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
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)
Computed by Diagonalization method (with fixpoint-shift to fixpoint 2)
compare:
\( \hspace{24} y_k = f^{\circ k+0.5}(x) \) (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: 1.24362162766852180429509977687Computed by Diagonalization method (with fixpoint-shift to fixpoint 2)
Code:
diag: 1.24362162766852180429509898361Code:
_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