guys, this question is kinda bothering me since I'm having trouble with one of the steps in proving it. The question reads. Assume funciton f is continuous on an interval [0,1], such that the range of f is contained within or equal to [0,1]. Show that for a value of c contained within...
  50. G

    Numerical Analysis: Fixed Point Iteration

    Consider the fixed point iteration formula: *x_(n+1) = (2/3)[(x_n)^3 - 1] - 3(x_n)^2 + 4x_n = g(x) *Note: "_" precedes a subscript and "^" precedes a superscript (a) Find an interval in which every starting point x_0 will definitely converge to alpha = 1. (b) Show that the order of the...
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