1
vote
0answers
33 views

Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
0
votes
0answers
26 views

The constant of fixed-point iteration

I saw some sentences and proofs related to fixed-point iteration in my numerical method textbook: $$e_{k+1} = g'(\theta) e_{k}$$ if $|g'(x^\ast)|<1$, there is a constant such that $|g'(\theta)|\le ...
1
vote
0answers
46 views

Aitkens Extrapolation

The Aitken's extrapolation can be written as $$X^n = X_{n-2} + \dfrac{(X_{n-1}- X_{n-2})^2}{(X_{n-1}- X_{n-2})-(X_n- X_{n-1})}$$ Verify it? And $X^n$ can be viewed being defined recursively by ...
0
votes
0answers
23 views

Heuristics for sequence convergence

Having a finite sequence of double precision floating point numbers (obtained using the fixed point iteration of a function), is there any algorithm which can be used to determine that this sequence ...
0
votes
0answers
25 views

Show that $x_{k+1}:=g^{-1}(g(x_k)-f(x_k))$ converges to a root of $f$

If $f:[-1,1]\to\mathbb R$ continously differentiable and $g:[-1,1]\to[-2,2]$ continuously differentiable and bijective such that, $|f'(x)-g'(x)|\le 1/2 \inf\limits_{y\in[-1,1]}g'(y)$ ...
1
vote
1answer
65 views

Root of a function? (Proof with Banach theorem)

Given is a function $f$: $\left [ -1,1 \right ]\rightarrow \mathbb{R}$, which is continuous and differentiable. The function $g$: $\left [ -1,1 \right ]\rightarrow \left [ -2,2 \right ]$ is a ...
0
votes
1answer
70 views

Fixed-point iteration, Convergence of a sequence?

Given is the function $f(x)=x^{3}+x-1$ on $\mathbb{R}$. Use the Fixed-point iteration for $x\in \left [ 0.5 , 1 \right ]$ to show that the sequence $\left \{ x \right \}_{n}$ converges to the ...
2
votes
1answer
42 views

Show an iterative fixed point method does not converge

I was asked the following question: let $g(x)=\frac{30}{1+x}$. Notice that $g(5)=5$. Is there an $\epsilon >0$ such that the series $\{x_k\}_{k=0}^{\infty}$ defined by $x_{k+1}=g(x_k)$ and ...
0
votes
1answer
37 views

Trouble with fixed point iteration.

Find the fixed point(s) of $g(x) = (1/2)x^2 + (1/2)x$. Does the fixed point iteration(s) converge(s) to the the fixed point(s) if you start with a close enough approximation? Then choose $x_0 ...
0
votes
4answers
174 views

Fixed Point Iteration, does it converge?

Find the fixed point(s) of $g(x) = x^2 + 3x - 3$. Does the fixed point iteration(s) converge(s) to the fixed points if you start with a close enough first approximation? I set $g(x) = x$ and got ...
1
vote
3answers
63 views

why do we take this interval?

I am looking at an exercise,where given that $a_{n}=\sqrt{2+\sqrt{2+...+\sqrt{2}}}$,I have to show that $\lim_{n \to \infty}a_{n}=2$. We find that $a_{0}=0,a_{1}=\sqrt{2},a_{2}=\sqrt{2+a_{1}} \text{ ...
1
vote
1answer
58 views

Fixed point iteration for $\sqrt[3]{a}$

So I'm given the scheme for computing $\sqrt[3]{a}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this scheme is as fast as possible. Any hints ...
0
votes
0answers
58 views

This system is contractive?

I have a system which has a form of find point problem, described as following $$p_i=h_i(\mathbf{p})$$ where $$p_i\in[0,1]$$ is the $i$-th components of the $n$-dimensional column vector ...
0
votes
0answers
219 views

Newton iteration- estimate the error

I was wondering whether there are equations available to estimate the a priori and a posteriori error for newton's method? My idea was to use that it is a fixed point iteration and therefore one can ...
0
votes
1answer
47 views

Numerical Analysis, build a contractive function

I have a question regarding Numerical Analysis. I've never been asked these sorts of questions before and don't even know where to begin. The goal of this exercise is to find a value alpha such that: ...
0
votes
0answers
38 views

correctness of functional iteration and contraction proof

I need to prove: If $F$ is contractive from $[a, b]$ to $[a, b]$ and $x_{n+1} = F(x_n)$ with $x_0\in[a,b]$ then $|x_n -s| \leq C\lambda^n$ for an appropriate $C$ where $s$ is a fixed point of $F(x)$. ...
4
votes
1answer
221 views

How to prove that the following iteration process converges?

I have the following iteration process: $$ p_{n+1} = \frac{{p_{n}}^3 + 3 a p_{n} }{3 {p_{n}}^2 + a } , $$ where $a > 0$. Q1: How to prove that this iteration process converges for every number ...
6
votes
2answers
445 views

Convergence of fixed point iteration for polynomial equations

I'm looking for the solution $x$ of $$x^n+nx-n=0.$$ Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the ...
1
vote
2answers
827 views

Basic numerical analysis, fixed point iteration

I have just started numerical analysis so this question probably seems trivial. Say I have a function $f(x) = x^2 - x - 3$ I let $g(x) = x^2 - 3$ Then I want to find the roots of $f(x)$ so I have ...
2
votes
2answers
266 views

Fixed point iteration

Assume that $g$ is a continuously differentiable function and that the Fixed-Point Iteration $g(x)$ has exactly three fixed points, $-3, 1$ and $2$. Assume that $g '(-3) = 2.4$ and that FPI ...
4
votes
2answers
378 views

Question regarding upper bound of fixed-point function

The problem is to estimate the value of $\sqrt[3]{25}$ using fixed-point iteration. Since $\sqrt[3]{25} = 2.924017738$, I start with $p_0 = 2.5$. A sloppy C++ program yield an approximation to within ...
2
votes
3answers
358 views

Fixed point theorem

Is $|g'(x)|<1\ \forall x\in(a,b)$ is one of the hypothesis of the Fixed-Point Theorem? The answer is NO. Can someone please enlightened me about this? My teacher reason is this... Note that ...