MSE Half-iterate of x²+0.25

For MSE http://math.stackexchange.com/questions/208996/half-iterate-of-x2c

Table 1) of half-iterates of f(x) = x² + 1/4, for x=0.5 ... 5.5

We take 100 different x₀ in the stepwith of 0.05 beginning at x=0.5 (the fixpoint as lower bound). For each of that x₀ we determine the half-iterate x_0.5 using the Abel-function (fAbel as in the MSE-posting). For the Abel-function we need recentering of x (by -0.5); also to make the Abel-function best converging we iterate j-times towards the fixpoint 0.5 until x_-j is smaller than 0.51. Then we apply the Abel-function (fAbel) and get the value a_-j. Increasing by the height 0.5 the inverse Abelfunction gives us the half iterate of the (recentered) x. Also reversing the initial iteration leads then to the value x_0.5 in the second column.

	x₀	x_0.5	h	x_-j <=0.51	fAbel(x_-j-0.5)	x_-j+0.5
0	0.5	0.5	0.5	0	----	0
1	0.5500	0.551220221021	0.5000	0.509959586535	133.019941329	0.509910716366
2	0.6000	0.604772246757	0.5000	0.509927314669	133.349571564	0.509878758400
3	0.6500	0.660513076737	0.5000	0.509997805211	132.632308338	0.509948562011
4	0.7000	0.718321237354	0.5000	0.509991267832	132.698404498	0.509942088539
5	0.7500	0.778092005234	0.5000	0.509913168187	133.494739396	0.509864749203
6	0.8000	0.839733963915	0.5000	0.509994633448	132.664365697	0.509945421259
7	0.8500	0.903166460208	0.5000	0.509956452358	133.051861104	0.509907612719
8	0.9000	0.968317683702	0.5000	0.509905041674	133.578318349	0.509856701467
9	0.9500	1.03512318842	0.5000	0.509942006137	133.199246399	0.509893307091
10	1.000	1.10352473518	0.5000	0.509972605557	132.887563844	0.509923608474
11	1.050	1.17346937130	0.5000	0.509998449989	132.625793974	0.509949200483
12	1.100	1.24490868889	0.5000	0.509922188699	133.402125463	0.509873682197
13	1.150	1.31779821998	0.5000	0.509941122876	133.208271555	0.509892432420
14	1.200	1.39209693773	0.5000	0.509957789994	133.038235616	0.509908937326
15	1.250	1.46776684116	0.5000	0.509972605557	132.887563844	0.509923608474
16	1.300	1.54477260629	0.5000	0.509985886793	132.752874096	0.509936760072
17	1.350	1.62308129071	0.5000	0.509997880426	132.631548379	0.509948636490
18	1.400	1.70266208158	0.5000	0.509910562230	133.521526085	0.509862168515
19	1.450	1.78348607912	0.5000	0.509920333288	133.421161342	0.509871844794
20	1.500	1.86552610963	0.5000	0.509929310660	133.329122322	0.509880735005
21	1.550	1.94875656285	0.5000	0.509937596620	133.244318738	0.509888940450
22	1.600	2.03315324992	0.5000	0.509945276014	133.165848685	0.509896545163
23	1.650	2.11869327860	0.5000	0.509952419875	133.092959044	0.509903619500
24	1.700	2.20535494330	0.5000	0.509959088171	133.025015541	0.509910222857
25	1.750	2.29311762756	0.5000	0.509965331916	132.961479827	0.509916405760
26	1.800	2.38196171736	0.5000	0.509971194810	132.901891708	0.509922211487
27	1.850	2.47186852372	0.5000	0.509976714524	132.845855218	0.509927677352
28	1.900	2.56282021339	0.5000	0.509981923729	132.793027601	0.509932835710
29	1.950	2.65479974653	0.5000	0.509986850912	132.743110500	0.509937714774
30	2.000	2.74779082048	0.5000	0.509991521037	132.695842867	0.509942339269
31	2.050	2.84177781903	0.5000	0.509995956085	132.650995198	0.509946730965
32	2.100	2.93674576629	0.5000	0.509902123447	133.608364815	0.509853811514
33	2.150	3.03268028483	0.5000	0.509906066382	133.567771984	0.509857716245
34	2.200	3.12956755754	0.5000	0.509909829857	133.529056694	0.509861443242
35	2.250	3.22739429272	0.5000	0.509913427375	133.492075965	0.509865005877
36	2.300	3.32614769218	0.5000	0.509916871057	133.456701601	0.509868416157
37	2.350	3.42581542195	0.5000	0.509920171821	133.422818279	0.509871684893
38	2.400	3.52638558534	0.5000	0.509923339525	133.390321946	0.509874821852
39	2.450	3.62784669814	0.5000	0.509926383100	133.359118441	0.509877835877
40	2.500	3.73018766572	0.5000	0.509929310660	133.329122322	0.509880735005
41	2.550	3.83339776189	0.5000	0.509932129591	133.300255858	0.509883526552
