Fixed point Definition and 96 Threads

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...

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...

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...

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...

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...

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...

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 =...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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) =...

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...

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?

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)...

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...

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

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...

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...

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...

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...

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...

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...

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

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.

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...

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...

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.

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...

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...

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...

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...

Hi, I need to prove that f(x) has a fixed point, given that f'(x) >= 2 for all x. my problem is that I've reached the part in which g(x) = f(x) - x and g'(x) = f'(x) - 1 and therefore g'(x) >= 1 but now I'm completely stuck. I knwo that I need to use the mean value theorem, but i just...

I'm trying to work out how an existence of a fixed point is linked to the constraint on the differential of that function. For example, i need to prove f has a fixed point if f'(x)=>2. I understand that what I have is a monotone increasing function so it is 1-1. All the fixed points are...

I have read the following : "The usual proof of Brouwer's fixed-point theorem makes use of some machinery from simplical homology theory. First we establish that there does not exist a "retraction" of an n-cell onto its boundary, which is to say, there is no continuous mapping from an n-cell...