Iteration of function

Fig.1. Iterates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(z)=z^2~} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~y\!=\!T^n(x)\!=\!x^{2^n}~} for various $n$

Fig.2. Iterates of $T(z)=\exp(z)~$: $~y\!=\!T^{n}(x)\!=\!\exp ^{n}(x)~$ for various $n$

Fig.3. Iterates of $T(z)=z\exp(z)~$: $~y\!=\!T^{n}(x)\!=\!\mathrm {zex} ^{n}(x)~$ for various $n$

Fig.4. Iterates of $T(z)=z+\exp(z)~$: $~y\!=\!T^{n}(x)\!=\!\mathrm {tra} ^{n}(x)~$ for various $n$

Iterate of funciton , or iterate (итерация), is function, expressed as repetition of another (iterated) function, that may be called iterand.

Any function by itself is considered as its first iterate.

The zeroth iterate of a function is supposed to be Identity function.

The minus first iteration of a function is its inverse function

Iteration of a function $T$ is denoted with superscript; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^n} means the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} th iteration of function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} , id est,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle T^n(z) =T\!\Big(T\!\big( .. T(z).. \big)\Big)}

where function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} in the right hand side of equation is repeated Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} times.

This may be expressed with equations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~T^0(z)\!=\!z~} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ T^n(z)=T\Big( T^{n-1}(z) \Big)~} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\! > \!0}

Term iteration may have also another meaning, indicating repetition of some process (even if such a repetition does not define a new function).

Physical sense

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is transfer function of some device, say, amplifier, then, the transfer function of combination of two such devices can be expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^2} ; and the transfer function of combination of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} identical devices can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^n} .

Nest

In Mathematica, the special operation is incorporated, called Nest. Usually, Nest has 3 arguments. The first argument is name of iterand, which is function. The second argument is initial value of the iterations, and the last argument is number of iteration. With Nest, the iteration of function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} can be written as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^n(z)=\mathrm{Nest}[f,z,n]}

In Mathematica, all arguments of all functions should be written in squared parenthesis. Up to year 2012, the Nest is implemented only for the cases, when number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} of iterations can be simplified to a natural number, expressed with integer constant. Attempts to call Nest to evaluate a non–integer iterate of a function are interpreted as error.

Non–integer iterate

The non–integer iterate is simple and obvious in the case when the iterand is linear function; then the iteration can be expressed through the elementary function. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(z)=a+b z} . Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle T^n(z)=a \frac{1\!-\!b^n}{1\!-\!b} + b^n z}

This expression allows the straightforward interpretation for non–integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

The case of power function is also simple. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(z)=z^a} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\!>\!1} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^n(z)\!=\!z^{a^n}} . In this expression, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} has no need to be integer. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\!=\!2} , the iterations of quadratic function are shown in Figure 1.

For more general case, the non–integer iterates of a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} can be defined through its superfunction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and the Abel function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\!=\! F^{-1}} ; the iterand Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is treated as the transfer function, and the superfunction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} satisfies the transfer equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z\!+\!1)=T\!\Big( F(z) \Big)}

Then, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} th iteration of the transfer function can be expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^n(z)=F\Big(n+G(z) \Big)}

In this expression, parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} also has no need to be integer. Once holomorphic functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} are established, the function can be iterated even complex number of times. For a given transfer function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} , the establishment of the superfunciton Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and the Abel function $G$ is not trivial, but solvable problem. It had been formulated in 1950 by Helmuth Knezer for iterates of exponential, the half-iterate of the exponential appears in the title of his article. In century 21, such a function had been calculated and plotted ; the iterates of the exponential are shown in Figure 2. Then, the formalism had been extended to other functions , in particular, to the exponentials to other bases, factorial and the logistic sequence.

For the non–integer iterates, it should be assumed that $T$ is holomorphic function, and the superfunction $G$ is also expected to be holomorphic. In order to provide uniqueness of the superfunction, the certain requirements on its behavior should be assumed. Usually, $F$ approaches some fixed points of function $T$ at infinity (see the special articles Superfunction and Abel function). Usually, the physically–meaningful transfer functions have a real fixed point and the corresponding superfunciton and the Abel function can be evaluated with the regular iteration; if the fixed point are com[lex (not real), the superfunction and the Abel function can be constructed with the Cauchi iteration .

