10/25/2009, 12:09 PM

Recently I found Andrew's remark in "designing a tetration library", that the superrots were not yet well developed. Facts on superroots seem to be spread over various threads; so to have some collection under an expressive title I've put together some details, tending to compile more information from time to time as they appear.

(@admin: maybe that msg is better located in some related thread, for instance the tetration library thread)

Ok, I've put the text in plain text and am lazy to MimeTex it today, perhaps I'll rework it next days.

======================================================================

A short collection concerning superroots

======================================================================

Starting point is the nice powerseries for

Using the exponential-/logarithm-series for this we write first

and get

This series has the nice property, that the constant term vanishes and also, that the linear term has coefficient 1, so g(0)=0 and g'(0)=1, so we can do some common operations with it: inversion, iteration, ... getting exact coefficients.

Now we define higher orders by something like chaining, which is not exactly iteration of g().

The sequence of functions

gives similarly nice shaped powerseries, for instance

From this it is easy to define a sequence of functions for exponentialtowers of integer heights:

Note, that this is in principle all well known and is merely a restatement of known results.

The unusual aspect with that sequence of powerseries is, that the leading coefficients stabilize when the height increases, and thus we have a "strange" behave when the height increases to infinity.

Example: we get the following table of coefficients (where the rows contain the coefficients for one height and each column is associated with one power of x):

where the first column (containing zeros) represent the placeholders for the nonexistent constant terms. (The first row was appended to get a meaningfully interpretation for the "once"-iterate; it represents just g(x,1) = (1+x) -1 . The limit case for h->inf begins with the same coefficients as the last row of the table above)

------------------------------------------------------------------

Inversion

Since the g(x,h)-series have no constant term but a linear term with unit-coefficent, we can invert each of that g-series. Expressed by the appropriate f-function we get the superroot-powerseries for each integer height. Let's denote the inverse functions as gi() and fi(), then

for gi(x,2) we get

and a higher h, for instance

which again stabilizes for h->inf

======================================================================

The h'th superroot

======================================================================

The shown powerseries, formally seen, give the functons for the h'th superroots:

and the computation of the h'th superroot can be implemented by calls of the gi(x,h)-function:

This gives, if convergent, the base b, which must be exponentiated h times to equal the given value x.

-----------------------------------------------------------------------

Convergence:

Concerning the range of convergence I don't have an idea yet. For instance for g(x,2) we can guess a rate of decrease similar to µ/k^2 where k is the index and µ some constant, so we should have a range of convergence for |x|<=1 only. For f(x,2) consequently we had then 0<x<=2 .

It seems, that the occuring divergences are not "too strong" so that we can extend the domain for x using Euler-summation to get meaningful approximations even if only 64 or 128 terms are known.

-----------------------------------------------------------------------

Interpolation to fractional orders:

The special form of the powerseries, where each k'th coefficient becomes constant when chaining-height h>=k this poses a new challenge for the interpolation to fractional heights.

I have currently no idea how to proceed here...

-----------------------------------------------------------------------

(should be continued)

Gottfried

(@admin: maybe that msg is better located in some related thread, for instance the tetration library thread)

Ok, I've put the text in plain text and am lazy to MimeTex it today, perhaps I'll rework it next days.

======================================================================

A short collection concerning superroots

======================================================================

Starting point is the nice powerseries for

Code:

`´ g(x) = (1+x)^(1+x) - 1`

Using the exponential-/logarithm-series for this we write first

Code:

`´ g(x) = exp( log(1+x)*(1+x)) - 1`

and get

Code:

`´ g(x) = 1*x + 1*x^2 + 1/2*x^3 + 1/3*x^4 + 1/12*x^5 + 3/40*x^6`

- 1/120*x^7 + 59/2520*x^8 - 71/5040*x^9 + 131/10080*x^10

- 53/5040*x^11 + O(x^12)

This series has the nice property, that the constant term vanishes and also, that the linear term has coefficient 1, so g(0)=0 and g'(0)=1, so we can do some common operations with it: inversion, iteration, ... getting exact coefficients.

Now we define higher orders by something like chaining, which is not exactly iteration of g().

The sequence of functions

Code:

`´ g(x,1) = (1+x)-1 `

g(x,2) = (1+x)^(1+x) - 1

g(x,3) = (1+x)^(1+x)^(1+x) - 1

... = ...

gives similarly nice shaped powerseries, for instance

Code:

`´ g(x,3) = (1+x)^(1+x)^(1+x) -1`

= 1*x + 1*x^2 + 3/2*x^3 + 4/3*x^4 + 3/2*x^5 + 53/40*x^6

+ 233/180*x^7 + 5627/5040*x^8 + 2501/2520*x^9 + 8399/10080*x^10

+ 34871/50400*x^11 + O(x^12)

From this it is easy to define a sequence of functions for exponentialtowers of integer heights:

Code:

`´ f(x,h) = g(x-1,h) + 1 = x^x^x^...^x // h-occurences of x`

Note, that this is in principle all well known and is merely a restatement of known results.

