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 Tetration FAQ Discussion Gottfried Ultimate Fellow     Posts: 759 Threads: 117 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 « Next Oldest | Next Newest »

 Messages In This Thread Tetration FAQ Discussion - by bo198214 - 08/09/2007, 12:07 AM Literature - by Gottfried - 11/23/2007, 04:30 PM RE: Literature - by andydude - 12/17/2007, 07:31 PM RE: Tetration FAQ - by andydude - 01/13/2008, 08:43 AM RE: Tetration FAQ - by Ivars - 01/13/2008, 10:17 AM RE: Tetration FAQ - by GFR - 01/13/2008, 04:02 PM RE: Tetration FAQ - by andydude - 01/16/2008, 04:13 AM RE: Tetration FAQ Discussion - by Gottfried - 07/17/2008, 10:34 AM FAQ-discuss: What is a fixpoint? - by Gottfried - 07/17/2008, 10:45 AM FAQ-discuss: What does f o g mean? - by Gottfried - 07/17/2008, 10:47 AM FAQ-discuss: What does half-/ fractional-/continuous-/complex iteration mean? - by Gottfried - 07/17/2008, 10:48 AM RE: FAQ-discuss: What does half-/ fractional-/continuous-/complex iteration mean? - by andydude - 07/17/2008, 02:52 PM Intro and Formula sheet - by andydude - 11/13/2007, 01:21 AM RE: Intro and Formula sheet - by jaydfox - 11/13/2007, 02:39 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 03:15 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 03:25 AM RE: Intro and Formula sheet - by Gottfried - 11/13/2007, 06:08 AM RE: Intro and Formula sheet - by Gottfried - 11/13/2007, 06:12 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 08:37 AM RE: Intro and Formula sheet - by andydude - 11/13/2007, 09:09 AM RE: Intro and Formula sheet - by bo198214 - 11/13/2007, 10:42 AM RE: Intro and Formula sheet - by andydude - 11/15/2007, 03:16 AM RE: Intro and Formula sheet - by jaydfox - 11/15/2007, 03:38 AM RE: Intro and Formula sheet - by andydude - 11/30/2007, 05:32 PM [split] Tetration FAQ - by Ivars - 07/11/2008, 06:28 AM

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