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 Tetration FAQ Discussion bo198214 Administrator Posts: 1,394 Threads: 91 Joined: Aug 2007 02/25/2008, 10:11 AM Oh gosh, I didnt append the tex source. Must somehow gone wrong. Here it is now: Attached Files   20070809tetrationfaq.tex (Size: 29.55 KB / Downloads: 599) Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 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. bo198214 Administrator Posts: 1,394 Threads: 91 Joined: Aug 2007 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. Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 07/17/2008, 10:34 AM Hi - 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... Gottfried Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 07/17/2008, 10:45 AM Code:-- What is a fixpoint? We use the iteration-paradigm: a is a fixpoint if (,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: ] --- iterated decremented exponentiation ----------------------------------------   (dxp,b,a)°h = a   a = 0   for all bases b                     a =    for all bases b                             b^0 - 1 = 0               [see also: ]        -------------------------------------------------------------------------------- The aspect of fixpoints is an important tool for adapting powerseries with constant term to enable fractional iterations. [see fixpoint-shift] =================================================================================== -- What is a "repelling"/"attracting" fixpoint?    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 =================================================================================== Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 07/17/2008, 10:47 AM Code:-- What does f o g mean? We use the iteration paradigm. Given two functions f(x) and g(x), then this is a short-form for f(g(x)) Note, that f o g =/= g o f  in the general case (Non-commutativity) -- What does f°h o f = f o f°h mean? If the special case     g(x) = f°h(x) is given, then     g(f(x)) = f(g(x)) = f°(h+1)(x) =================================================================================== Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 787 Threads: 121 Joined: Aug 2007 07/17/2008, 10:48 AM Code:-- 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:   Assume f(x) = Ax + Bx^2 + Cx^3 + ...   target g(x) = ax + bx^2 + cx^3 + ...   satisfying g(g(x)) = f(x)   then      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          ]   [see : Faa di Bruno-formula, ... ]   [see further : iteration-theory, time-series, dynamical systems]    ==================================================================================== Gottfried Helms, Kassel andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 07/17/2008, 02:52 PM 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. Andrew Robbins « Next Oldest | Next Newest »

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