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Questions tagged [fractional-iteration]

The study of fractional self-iterations of a map. A basic example is the analysis of functional square roots of a map $g$, i.e. solutions $f$ to the functional equation $f\circ f=g$ are functional square roots and solutions to $f^n=g$ are functional nnth roots.

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Power Iteration algorithm for computing the eigenvalues

Recently I was looking for Power Iteration algorithm for computing the eigenvalues and I can't under stand what is it exactly and where it used. I would appreciate if you give me an explanation
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Real numbers arbitrary closeness (iterated fraction)

I was playing with some iterated functions/fractions when I stumbled across this one $$\frac{1+\frac{1+\frac{1+...}{1-...}}{1-\frac{1+...}{1-...}}}{1-\frac{1+\frac{1+...}{1-...}}{1-\frac{1+...}{1-...}}...
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How can I create currency note count calculator if I have currencies which has decimals and total amount which also has decimals?

If i have currencies that are in decimals {0.7,0.8,1.4,0.5} and total amount which also has decimals (example: 1279.6) and I tell you that you need to make total within certain number of currency note ...
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Fourier transform and fourth root

Given a well-behaved convex function $f(t):\mathbb{R}\to \mathbb{R}$, its Fourier transform (FT) $\hat{f}(\omega)=\mathcal{F}[f(t)](\omega)$ is positive (and decreasing) proof here. It follows that ...
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Find $f(x)$ satisfying $f(f(x))=x^x$

By inspection my attempts are always wrong. I really have no idea and given up. How to find $f(x)$ satisfying $f(f(x))=x^x$? My attempts: $f(x)=x^x$ $f(x)=x^{1/x}$ $f(x)=\frac{1}{x^x}$ My ...
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Is the Schröder Equation valid for higher dimensional iterated maps?

The Schröder's functional equation is the eigenfunction equation for the composition operator given as: $$ \psi \circ y (x) = s \cdot \psi(x) ~~~~~~~~~~~ (1) $$ The interesting bit about it (at ...
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Is there any definition of the sum of $n$ summands, when $n$ is not a natural number?

I think, from the algebraic point of view, that this doesn't make sense, since the addition of $n$ terms is defined "inductively" from the binary operation of the addition. This even yields the ...
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Find $f(f(\cdots f(x)))=p(x)$

$\newcommand{\nest}{\operatorname{nest}}$Let's define a function $\nest(f, x, k)$, which takes a function $f$, an input $x$, and a non-negative integer $k$, and calls $f$ on $x$ repeatedly ($k$ times)....
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Functional Square Root of Digit Sum?

Define $\text{sdig}(n)$ to be the sum of the decimal digits of $n$, where $n$ is a positive integer. My question is as follows: Does there exist a function $h:\mathbb Z^+\mapsto\mathbb Z^+$ such ...
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Do two exponential spirals intersect?

I have lists of complex points: orbit of complex point z under quadratic function f(z) = z*z I know that lists are: z, z^2, z^4, z^8, ... (r,t), (r^2, 2*t), .....
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Golden ratio - almost integer

Let, $a_1=1, b_1=3, c_1=3$ and $d_1=1$ Where $a_2=a_1+c_1,b_2=b_1+d_1, c_2=a_1+b_1+c_1 $ and $d_2=a_1+d_1$ and we defined $$a_n=a_{n-1}+c_{n-1}$$ $$b_n=b_{n-1}+d_{n-1}$$ $$c_n=a_{n-1}+b_{n-1}+c_{...
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Almost-half iterate of $x^2+1$

Because I can't find a function $h:\mathbb R\mapsto\mathbb R$ with the property $$h^{\circ 2}(x)=x^2+1$$ I'm looking for a function that almost has that property - that is, I would like to find a ...
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If $q(x)=x^2+1$, does $q^{\circ 1/2}$ exist? [duplicate]

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 $$q(x)=x^2+1$$ ...
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Functional Square Root of Piecewise Functions

Let $s:\mathbb R\to\mathbb R$ be the function defined as $$s(x)=x+(-1)^{\lfloor x\rfloor}$$ Find a function $t:\mathbb R\to \mathbb R$ such that $t^{\circ 2}=s$, or prove that no such function ...
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Partial iteration and inverse iteration of $e^x$

Find the general $f^n(x)$ where $f^1(x)=e^x$ $f^{a}(f^{b}x)=f^{(a+b)}(x)$ where $n,x\in\mathbb R$ I'm fairly confident that no two $f^n(x)$ with different values of $n$ will intersect, ...