Questions tagged [fixed-point-iteration]
126 questions
13
votes
2 answers
Prove existence of unique fixed point
Let $f(x)$ be a strictly decreasing function on $\mathbb{R}$ with $|f(x)-f(y)|<|x-y|$ whenever $x\neq y$. Set $x_{n+1}=f(x_n)$. Show that the sequence $\{x_n\}$ converges to the root of $x=f(x)$.
Note that the condition is weaker than what is…
Zhang Edison
- 337
10
votes
3 answers
How to determine the outcome of the recursive sequence $a_n=\frac{1}{\operatorname{abs}\left(a_{n-1}\right)-1}$
Depending on the starting value, the end result of this iterative sequence appears to be very variable, for example, if the starting value $a_0\ =\frac{b}{c}$ where $b$ and $c$ are integers, then eventually the sequence appears to always reach 0,…
Cubbs
- 829
9
votes
0 answers
does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?
I've recently discovered that modifying the standard Newton-Raphson iteration by "squashing" $\frac{f (t)}{\dot{f} (t)}$ with the hyperbolic tangent function so that the iteration function is
$$N_f (t) = t - \tanh \left( \frac{f (t)}{\dot{f} (t)}…
crow
- 1
5
votes
1 answer
If $q(x)=x^2+1$, does $q^{\circ 1/2}$ exist?
I've been doing a lot of research about functional half-iteration, and I posed the following question to myself:
Consider the function $q:\mathbb R\mapsto\mathbb R$ defined as
$$q(x)=x^2+1$$
Does $q^{\circ 1/2}$ exist? Does a continuous…
Franklin Pezzuti Dyer
- 40,930
- 9
- 80
- 174
5
votes
2 answers
Local Truncation Error of Implicit Euler
The LTE of an implicit Euler method is $O(h^2)$ because the method has order $O(h)$, but I'm not sure where to get started in proving this arithmetically. Any help would be appreciated. Thank you!
mil10
- 321
4
votes
1 answer
Algorithm faster than Newton's for calculating $\sqrt{2}$
It is well known that the iteration scheme
$$ x_{n+1} = \frac{1}{2} (x_n + \frac{2}{x_n })$$
converges to $\sqrt{2}$ very fast. More precisely, it converges quadratically. The problem is, is there any even faster algorithm? Namely, can we find…
poisson
- 1,075
4
votes
1 answer
What is the value of $z$ for any Julia set? Does it influence the graphical result?
I'm trying to understand how Julia sets iterations are done and how those iterations differ from the ones that generate the Mandelbrot set.
Both of them use the following function: $f(z) = z^2 + C$
For the Mandelbrot set, one would just start with…
78dtat78da
- 95
4
votes
1 answer
Approximating the implicit function
Last week in our Calculus II course we learned about the implicit function theorem, to prove this theorem we used the inverse function theorem which we proved using using the Banach fixed-point theorem:
In the proof of the inverse function theorem…
user337745
4
votes
3 answers
Prove $(x_n)$ defined by $x_n= \frac{x_{n-1}}{2} + \frac{1}{x_{n-1}}$ converges when $x_0>1$
$x_n= \dfrac{x_{n-1}}{2} + \dfrac{1}{x_{n-1}}$
I know it converges to $\sqrt2$ and I do not want the answer. I just want a prod in the right direction.
I have tried the following and none have worked: $x_n-x_{n-1}$ and this got me nowhere.
I have…
stackdsewew
- 1,047
4
votes
2 answers
Find if a fixed-point iteration converges for a certain root
I'm asked to find if the fixed-point iteration
$$x_{k+1} = g(x_k)$$
converges for the fixed points of the function $$g(x) = x^2 + \frac{3}{16}$$ which I found to be $\frac{1}{4}$ and $\frac{3}{4}$.
In this short video by Wen…
user168764
4
votes
2 answers
When to use Newtons's, bisection, fixed-point iteration and the secant methods?
I've been introduced more or less to these methods of finding a root of a function (a point where it intersects the $x$ axis), but I'm not sure when they should be used and what are the advantages of one method over the other.
I think that it would…
user168764
3
votes
1 answer
Find all $k_0$'s such that $x_{n+1}=f(x_n)$ will remain in $[0,1]$ for the "tent map" of height $3$
Let $f: [0,1]\rightarrow \mathbb{R}$:
$f(x)=3x \, \, $ if $0\le x\le \frac{1}{2}$;
$f(x)=3-3x \, \, $ if $\frac{1}{2}
L.A.
- 335
3
votes
0 answers
Kth derivative of the nth iterate of a function evaluated at a fixed point $a$?
Something that has interested me is this wikipedia article: https://en.wikipedia.org/wiki/Iterated_function. Now, In the section "Some formulas for fractional iteration", a technique is provided in which you evaluate a taylor series of the function…
Sam
- 920
3
votes
1 answer
Understanding Baker and Rippon's proof of a result in iteration theory
Let $a \in \mathbb{C}, b = e^a, T(z) = e^{az}$. Define $W_n = T^n(1)$ where $T^n$ is the nth iterate of $T$. The main result that motivated asking this question is Theorem 1 below.
Note: in Theorems 2,3,4 and in the proof of Theorem 1, $f$ is a…
cpiegore
- 1,556
3
votes
0 answers
What is this numerical method for solving Ax=b called?
Following is the pseudo-code of a simple iterative method of solving $Ax=b$ where $A$ is an $n\times n$ matrix.
for i = 0 .. iteration_count
for j = 0 .. n
x = x + A.row(j) * (b[j] - A.row(j) * x) / (A.row(j) * A.row(j)^T)
Note that…
badatmath
- 31