Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
matrix function like iteration without power series expansion
#11
Gottfried Wrote:Henryk's formula is (I inserted (x) at f°t)

Hmm ... you mean Woon's formula ...
Reply
#12
I guess I should be more specific. The key observation to show the equivalence is that

which means Woon's formula simplifies to:

and letting this simplifies to

and assuming this simplifies to

and combining (-1) terms, this simplifies to

and that is, as you call it, "Henryk's formula" from Woon's formula.

Andrew Robbins
Reply
#13
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:



Woon just applied this to linear operators instead of . 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. but generally . The binomial formula however relies on full (both side) distributivity. Especially generally


The interesting thing however is that both expansions together are then again valid:
for each 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).
Reply
#14
I've got better convergence for the binomial (Woon-) method when computing f°0.5(x) by

(using Woon-method)
(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...
Gottfried Helms, Kassel
Reply
#15
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
Reply
#16
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.
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Is there a function space for tetration? Chenjesu 0 245 06/23/2019, 08:24 PM
Last Post: Chenjesu
  A Notation Question (raising the highest value in pow-tower to a different power) Micah 8 1,737 02/18/2019, 10:34 PM
Last Post: Micah
  Degamma function Xorter 0 720 10/22/2018, 11:29 AM
Last Post: Xorter
Question Taylor series of i[x] Xorter 12 10,553 02/20/2018, 09:55 PM
Last Post: Xorter
  Functional power Xorter 0 1,193 03/11/2017, 10:22 AM
Last Post: Xorter
  Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 1,470 12/26/2016, 03:01 AM
Last Post: tommy1729
  2 fixpoints , 1 period --> method of iteration series tommy1729 0 1,329 12/21/2016, 01:27 PM
Last Post: tommy1729
  2 fixpoints related by power ? tommy1729 0 1,324 12/07/2016, 01:29 PM
Last Post: tommy1729
  Taylor series of cheta Xorter 13 11,349 08/28/2016, 08:52 PM
Last Post: sheldonison
  Tommy's matrix method for superlogarithm. tommy1729 0 1,522 05/07/2016, 12:28 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)