07/11/2008, 06:28 AM (This post was last modified: 07/11/2008, 09:16 AM by bo198214.)
Hi Andrew,
You have a pedagogical talent. I like that this FAQ is not overformalized, and there are verbal explanations, but its still concise.
One suggestion would be to mention on page 32 , before table, explicitly that this table as it is is valid for real x.(since some functions in the table like W could be definitely perceived by some readers as functions of complex variable) . In general, such clarifications ( where x is real, where z is complex, range of t) would help (me), but may be not needed for practicionists in computing? Or than in the beginning of FAQ there could be a notation table where it is stated that x is always real, z always complex , t always ? ,etc.
In total, Great stuff. Do not aim for reducing explanations, though, so it becomes just formulas and manipulations with logical symbols, as most modern mathematical texts. E.g this is now clear in regard to superlogarithm on p.26 ,and may be can be made even more clear with examples:
Quote: Using the third interpretation of super-logarithms (i.e. the number of times you
must apply a logarithm before one is obtained)
I would though suggest there "before value=1 is obtained".
More examples in general would help, but I understand it is such a big job, since in Your programm there are still many places You want to fill theoretically.
Ivars
Moderator's note: Moved from thread "Tetration FAQ". Ivars, it was explicitely written that the thread "Tetration FAQ" is not for the discussion of the FAQ.
07/11/2008, 09:21 AM (This post was last modified: 07/11/2008, 09:31 AM by bo198214.)
Hey Andrew,
first a lot of thanks to merge/split the documents.
For further joint editing I started another git repository, with the contents provided in your zip:
git@github.com:bo198214/tetration-faq.git
and added you as collaborator. As usual you just start with a
git clone git@github.com:bo198214/tetration-faq.git
And afterwards you can pull and push with
git pull origin master
git push origin master
where origin is an alias for git@github.com:bo198214/tetration-faq.git, which you can list by "git remote"
I think we should use the github e-mail feature to discuss further changes that are too technical to be discussed here.
unfortunately I've little time this week due to intense handcraft-homework. However, I tried to collect some more ideas for the FAQ in the evening, which I want to propose here.
Andrew, Henryk - I hope you don't feel pissed from the fact, that I didn't include the already present material appropriately - I wanted first expose an own style of explanation of some matters (as far as I got them). Perhaps you like this or that, this or that example or this or that way of explanation and possibly take some thing into the faq.
I tried to get more used to latex and downloaded a Texmacs-version, but I haven't things not handy yet, so I just post it here in plain ascii.
I separate the main points into several posts with an appropriate heading - perhaps this is also an idea, how to expose some FAQ-points in the forum with direct access via its subject-line.
Well... I'll wait for response before proceeding...
We use the iteration-paradigm:
a is a fixpoint if (<op>,b,a)°h = a for all h
Examples:
--- iterated addition ----------------------------------------------------
(+,b,a)°h = a no fixpoints a, except for base b=0
a + 0 + 0 + ... + 0 = a
polynomial expression:
f_b(x) = b + x
f_b°h(a) = b*h + a ==> if b=0 then any a is a fixpoint
[for extension to the ring of powerseries see matrix-approach]
--- iterated multiplication ----------------------------------------------------
(*,b,a)°h = a a = 0 for all bases b
a = arbitrary for base b=1
0 * b * b * ... * b = 0
a * 1 * 1 * ... * 1 = a
polynomial expression:
f_b(x) = 0 + b*x
f_b°h(a) = a*b^h ==> if b=1 then any a is a fixpoint
==> for other b then a=0 is a fixpoint
[for extension to the ring of powerseries see matrix-approach]
--- iterated exponentiation ----------------------------------------------------
(^,b,a)°h = a b = a^(1/a) for all a<>0
multiple a for the same b
b^a = (a^(1/a))^a = a^1 = a
Series expression:
f_b(x) = 1 + log(b)*x + log(b)^2*x^2/2! + ...
f_b°h(a) ==> if log(b) = log(a)/a then a is a fixpoint
using b=a^(1/a) :
f_b(a) = 1 + log(a)*(a/a) + log(a)^2*(a/a)^2/2! + ...