In 2011, Henryk Trappmann had suggested an example of a transfer function that has no fixed points, let it be called Trappmann function or tra; $\mathrm {tra} (z)=z\!+\!\mathrm {e} ^{z}$. Such a transfer function was supposed to be difficult to iterate. However, the superfunction (id est, "SuperTrappmann") can be expressed through that for function zex, $\mathrm {zex} (z)\!=\!z\exp(z)$, and then the "SuperTrappmann" can be called simply hen, and the iterate can be expressed as usually through the hen and the "AbelTrappmann" ahe$~=\mathrm {hen} ^{-1}$. The iterates of zex are shown in Figure 3, and the iterates of the Trappmann function tra are shown in Figure 4.

It seems, that the non-integer iterates can be defined for any holomorphic function, but the rigorous mathematical proof of such conjecture is required.

In some cases, the non–integer iterate can be defined also throgh the Schroeder function.

In general, the non-integer iterate is not unique, and the specific behavior of the superfunction $F$ should be assumed for the unigieness.

Examples

This section offers a little bit more detailed description of figures, mentioned above.

Elementary functions

Let

$\displaystyle T(z)\!=\!{\frac {a^{2}+(2a\!+\!b)z}{b-z}}$

The corresponding superfunction and Abel function are

$\displaystyle F(z)={\frac {az\!+\!b}{1\!-\!z}}~$, $~\displaystyle G(z)={\frac {z\!-\!b}{a\!+\!z}}$

give the iterate

$\displaystyle T^{n}(z)\!=\!{\frac {a^{2}+{\Big (}a(1\!+\!n)+b{\Big )}z}{a+b-an-nz}}$

More "solvable" tranfer functions can be constructed using Table of superfunctions. However, for some cases the solution cannot be simply expressed with elementary functions. Few of them are mentioned below.

Special functions

Figure 1 shows iterates of quadratic function, $~T(z)\!=\!z^{2}~$. In this case, the $n$th the iterate $~T^{n}(z)\!=\!z^{2^{n}}~$ can be expressed through the power function.

Figure 2 shows iterates of the exponential, $~T(z)\!=\!\exp(z)$. In this case, for non–integer $n$, the $n$th iterate $~T^{n}(z)\!=\!\mathrm {tet} {\Big (}n\!+\!\mathrm {ate} (z){\Big )}~$ is expressed through tetration tet and ArcTetration ate, and cannot be easy represented through elementary functions.

Figure 3 shows iterates of function zex, $~T(z)\!=\!\mathrm {zex} (z)\!=\!z\exp(z)~$. In this case, for non–integer $n$, the $n$th iterate

$~T^{n}(z)\!=\!\mathrm {SuZex} {\Big (}n\!+\!\mathrm {AuZez} (z){\Big )}~$

can be expressed through functions SuZex and AuZex, which are, of course, superfunction and Abel function for the transfer function zex.

Figure 4 shows iterates of the Trappmann function, $~T(z)\!=\!\mathrm {Tra} (z)\!=\!z\!+\!\exp(z)~$. In this case, for non–integer $n$, the $n$th iterate

$~T^{n}(z)\!=\!\mathrm {Tra} ^{n}(z)\!=\!\mathrm {hen} {\Big (}n\!+\!\mathrm {ahe} (z){\Big )}~$

is expressed through the Henryk function hen and the ArcHenryk function ahe , and cannot be easy represented through elementary functions. However functions hen and ahe can be expressed in terms of SuZex and AuZex mentioned above in the following way:

$\mathrm {hen} (z)=\ln {\Big (}\mathrm {SuZex} (z){\Big )}$

and

$\mathrm {ahe} (z)=\mathrm {AuZex} {\Big (}\exp(z){\Big )}$

The number of examples can be greatly extended, using the Table of superfunctions, that offers triads $T,F,G$, and general expression $T^{n}(z)=F{\Big (}n\!+\!G(z){\Big )}$. In this way, one can plot the iterates of exponential to various bases, iterates of factorial, iterates of the logistic operator, and iterates of any functions, assuming, that we can choose the appropriate superfunction and the Abel function. In principle, any holomorphic function can be iterated any number $n$ of times, even complex number of times.

This article is adopted from TORI .

References

http://en.wikipedia.org/wiki/Iteration

Keywords

Superfunction, Abel function, Transfer equation, Abel equation