Linked Questions

7
votes
5answers
2k views

If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent.

If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent. What is the limit? The back of my textbook says that $\lim(x_n)=\frac{...
4
votes
2answers
10k views

When does Newton-Raphson Converge/Diverge?

Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? I am aware that Newton-Raphson is a special case of fixed point iteration, where: $...
3
votes
4answers
7k views

How to Determine Interval $[a,b]$ for a Fixed-Point Iteration?

Determine an interval $[a,b]$ on which the fixed-point ITERATION will converge. $x = g(x) = (2 - e^x + x^2)/3$ I've determined that $g'(x) = (2x -e^x)/3$, but I don't know how to determine ...
4
votes
2answers
2k views

Using the Banach Fixed Point Theorem to prove convergence of a sequence

Use the Banach fixed point theorem to show that the following sequence converges. What is the limit of this sequence? $$\left(\frac{1}{3}, \frac{1}{3+\frac{1}{3}}, \frac{1}{3+\...
2
votes
5answers
521 views

Prove convergence and find limit of $a_{n+2}=\frac{1}{2}(a_n+a_{n+1})$

I need to prove the convergence and find the limit of the following recursive sequence: $$a_1=2,a_2=5$$ $$a_{n+2}=\frac{1}{2}(a_n+a_{n+1})$$ *Similar problems were solved by showing the sequence is ...
8
votes
2answers
2k 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 solution is ...
0
votes
1answer
2k views

Intuition for convergence iterative formula

A convergence iterative formula , $g(x)$ , holding that $|g'(z)|<1$ . In a case which the equation is given and I have to evaluate iterative formula in order to find its fixed point . For ...
1
vote
2answers
162 views

Show that, if $g(x)=\frac{1}{1+x^2}$ and $x_{n+1}:=g(x_n)$ then $x_n$ converges

Why does the sequence $\{x_n\}$ converge ? If $x_{n+1}:=g(x_n)$, where $g(x)=\frac{1}{1+x^2}$ (We have a startpoint in $[0.5,1]$) The sequence is bounded by $1$ independant of the ...
3
votes
3answers
319 views

Fixed point iteration contractive interval

Consider the function $F(x) = x^2-2x+2$. Find an interval in which the function is contractive and find the fixed point in this interval. What is the convergence rate of the fixed point iteration: $...
1
vote
1answer
292 views

Iteration convergence.

How can I solve this problem? Let $$x(n+1)=-\frac{\exp(x(n)/2)}{5}$$ be a given sequence. Prove using the Banach contraction principle that this sequence converges to some fixed point $X$ with $x(0)...
4
votes
2answers
176 views

Fixed point iteration problem of $f(u)=u^3-u-1$

I was thinking about the following problem: Let $f(u)=u^3-u-1$. Then I have to verify whether the following statements are true/false? 1.Starting with the initial guess $u^{(0)}=1.5,$ the ...
3
votes
1answer
118 views

$x_{n+1}=\frac{2x_n+3f(x_n)}{5}$ showing $f$ has a fixed point

Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a differentiable function , and suppose that there is a constant $A<1$ such that $|f'(t)|\le A$ for all real $t$. Define a sequence $\{x_n\}$ by $ $$$x_{n+...
1
vote
2answers
342 views

Showing that Newton's method converges

I'm studying newton's method for the equation $\cos(x) = x$ over $[0, \pi/2]$ and I'm asked to support the argument that given the values $x_0 = 0.78539816$, $x_1 = 0.73953613$, $x_2 = 0.73908518$ ...