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andydude · 07/09/2008, 02:38 PM

Gottfried Wrote:Henryk's formula is (I inserted (x) at f°t)

Hmm ... you mean Woon's formula ...

andydude · 07/09/2008, 02:57 PM

I guess I should be more specific. The key observation to show the equivalence is that
$n!\left({s \atop n}\right) = \prod_{k=0}^{n-1}(s-k)$
which means Woon's formula simplifies to:
$ A^s = w^s \left( 1 + \sum_{n=1}^{\infty} (-1)^n \left({s \atop n}\right) \left[ 1 + \sum_{m=1}^{n} (-1)^m w^{-m} \left({n \atop m}\right) A^m \right] \right) $
and letting $w=1$ this simplifies to
$ A^s = \left( 1 + \sum_{n=1}^{\infty} (-1)^n \left({s \atop n}\right) \left[ 1 + \sum_{m=1}^{n} (-1)^m \left({n \atop m}\right) A^m \right] \right) $
and assuming $0^0 = 1$ this simplifies to
$ A^s = \left( \sum_{n=0}^{\infty} (-1)^n \left({s \atop n}\right) \left[ \sum_{m=0}^{n} (-1)^m \left({n \atop m}\right) A^m \right] \right) $
and combining (-1) terms, this simplifies to
$ A^s = \sum_{n=0}^{\infty} \left({s \atop n}\right) \sum_{m=0}^{n} (-1)^{n+m} \left({n \atop m}\right) A^m $
and that is, as you call it, "Henryk's formula" from Woon's formula.

Andrew Robbins

***bo198214*** · 07/09/2008, 06:18 PM

andydude Wrote:
Gottfried Wrote:Henryk's formula is (I inserted (x) at f°t)

Hmm ... you mean Woon's formula ...

You can also call it Newton's formula:

$x^t = (x-1+1)^t = \sum_{n=0}^\infty \left(t\\n\right) (x-1)^n = \sum_{n=0}^\infty \left(t\\n\right)\sum_{k=0}^n \left(n\\k\right)(-1)^{n-k} x^k$

Woon just applied this to linear operators $A$ instead of $x$ . This is possible because you dont need the commutativity for those formulas to stay true.

However we cant calculate with iterations as with powers, because the composition is no more right distributive. $(f+g)\circ h=f\circ h+g\circ h$ but generally $f\circ (g+h)\neq f\circ g + f\circ h$ . The binomial formula however relies on full (both side) distributivity. Especially generally
$(f-\text{id})^{\circ n} \neq \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} f^{\circ k}$

The interesting thing however is that both expansions together are then again valid:
$f^{\circ t} = \lambda^t \sum_{n=0}^\infty \left(t\\n\right) \sum_{k=0}^n \left(n\\k\right) (-1)^{n-k} \frac{f^{\circ k}}{\lambda^k}$ for each $\lambda$ as long as the right side converges.

And I think this is new. Especially that the iteration by this formula is the same as regular iteration (which is sure for elliptic iteration, other cases are still to prove).

Gottfried · (This post was last modified: 07/10/2008, 07:32 AM by Gottfried.)

I've got better convergence for the binomial (Woon-) method when computing f°0.5(x) by

$\hspace{24} y_k = f^{\circ k+0.5}(x)$ (using Woon-method)
$\hspace{24} f^{\circ 0.5}(x) = f^{\circ -k}(y)$ (using integer iteration)

(but not yet for the Stirling-transformation. This became even worse)

Also, the value, computed by diagonalization is verified (and seems to be the best approximation, see below)

Here is the list of partial sums of the series for different k+0.5 (with a slight Euler-acceleration)

Table of partial sums of series-terms for y_k = f°{k+0.5}(1), f(x)= sqrt(2)^x
using 96 terms for all computations

Code:
k+0.5=  0.5    ...     5.5              6.5              7.5              8.5             9.5

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

  1.25000000000   1.25000000000   1.25000000000   1.25000000000   1.25000000000      1.25000000000

  1.26110434560   4.49714780164   5.14435649285   5.79156518406   6.43877387527      7.08598256648

  1.22525437136   1.93944375677  -3.72023614787  -5.88364612747  -8.42967369556     -11.3583188521

  1.24668751623   5.99235259508   9.11068722540   13.5553576094   19.5849074109      27.4578802935

  1.24076737182   1.49979923591  -5.23653818585  -11.6346957756  -21.7708420705     -36.9172871478

  1.24335352318   4.28644727119   7.88117550221   15.1234185961   28.4432394460      51.2082273200

  1.24296927970   0.47806011620  -2.27471644838  -8.89792678520  -23.1008859405     -50.9291782428

  1.24330148390   2.68540223884   4.57545778450   9.77997787357   22.6057608416      51.2598956353

  1.24334689776   1.52931366136  0.451609309727  -3.07391363987  -13.0838077583     -38.5219403266

  1.24342527754   2.08739469898   2.71206223922   4.90465898282   11.8835163557      31.8389560094

  1.24346627727   1.83531664757   1.56559795649  0.365479208323  -4.00667177875     -18.0519511170

  1.24349944650   1.94309092062   2.11054106819   2.76424236321   5.31674044248      14.3664345470

  1.24352286906   1.89908422835   1.86530727995   1.57378990702  0.216343075033     -5.15411825349

  1.24354080559   1.91636602471   1.97064237771   2.13360031601   2.84254373454      5.84385041401

  1.24355452136   1.90980233711   1.92717452801   1.88231915313   1.55966874840  -0.00183357885750

  1.24356526803   1.91222417243   1.94449952880   1.99063473377   2.15814548109      2.95012751435

  1.24357379457   1.91135286176   1.93780095156   1.94557320516   1.89002762276      1.52540063456

