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some general thoughts
We had a short discussion in the newsroup sci.math recently, which motivated me to put together some thoughts, which came up recently.
I got some nice responses, so I thought it might be interesting for this forum, too.
I had no time to answer to the reply of Ioannis up to now, I hope, Ioannis, you forgive me that delay. (I didn't copy your answer here and hope, you don't take this as an affront. I'll answer to it, once I have something substancial to ask or to say)


I'd like to add two comments. May be, I'm on a wrong track,
because this requires to change view of things a bit and
introduces other weaknesses, of which I'm not aware currently.

The first idea is even a change of a very common view, so
this is a special slippery path. Let me put it here anyway.

a) -----------------------------------------------------
I tend to the opinion, that we should change the view
of tetration, going away from the assumption, that it

   b, b^b, b^b^b, b^b^b^...

with the dotted line *at the right* and

   b^^n = b^b^...^b (n times)

There is already a notation, coined by Andrew Robbins (and I
actually don't know from which sources this notation stems)
which supports my other direction of view:
He defines
   {b,x}^^n :=  b^b^b...^b^x

The idea is, that x is sort of starting value, and tetration
recursively appends bases (not exponents), so a coherent
notation is then for the sequence of partial expressions for
{b,x}^^inf is then
  x, b^x, b^b^x, ...^b^b^x
with the dotted line at the left.

This seems to be an super-artificial difference, but has its
own impact. It means, for instance, that the evaluation of
partial expressions is different from (and in opposite direction to)
the common view of partial evaluation, which was discussed
in a thread here already.
The infinite powertower b^b^b^.... cannot take a special
value "at the top" - since there is no top. And in this view
it is true, that (using r=sqrt(2)) the infinite expression
r^r^r^.... approximated by partial expressions converges to 2
and nothing else, so 2 is the only solution (which also reflects
the view of Lambert, Euler and the descendent discussion).

But with this convention we have no tools to include the
occurence (multiple) "fixpoints" in our formula, and must
leave this problem open.

If we redefine tetration as appending bases to a certain
starting value, as in

   {b,x}^^n :=  b^b^b...^b^x
   {b,x}^^inf := ...^b^b^x

then also the infinite expression (for n->inf) makes sense
and is better suited to the concept of evaluation of
partial expressions. We may then legally insert the various
fixpoints into x and always have valid expressions.

   {b,x0}^^inf := ...^b^b^x0 = x0
   {b,x1}^^inf := ...^b^b^x1 = x1
where x0,x1,... are the fixpoints.
Note, that this convention is also more coherent with the
lot of research in iteration-theory and theory of dynamical
systems, where always an "initial state" is discussed, to
which then an operator is applied - one time, two times, and
in generalization even fractional or complex "times".

*Only* if this notation is accepted, then the following is
  {sqrt(2),2}^^inf = 2
  {sqrt(2),4}^^inf = 4

This is *not* allowed (and we even cannot notate a second solution)
if we write
   sqrt(2)^sqrt(2)^... = ??
then we can have only one solution and we have no second
parameter to refer to the different fixpoints.
(See my posting some monthes ago
   subject: "sqrt(2)^sqrt(2)^sqrt(2)^... = 2 ? or 4 ?"
where I initiated a discussion of this)

Another spin-off of a redefinition is then the following.
  {b,x}^^h = y
then, if x is already x = b^b^1 = {b,1}^^2, or more
general x = {b,z}^^g then we may do a bit of arithmetic

   {b,x}^^h = {b,{b,z}^^g}^^h = {b,z}^^(g+h)

*iff* the bases for y and x are the same b.
ANd such an approach complies then easily to the idea
of a dynamical system, where an initial state (here: z)
is modified by an operator g and h times.

Well, *redefining* a common definition is easily a
crankish behave (see the discussion with A.Plutonium, for
instance), but in this case I'll take this risk
and propose a review of our definition.

b) ----------------------------------------------------

If a) is settled, then we may solve for x in
{b,x}^^inf = x
with multiple solutions, assigning branches to the h()-
function this way.

To find the various "fixpoints" then can be done by
a process, which reflects the idea of the newton-approximation,
for instance for finding the square-root of a number.
If we want to find x=sqrt(Z) we use an initial guess, say x0,
compute x1 = (Z/x0 + x0)/2 and iterate. This gives diminuishing
intervals for the error and approximates the solution to
arbitray precision.

With complex initial guesses for a fixpoint in the a)-definition
of infinite tetration, we can apply this idea equivalently.
The difference is here, that iterations spiral into the fixpoint
or spiral away - but seemingly always do spirals.
Assume an initial value, say x0, and apply an appropriate
function f(b,x0) and average
    x1 = (f(g,x0) + x0)/2
    x2 = (f(g,x1) + x1)/2
then a partial evaluation of such a spiral up to one approximate
circle gives a center-point near the "mean" of the  spiral
(or say: the assumed convergence-point).
    y1 = (x1+x2+x3.... xk)
which can then be a better approximation.
Then use  x0 = y1 and iterate again up to one complete rotation
and iterate. This is somehow like throwing a lasso, whose
diameter tightens radically.

Depending on the initial guess we may find then the different
fixpoints for b (well, we may implement the Lambert-W-function
allowing different branches as well)

c) This "lasso"-method has another useful consequence: the
initial spiral may even diverge : it will still circle around a
center - and this center may then be nearer to the fixpoint
than the initial guess.
This may then be seen as an equivalent of Euler-summation of
alternating divergent series: the angle of "rotation" here is just
pi, and finding the "center" of a real-valued alternating series
is then just a special case of this "lasso"-process with two
    x1 = something(x0)       (negative value)
    x2 = something(x1)       (positive value again, "circle" closed)
  (which is only a sketch here, since Euler-summation employs
   a binomial-transform of the values)

So a),b) and c) may be a useful framework for the discussion of
fixpoints for complex tetration. I would like to see a better
formal description here (which I cannot supply due to lack of
knowledge) and then a check, whether such description agrees
with the needs of compatibility of the assumptions/results
with the whole surrounding scene of theory and application of

Gottfried Helms
Gottfried Helms, Kassel

Messages In This Thread
some general thoughts - by Gottfried - 11/03/2007, 03:08 PM
RE: some general thoughts - by bo198214 - 11/03/2007, 09:08 PM
RE: some general thoughts - by Gottfried - 11/03/2007, 10:08 PM
RE: some general thoughts - by GFR - 11/04/2007, 03:23 PM
RE: some general thoughts - by Gottfried - 11/06/2007, 07:17 AM
RE: some general thoughts - by bo198214 - 11/06/2007, 06:15 PM
RE: some general thoughts - by Gottfried - 11/06/2007, 07:16 PM
RE: some general thoughts - by andrewan - 01/06/2010, 05:31 AM
RE: some general thoughts - by Gottfried - 01/06/2010, 07:38 AM

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