All Questions
Tagged with fixed-point-iteration or fixed-point-theorems
2,225 questions
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Central Limit Theorem via Fixed Point Theorem and Entropy
I'd like to work out the details of a proof of the Central Limit Theorem that utilizes the Banach Fixed Point Theorem and possibly also entropy. The rough idea is:
The average $\bar{X} = \frac{1}{n} \...
- 1,987
1
vote
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80
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uniqueness of fixed point using a non-linear function [closed]
I have some function from $R^n$ to $R^n$ defined as:
$$T(v) = u + f(v) F$$
where u is some vector, f is some non-linear function and F is a n,n matrix.
i want to prove that there is a unique fixed ...
- 11
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79
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Unclear step in van Maaren's theorem proof by Schechter
I am trying to disentangle the proof of Brouwer's fixed point theorem via van Maaren's geometry-free Sperner lemma in Eric Schechter's Handbook of Analysis and its Foundations (sections 3.28-3.37). ...
- 61
3
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1
answer
205
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Iterating the arithmetic-derivative map $U(n)=n+D(n)−1$
For integers $n$, the arithmetic derivative $D(n)$ is defined as follows:
$D(p) = 1$, for any prime $p$.
$D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule).
$D(-n) = -D(n)$.
The ...
- 3,054
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346
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A sequence based on arithmetic derivative that always converges to prime numbers
For integers $n$, the arithmetic derivative $D(n)$ is defined as follows:
$D(p) = 1$, for any prime $p$.
$D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule).
$D(-n) = -D(n)$.
The ...
- 3,054
4
votes
1
answer
166
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Question About Milnor's proof of the Brouwer's fixed-point theorem
I'm currently going through Milnor's proof of the Brouwer's fixed-point theorem, which I've linked here. I am able to follow the proof up to theorem 2 - afterwards, I've having a bit of trouble ...
- 2,934
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80
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What are the problems in math that are equivalent to fixed point problems?
I am not sure if this question makes some sense. But I have been thinking for a while now. Let us start with simple problems; let us say that we want to solve the following equation for $x$
$$f(x)=y.$$...
- 723
2
votes
2
answers
202
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Proving the existence of a fixed-point for the interpolation problem
This is a rephrasing of the original post in (Interpolation problem with varying nodes)
Let $\{f_i\}^{M}_{i=0}$ be a set of real numbers satisfying either $$f_0>f_1<f_2>f_3 \dots$$ or $$f_0&...
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A function $f$ such that repeated application gives you identity? i.e. $f^n(x) = x$ [duplicate]
Given a fixed $n$, I wanted a continuous function $f$ (preferably whose domain is $\mathbb{R}$) such that, applying $f^n$ (i.e. applying it $n$ times) to $x$ gives you back $x$ for any $x$ in the ...
- 2,506
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33
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Why is R-convergence a useful notion?
Consider the definition of $R$-convergence as given in Definition 9.2.1 of "Iterative solutions of nonlinear equations in several variables" by Ortega and Rheinbolt.
Let $A$ be a fixed-...
5
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98
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Convergence of fixed-point iteration $x = \frac{1 + Px}{2 + Px + Nx}$
I am studying a problem related to trust propagation / rating in a network. Users rate each other either negatively or positively. From this I want to compute an overall rating vector $\boldsymbol{r} \...
- 51
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51
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Householder versus Halley speedtest?
Consider this Householder's method :
https://mathworld.wolfram.com/HouseholdersMethod.html
versus this Halley's method :
https://mathworld.wolfram.com/HalleysMethod.html
Which method is the most ...
- 18.3k
-1
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1
answer
127
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Why fixed point iteration of $x^3 = 1-x^2$ doesn't converge when $x_0 = 0$?
I am new to numerical methods and am currently learning Fixed point iteration. I have learned that if you can express $x = g(x)$, and $|g'(x_0)|<1$, then the sequence, $x_{n+1} = g(x_n)$ converges ...
- 53
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86
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Convergence of discrete replicator equation
One of the evolution equations used in evolutionary game theory is the Replicator type II dynamics
$$
x_i(t+1)=x_i(t)\frac{(Ax(t))_i}{x^T(t)Ax(t)}
$$
where $A$ is an $M\times M$ payoff matrix with ...
- 23
1
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0
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74
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Proving Uniqueness of Nonlinear Equation Solution
Consider the following system of two nonlinear equations:
$h'_{x}( x) -C y -C_{1}=0$
$h'_{y}( y) -C x -C_{2}=0$
With $x,y\in[0,1]$
The $h$ functions both satisfy $h(0)=0$, $\lim _{t\rightarrow 1} h( t)...
- 182