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Superroots (formal powerseries) - Gottfried - 10/25/2009
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
Code:
´ g(x) = (1+x)^(1+x) - 1Using the exponential-/logarithm-series for this we write first
Code:
´ g(x) = exp( log(1+x)*(1+x)) - 1and 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 xNote, 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) = xand 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) + 1This 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
RE: Superroots (formal powerseries) - andydude - 10/26/2009
Nice job. Might I also link to this thread, where I posted a document with some super-root stuff.
RE: Superroots (formal powerseries) - Gottfried - 10/26/2009
(10/26/2009, 01:50 AM)andydude Wrote: Nice job. Might I also link to this thread, where I posted a document with some super-root stuff.
Thanks! Yes, my idea were to just collect the links to relevant threads/posts here to have a starting place for readers who want to step into the problem of the superroot.
I'll just start a list of links; would be good to be extended.
Gottfried
Superroot: List of relevant msg's/threads - Gottfried - 10/26/2009
A.Robbins On Analytic iteration... (2009) See pages 18-21; some more formulae are derived there
"limit of self-superroots is exp(exp(-1))..."
(msg 2009-10) initial conjecture
(msg 2009-10) proof
a curiosity/paradox(?) with limit of consecutive orders of superroots
(msg 2009-02) superroots of real numbers x>e
Connection between superroot and Lambert-W:
(msg 2008-03) Lambert W function and the Super Square Root
(msg 2008-03) Deriving tetration from selfroot?
I.Galidakis Hyperroot with Lambert W (2009) This discussion gives also code for maple for some investigations
Ideas for superroots of higher order
2019 Aug: MSE-question on superroots
2019 Aug: My own answer for generalization of the Lambert-W-formula for the higher order superroots
A small treatize on the generalization of Lambert-W on my webspace (2015/2017) (referred to in my MSE-answer)
RE: Superroots (formal powerseries) - robo37 - 10/28/2009
Dudes can someone put some of this information on Wikipedia? On this page? You see people seem to want that article to be merged with the one about tetration and as I can't think of any more information about the function I seem to be fighting a loosing argument.
RE: Superroots (formal powerseries) - Gottfried - 10/28/2009
(10/28/2009, 09:20 AM)robo37 Wrote: Dudes can someone put some of this information on Wikipedia? On this page? You see people seem to want that article to be merged with the one about tetration and as I can't think of any more information about the function I seem to be fighting a loosing argument.
But this is so far only "original research" while the encyclopedic approach of wikipedia focuses on citations of standard material (refereed journals, books which have undergone a full criticism and publishing process).
As far as we cannot locate such material concerning "superroot" independently of the broader context of tetration we cannot help in this case.
I myself am unfortunately especially useless for this since I have little access to relevant literature (resp time to acquire relevant articles & to understand the professional lingo) and only recall only marginally that/where I've read here&there about such x^x- and superroot-applications in praxi (as I stated in wikipedia the text about "wexzal" and the superroot for the calculation of explosion-driven projectile- and automobile-moving).
So, well, having said this I'll better hand your request over to the next one...
Gottfried
Superroots article "WexZal" - Gottfried - 10/29/2009
I've uploaded that "WexZal"-article; it is originally in plain text (around 1998, if I recall right) but I thought it might be better to have it in .pdf. In a private mail the author welcomed to distribute this text. I'll add that statement later (I'll put it into Henryk's lib, too)
WexZal.pdf (Size: 666.98 KB / Downloads: 1,707)
A treatize of iterated exponentiation and inverse, seemingly initially inspired by the question of superroot of order 2. Mentions two practical applications
RE: Superroots article "WexZal" - bo198214 - 10/29/2009
(10/29/2009, 05:56 AM)Gottfried Wrote: I've uploaded that "WexZal"-article; it is originally in plain text (around 1998, if I recall right) but I thought it might be better to have it in .pdf. In a private mail the author welcomed to distribute this text. I'll add that statement later (I'll put it into Henryk's lib, too)
Thanks a lot Gottfried!
Why is this not published as a proper book?
I think there should be included some more data,
like finishing date and some jacket text about the authors.
And great: it contains appliations, as this really often asked and some ignorant physicists just think that there can not be applications.