  1.24358065583   1.91165944033   1.94032249372   1.96374161889   2.00591813819      2.18595121356

  1.24358624145   1.91155367278   1.93939558496   1.95661723354   1.95739708093      1.89047738279

  1.24359083767   1.91158952482   1.93972916723   1.95934236079   1.97714102763      2.01847035148

  1.24359465614   1.91157756208   1.93961137808   1.95832298338   1.96931015585      1.96460639324

  1.24359785628   1.91158149694   1.93965226197   1.95869669466   1.97234499833      1.98668941734

  1.24360055956   1.91158021912   1.93963829064   1.95856216865   1.97119328258      1.97784819071

  1.24360285978   1.91158062912   1.93964299792   1.95860979575   1.97162207811      1.98131215334

  1.24360483013   1.91158049890   1.93964143237   1.95859318842   1.97146519999      1.97998157171

  1.24360652831   1.91158053980   1.93964194690   1.95859889909   1.97152168166      1.98048347876

  1.24360800024   1.91158052704   1.93964177964   1.95859696045   1.97150164438      1.98029729442

  1.24360928281   1.91158053097   1.93964183347   1.95859761081   1.97150865646      1.98036530182

  1.24361040586   1.91158052975   1.93964181631   1.95859739502   1.97150623337      1.98034081358

  1.24361139372   1.91158053011   1.93964182174   1.95859746589   1.97150706093      1.98034951501

  1.24361226636   1.91158052999   1.93964182004   1.95859744283   1.97150678136      1.98034646112

  1.24361304032   1.91158053002   1.93964182057   1.95859745027   1.97150687485      1.98034752063

  1.24361372931   1.91158053001   1.93964182041   1.95859744789   1.97150684388      1.98034715699

  1.24361434483   1.91158053001   1.93964182046   1.95859744865   1.97150685405      1.98034728054

  1.24361489654   1.91158053001   1.93964182044   1.95859744841   1.97150685074      1.98034723896

  1.24361539259   1.91158053001   1.93964182045   1.95859744848   1.97150685181      1.98034725283

  1.24361583993   1.91158053001   1.93964182044   1.95859744846   1.97150685147      1.98034724824

  1.24361624445   1.91158053001   1.93964182045   1.95859744847   1.97150685158      1.98034724975

  1.24361661124   1.91158053001   1.93964182045   1.95859744846   1.97150685154      1.98034724926

  1.24361694464   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724942

  1.24361724842   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724937

  1.24361752584   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361777974   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361801259   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361822655   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361842353   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361860521   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361877306   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361892839   1.91158053001   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361907236   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361920601   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361933026   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361944593   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361955376   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938

  1.24361965442   1.91158053000   1.93964182045   1.95859744846   1.97150685155      1.98034724938

 ....

We see the improving of convergence for higher k.

Here are the results of the second part of computation

Table for f°0.5(1) computed by f°{-k}(y_k)

Code:
using   value for

k+0.5   f°0.5(1)

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

_0.5:  1.24362081784741256884404697491

_1.5:  1.24362164528491826158633097226

_2.5:  1.24362162704002978026654871300

_3.5:  1.24362162769989241325230973243

_4.5:  1.24362162766649448252099408767

_5.5:  1.24362162766868372429248670409

_6.5:  1.24362162766850631390326437050

_7.5:  1.24362162766852353956309027978

_8.5:  1.24362162766852158056624384012

_9.5:  1.24362162766852183704563622465

10.5:  1.24362162766852179891157968560

11.5:  1.24362162766852180527979917972

12.5:  1.24362162766852180409621393150

13.5:  1.24362162766852180433916726086

14.5:  1.24362162766852180428444683747

15.5:  1.24362162766852180429789422049

16.5:  1.24362162766852180429430608135

17.5:  1.24362162766852180429534120427

18.5:  1.24362162766852180429501955967

19.5:  1.24362162766852180429512685495

20.5:  1.24362162766852180429508854399

21.5:  1.24362162766852180429510314755

22.5:  1.24362162766852180429509721882

23.5:  1.24362162766852180429509977687

Computed by Diagonalization method (with fixpoint-shift to fixpoint 2)

Code:
diag:  1.24362162766852180429509898361

compare:

Code:
_0.5:  1.243620...

_2.5:  1.2436216270...

18.5:  1.24362162766852180429502...

20.5:  1.24362162766852180429508...

22.5:  1.24362162766852180429509 | 721882...

diag:  1.24362162766852180429509 | 898361...

23.5:  1.24362162766852180429509 | 977687...

21.5:  1.24362162766852180429510...

19.5:  1.24362162766852180429512...

_1.5:  1.24362164...

andydude · 07/14/2008, 06:21 PM

bo198214 Wrote:And I think this is new. Especially that the iteration by this formula is the same as regular iteration (which is sure for elliptic iteration, other cases are still to prove).

Hmm. Awhile back, I did some test with parabolic iteration, and convinced myself that Woon's formula and Jabotinsky's formula produce exactly the same results for parabolic iteration (for symbolic coefficients). So I could write up something about this if needed.

Andrew Robbins

***bo198214*** · 07/14/2008, 09:55 PM

andydude Wrote:Awhile back, I did some test with parabolic iteration, and convinced myself that Woon's formula and Jabotinsky's formula produce exactly the same results for parabolic iteration (for symbolic coefficients). So I could write up something about this if needed.

The problematic thing about parabolic iteration is convergence. You usually only have an asymptotic development at the fixed point. I.e. the series is not developable at the fixed point however in every neighborhood and all the coefficients converge (towards the fixed point in a certain sector) to that of the asymptotic development.

I also see that I made mistake in my first post about the convergence of that series. It is not true that the series always converges if the function has a finite attracting fixed point.

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