08/24/2010, 02:29 PM
Ok, I think I found it, where the multivaluedness of log comes into the play.
For the following assume the notations:
e = exp(1)
b : for base (here b=e )
t : first fixpoint, imag(t)>0
u : ln(t)
x' := x/t-1
x" := (x+1)*t
For simpliness I misuse the common notation with the double-caret in the following way:
y=x^^h : meaning y=exp°h(x) for the current base b=e, so (x^^h)^^k = x^^(h+k)
x=y``h : meaning x=log°h(y) for the current base b=e
So actually t can be approximated by t = limit{h->infty} ( z``h ), where z is an appropriate nearly arbitrary initial value.
If I set x0 = <some complex value> and x1 = x0^^1 and want to find the fractional iterates in that interval, I look first, whether the powerseries of the schroeder-function schroeder(x') converges for x0' and x1' .
If not I use w0 = x``h0 for some h0 so that for some eps, for instance eps=1e-4 according to my wished accuracy, we have abs(w0-t) < eps
Then I compute the value of the schroeder-function for this w0, say s0=schroeder(w0')
Then I compute a set of values S such that S(h) = s0*u^h for h=0..1 . The assumption behind this is, that in the limit S is dense and continuous for the continuous parameter h.
Then the inverse Schroeder-function is used for all values S(h) of the set S:
T(h) = schroederInv(S(h))" = (schroederInv(S(h)) +1)*t
and finally I have to correct for the initial integer-height-reduction by h0:
X(h) = T(h)^^h0
Again we assume, that X(h) is in the limit continuous in that interval h=0..1, so that X(h) covers the interval x0^0 .. x0^^1 = x0..exp(x0) , the fractional iterates for one unit-height-interval.
Having that interval, the other unit-intervals for higher iteration-heights are then computed by integer-height iteration
X(m+h) = X(h)^^m
And now the problem of branch-selection becomes visible:
if I compute the set X(2+h) this way I get for the chosen initial value x0 some strongly varying imaginary values. If I do Y(2+h) = log(exp(X(2+h))) then the imaginary values of Y(2+h) are truncated modulo 2*Pi, and also that truncation removes "arbitrarily" multiples of 2*Pi. This is done by the exp-function, and thus the inversion by log cannot recover that integer multiples of 2*Pi.
Hmmm. Here I am reminded of my discussion of the tetration in log-polar-representation of complex numbers and the concept of a "winding-number" which should keep track of the integer multiples of 2*Pi when exponentiation takes place...
Also this suggests, that for a initial set X computed by fractional iterations of some x0 we must choose x0 and x1 such, that all X(h) for h=0..1 do not exceed 2*Pi*I in their imaginary component...
--------
Ah, all this seems to make things clearer now. (Perhaps I'm also going to understand the Cauchy-integral method this way)
And the special good news for me is, that I can now extend my Pari/GP-procedures for regular tetration also for bases having a complex fixpoint. At least some more partial solutions.
That this was not possible before has really been frustrating...
Gottfried
For the following assume the notations:
e = exp(1)
b : for base (here b=e )
t : first fixpoint, imag(t)>0
u : ln(t)
x' := x/t-1
x" := (x+1)*t
For simpliness I misuse the common notation with the double-caret in the following way:
y=x^^h : meaning y=exp°h(x) for the current base b=e, so (x^^h)^^k = x^^(h+k)
x=y``h : meaning x=log°h(y) for the current base b=e
So actually t can be approximated by t = limit{h->infty} ( z``h ), where z is an appropriate nearly arbitrary initial value.
If I set x0 = <some complex value> and x1 = x0^^1 and want to find the fractional iterates in that interval, I look first, whether the powerseries of the schroeder-function schroeder(x') converges for x0' and x1' .
If not I use w0 = x``h0 for some h0 so that for some eps, for instance eps=1e-4 according to my wished accuracy, we have abs(w0-t) < eps
Then I compute the value of the schroeder-function for this w0, say s0=schroeder(w0')
Then I compute a set of values S such that S(h) = s0*u^h for h=0..1 . The assumption behind this is, that in the limit S is dense and continuous for the continuous parameter h.
Then the inverse Schroeder-function is used for all values S(h) of the set S:
T(h) = schroederInv(S(h))" = (schroederInv(S(h)) +1)*t
and finally I have to correct for the initial integer-height-reduction by h0:
X(h) = T(h)^^h0
Again we assume, that X(h) is in the limit continuous in that interval h=0..1, so that X(h) covers the interval x0^0 .. x0^^1 = x0..exp(x0) , the fractional iterates for one unit-height-interval.
Having that interval, the other unit-intervals for higher iteration-heights are then computed by integer-height iteration
X(m+h) = X(h)^^m
And now the problem of branch-selection becomes visible:
if I compute the set X(2+h) this way I get for the chosen initial value x0 some strongly varying imaginary values. If I do Y(2+h) = log(exp(X(2+h))) then the imaginary values of Y(2+h) are truncated modulo 2*Pi, and also that truncation removes "arbitrarily" multiples of 2*Pi. This is done by the exp-function, and thus the inversion by log cannot recover that integer multiples of 2*Pi.
Hmmm. Here I am reminded of my discussion of the tetration in log-polar-representation of complex numbers and the concept of a "winding-number" which should keep track of the integer multiples of 2*Pi when exponentiation takes place...
Also this suggests, that for a initial set X computed by fractional iterations of some x0 we must choose x0 and x1 such, that all X(h) for h=0..1 do not exceed 2*Pi*I in their imaginary component...
--------
Ah, all this seems to make things clearer now. (Perhaps I'm also going to understand the Cauchy-integral method this way)
And the special good news for me is, that I can now extend my Pari/GP-procedures for regular tetration also for bases having a complex fixpoint. At least some more partial solutions.
That this was not possible before has really been frustrating...
Gottfried
Gottfried Helms, Kassel