RE: Superroots article "WexZal" - Gottfried - 10/29/2009
(10/29/2009, 12:08 PM)bo198214 Wrote: Why is this not published as a proper book?Well I know nearly nothing around that text; if I google today for the authors I only find a reference at Rob. Munafo's site without presenting further reference - that's all.
I think there should be included some more data,
like finishing date and some jacket text about the authors.
And great: it contains appliations, as this really often asked and some ignorant physicists just think that there can not be applications.
Here is a msg of si.math, which I saved when I came across it:
see also google groups
Code:
Betreff: Online "book" about solution of y=x^x
Von:"L" <believe@ptw.com>
Datum:Thu, 20 Jul 2000 17:47:57 -0700
Hello SCI.MATH,
I have some items on my web site that might be of interest to those
researching equations in the form of y=x^x and related functions. This is
a topic that appears from time to time in SCI.MATH. Myself and a friend
began researching the solution of y=x^x back in 1975 while we were still
in high school. The ASCII-based "book" on my web site is a compilation
of our results from 1975 to 1995. It is written in a very informal style
that we hope is informative to users of this news group. We have included
two FORTRAN-77 files, SKRFIT.F and WEXZAL.F which are included with the
book as they implement many of the topics in the book. The web site is:
http://www.networkone.net/~believe
Questions and comments can be sent to believe@networkone.net and we will
try to answer questions as time permits.
The outline of the book is:
Chapter 01 - Basic discussion of y=x^x.
Chapter 02 - Inverse of x^x. This is called the Coupled Root [crt(x)].
Chapter 03 - Coupled Root of large numbers. The "Wexzal" = crt(10^x).
Chapter 04 - Closed form solution of equations via Wexzals.
Chapter 05 - Integrals involving Wexzals.
Chapter 06 - Asymptotics and limits with Wexzals.
Chapter 07 - Numerical computation of wzl(x). Calculator fans... take note!
Chapter 08 - Misc items involving Wexzals. Incomplete.
Chapter 09 - Curve-fitting with Wexzals. Incomplete.
Chapter 10 - New type of graphs involving 1/wzl(1/x).
Chapter 11 - Math model of projectile deacceleration via Wexzals.
Chapter 12 - Barrel length vs. velocity question answered with Wexzals.
Chapter 13 - Car acceleration model via Wexzals.
Chapter 14 - Equalities and inequalities of Coupled Roots & other functions.
Chapter 15 - Closed form solution of equations via Coupled Roots.
Chapter 16 - Table of integrals involving Wexzals & Coupled Roots.
Chapter 17 - Table of Asymptotic expansions and limits.
Chapter 18 - Special values and table of functions from Chapters 11-13.
The file WEXZAL.F is a collection of "home-brewed" functions and routines
that are discussed in the book. Included is the computation of the
solution of y=x^x and y=x^(x^x) along with the first 4 derivatives of
y=wzl(x).
The program SKRFIT is a non-linear 2 parameter curve fitting program that
was written to solve the program of barrel lengths vs. muzzle velocity of
firearms (See chapter 12). It uses routines from WEXZAL.F which contains
the computation of WZL(x) and other items as discussed in the book.
To use the program SKRFIT after it has been compiled, one creates a
datafile that contains free-formatted (x,y) data. This data is to be fitted
in the form of:
A
y = ------
B
WZL(-)
x
where A and B are to be determined. An initial value of B is entered. This
is called B0 by the program. The value of B0 should be higher than the
expected value of B otherwise the program might not converge. An example
data set and a run of the program follows.
[...<snipped>, G.H.]
Anyway, we hope you find the "book" informative and entertaining.
-JayCode:
Betreff:Re: Wexzal book
Von:"lori" <lori1@antelecom.net>
Datum:Sun, 24 Nov 2002 09:15:37 -0800
[...]
Thank you for your quick reply. I just wanted to say again that
you have my permission to quote and use the Wexzal book in
anyway you see fit (just please quote the source). I am pleased
that you find it useful.
For your interest, I am currently researching tetration for real number
arguments.
See TETRATION in google for more info.
[...]
-JayRE: Superroots article "WexZal" - bo198214 - 10/29/2009
Thank you Gottfried for the additional information.
However in the book the relation between Lambert W and superroot/coupled root is not described, which was mentioned by James Knight here.