The unusual aspect with that sequence of powerseries is, that the leading coefficients stabilize when the height increases, and thus we have a "strange" behave when the height increases to infinity.

Example: we get the following table of coefficients (where the rows contain the coefficients for one height and each column is associated with one power of x):

Code:

`´ 0 1 0 0 0 0 0 0 0 ...`

0 1 1 1/2 1/3 1/12 3/40 -1/120 59/2520 ...

0 1 1 3/2 4/3 3/2 53/40 233/180 5627/5040 ...

0 1 1 3/2 7/3 3 163/40 1861/360 33641/5040 ...

0 1 1 3/2 7/3 4 243/40 3421/360 71861/5040 ...

0 1 1 3/2 7/3 4 283/40 4321/360 102941/5040 ...

0 1 1 3/2 7/3 4 283/40 4681/360 118061/5040 ...

0 1 1 3/2 7/3 4 283/40 4681/360 123101/5040 ...

...

where the first column (containing zeros) represent the placeholders for the nonexistent constant terms. (The first row was appended to get a meaningfully interpretation for the "once"-iterate; it represents just g(x,1) = (1+x) -1 . The limit case for h->inf begins with the same coefficients as the last row of the table above)

------------------------------------------------------------------

Inversion

Since the g(x,h)-series have no constant term but a linear term with unit-coefficent, we can invert each of that g-series. Expressed by the appropriate f-function we get the superroot-powerseries for each integer height. Let's denote the inverse functions as gi() and fi(), then

for gi(x,2) we get

Code:

`´ gi(x,2) = x - x^2 + 3/2*x^3 - 17/6*x^4 + 37/6*x^5 - 1759/120*x^6 `

+ 13279/360*x^7 - 97283/1008*x^8 + 654583/2520*x^9 - 10800299/15120*x^10

+ 75519317/37800*x^11 + O(x^12)

and a higher h, for instance

Code:

`´ gi(x,12) = x - x^2 + 1/2*x^3 + 1/6*x^4 - 3/4*x^5 + 131/120*x^6`

- 9/8*x^7 + 1087/1260*x^8 - 271/720*x^9 - 2291/10080*x^10

+ 523/630*x^11 + O(x^12)

which again stabilizes for h->inf

Code:

`Table of coefficients for gi(x,h), h=1..`

------------------------------------------------------------------------------------

0 1 0 0 0 0 0 0 0 0 ...

0 1 -1 3/2 -17/6 37/6 -1759/120 13279/360 -97283/1008 654583/2520 ...

0 1 -1 1/2 7/6 -17/4 821/120 -25/12 -56269/2520 52079/720 ...

0 1 -1 1/2 1/6 1/4 -349/120 161/24 -2642/315 677/72 ...

0 1 -1 1/2 1/6 -3/4 251/120 -45/8 13897/1260 -10891/720 ...

0 1 -1 1/2 1/6 -3/4 131/120 -1/8 -5213/1260 8909/720 ...

0 1 -1 1/2 1/6 -3/4 131/120 -9/8 2347/1260 -4231/720 ...

0 1 -1 1/2 1/6 -3/4 131/120 -9/8 1087/1260 449/720 ...

0 1 -1 1/2 1/6 -3/4 131/120 -9/8 1087/1260 -271/720 ...

0 1 -1 1/2 1/6 -3/4 131/120 -9/8 1087/1260 -271/720 ...

....

-----------------------------------------------------------------------------------

======================================================================

The h'th superroot

======================================================================

The shown powerseries, formally seen, give the functons for the h'th superroots:

Code:

`´ fi(x^x,2) = x = gi(x^x-1,2)+1`

fi(x^x^x,3) = x

...

fi(x^^h,h) = x

and the computation of the h'th superroot can be implemented by calls of the gi(x,h)-function:

Code:

`´ ssrt(x,h) = fi(x,h) = gi(x-1,h) + 1`

This gives, if convergent, the base b, which must be exponentiated h times to equal the given value x.

-----------------------------------------------------------------------

Convergence:

Concerning the range of convergence I don't have an idea yet. For instance for g(x,2) we can guess a rate of decrease similar to µ/k^2 where k is the index and µ some constant, so we should have a range of convergence for |x|<=1 only. For f(x,2) consequently we had then 0<x<=2 .

It seems, that the occuring divergences are not "too strong" so that we can extend the domain for x using Euler-summation to get meaningful approximations even if only 64 or 128 terms are known.

-----------------------------------------------------------------------

Interpolation to fractional orders:

The special form of the powerseries, where each k'th coefficient becomes constant when chaining-height h>=k this poses a new challenge for the interpolation to fractional heights.

I have currently no idea how to proceed here...

-----------------------------------------------------------------------

(should be continued)

Gottfried