= exp(log(a))
= a
[see also: matrix-approach]
[see also: <literature>]
--- iterated decremented exponentiation ----------------------------------------
(dxp,b,a)°h = a a = 0 for all bases b
a = <multiple values> for all bases b
For an iterable function f(x) the fixpoint is given (if it exists)
by
f°h(a) = a
for any h.
If a is unknown, then we may try to *find* it simply by iteration,
beginning with a suitable init-value a0:
a0
f(a0) = a1
f(a1) = a2 = f°2(a0)
f(a2) = a3 = f°3(a0)
...
If this converges to a fixed value a, then we have
f(a) = a = f°inf(a0)
and a is an attracting fixpoint.
For instance, Euler showed, that - using b=sqrt(2) and f(x) = b^x - the sequence
b^1, b^b^1,... or
f°1(1),f°2(1),f°3(1),...
converges to 2 so that
f(2) = 2
Since the fixpoint could be find by iteration with a different initial
value, a=2 is an attracting fixpoint of f(x)
But he also discussed, that another fixpoint is a=4, such that f(4) = 4.
However, this fixpoint cannot be found by iteration from another
initial value; if the difference delta from delta = 4 - a0 greater than
zero, the iteration leads to increasing delta - the iteration either
converges to a=2 (the attractin fixpoint) or diverges.
So in this case, a=4 is called a "repelling" fixpoint.
In general,
if |f'(a)| < 1 then a is an attracting fixpoint
if |f'(a)| > 1 then a is a repelling fixpoint
-- What does half-/ fractional-/continuous-/complex iteration mean?
This problem can only be expressed in terms of the series-paradigm, although
the Reihenalgebra-concept can possibly be seen as an equivalent approach.
The question is - for example -
given a function f(x) = y
what is the function g(x) such that g(g(x)) = y
g(x) is then called the half-iterate of f(x) and is a fractional-iterate
The terms half,fractional and continuous are used if the iterator-parameter
is thought as real, but continuous; if the iterator is thought as a general
complex number, sometimes the term continuous is as well used.
For real iterator h
f°h(x) = f°(n+r)(x) = f°n(f°r(x)) where n is integer and r is fractional
Example using powerseries:
For a function f(x), defined by powerseries, with constant term=0 (f(0)=0) and f'(0)=/=0
it is easy to find the half iterate g(x) by manipulation of the powerseries and
equating coefficients at like powers of x:
g(g(x)) = a g(x) + b g(x)^2 + c g(x)^3 + ...
= a*( ax + bx^2 + cx^3 + ...)
+ b*( ax + bx^2 + cx^3 + ...)^2
+ ...
= a^2 x + (ab + ba^2) x^2 + ...
= f(x) = A x + B x^2 + ...
then by equating coefficents at like powers of x , either a=+sqrt(A) or a=-sqrt(A)
and all other coefficients can then uniquely be determined, so we get
g(x) = sqrt(A) x + B/(sqrt(A) + A)*x^2 + ...
For general fractional iteration-heights the handling of the appropriate powerseries
is much more complicated and suggests the tools of algebra of infinitely-sized matrices.
Formal analytical handling for general functions is much developed and mostly
based on
(see:) Abel - functional relation
Schröder - functional relation
[see : matrix-approach, matrix-logarithm, matrix-diagonalization,
binomial-expansion using functions, ~ using matrix-operators,
function-logarithm (ILog) , exponential polynomial interpolation
<literature>]
[see : Faa di Bruno-formula, ... ]
[see further <literature>: iteration-theory, time-series, dynamical systems]
I like it! Your first post would go well in the FAQ, and the other three posts could all go under the "Iteration" chapter of the Ref. Thank you very much for putting this discussion together.