42	2.600	3.93746660925	0.5000	0.509934846635	133.272448159	0.509886217194
43	2.650	4.04238416106	0.5000	0.509937467963	133.245634420	0.509888813043
44	2.700	4.14814068433	0.5000	0.509939999231	133.219755269	0.509891319701
45	2.750	4.25472674407	0.5000	0.509942445635	133.194756197	0.509893742316
46	2.800	4.36213318872	0.5000	0.509944811963	133.170587055	0.509896085626
47	2.850	4.47035113643	0.5000	0.509947102628	133.147201618	0.509898354005
48	2.900	4.57937196232	0.5000	0.509949321709	133.124557201	0.509900551491
49	2.950	4.68918728652	0.5000	0.509951472982	133.102614316	0.509902681825
50	3.000	4.79978896302	0.5000	0.509953559949	133.081336371	0.509904748475
51	3.050	4.91116906916	0.5000	0.509955585861	133.060689402	0.509906754660
52	3.100	5.02331989580	0.5000	0.509957553745	133.040641839	0.509908703378
53	3.150	5.13623393805	0.5000	0.509959466417	133.021164289	0.509910597419
54	3.200	5.24990388660	0.5000	0.509961326507	133.002229350	0.509912439386
55	3.250	5.36432261947	0.5000	0.509963136469	132.983811437	0.509914231711
56	3.300	5.47948319435	0.5000	0.509964898602	132.965886634	0.509915976669
57	3.350	5.59537884124	0.5000	0.509966615055	132.948432557	0.509917676389
58	3.400	5.71200295561	0.5000	0.509968287845	132.931428223	0.509919332871
59	3.450	5.82934909190	0.5000	0.509969918867	132.914853949	0.509920947988
60	3.500	5.94741095729	0.5000	0.509971509899	132.898691242	0.509922523503
61	3.550	6.06618240594	0.5000	0.509973062616	132.882922714	0.509924061074
62	3.600	6.18565743345	0.5000	0.509974578594	132.867531993	0.509925562263
63	3.650	6.30583017156	0.5000	0.509976059321	132.852503653	0.509927028542
64	3.700	6.42669488324	0.5000	0.509977506199	132.837823140	0.509928461301
65	3.750	6.54824595795	0.5000	0.509978920554	132.823476715	0.509929861852
66	3.800	6.67047790709	0.5000	0.509980303639	132.809451389	0.509931231436
67	3.850	6.79338535980	0.5000	0.509981656640	132.795734879	0.509932571227
68	3.900	6.91696305883	0.5000	0.509982980679	132.782315554	0.509933882339
69	3.950	7.04120585671	0.5000	0.509984276822	132.769182392	0.509935165825
70	4.000	7.16610871201	0.5000	0.509985546079	132.756324940	0.509936422686
71	4.050	7.29166668584	0.5000	0.509986789410	132.743733277	0.509937653872
72	4.100	7.41787493848	0.5000	0.509988007725	132.731397981	0.509938860286
73	4.150	7.54472872617	0.5000	0.509989201891	132.719310093	0.509940042785
74	4.200	7.67222339803	0.5000	0.509990372733	132.707461093	0.509941202187
75	4.250	7.80035439313	0.5000	0.509991521037	132.695842867	0.509942339269
76	4.300	7.92911723769	0.5000	0.509992647551	132.684447690	0.509943454773
77	4.350	8.05850754237	0.5000	0.509993752988	132.673268193	0.509944549406
78	4.400	8.18852099970	0.5000	0.509994838030	132.662297351	0.509945623841
79	4.450	8.31915338165	0.5000	0.509995903327	132.651528457	0.509946678723
80	4.500	8.45040053722	0.5000	0.509996949500	132.640955106	0.509947714668
81	4.550	8.58225839018	0.5000	0.509997977143	132.630571175	0.509948732262
82	4.600	8.71472293692	0.5000	0.509998986825	132.620370811	0.509949732069
83	4.650	8.84779024432	0.5000	0.509999979088	132.610348414	0.509950714628
84	4.700	8.98145644779	0.5000	0.509902887276	133.600498622	0.509854567943
85	4.750	9.11571774931	0.5000	0.509903827618	133.590816299	0.509855499174
86	4.800	9.25057041558	0.5000	0.509904752345	133.581296525	0.509856414942
87	4.850	9.38601077623	0.5000	0.509905661910	133.571934580	0.509857315692
88	4.900	9.52203522214	0.5000	0.509906556742	133.562725938	0.509858201853
89	4.950	9.65864020375	0.5000	0.509907437257	133.553666253	0.509859073834
90	5.000	9.79582222949	0.5000	0.509908303852	133.544751352	0.509859932029
91	5.050	9.93357786422	0.5000	0.509909156910	133.535977228	0.509860776818
92	5.100	10.0719037278	0.5000	0.509909996798	133.527340028	0.509861608564
93	5.150	10.2107964937	0.5000	0.509910823871	133.518836045	0.509862427618
94	5.200	10.3502528874	0.5000	0.509911638468	133.510461718	0.509863234317
95	5.250	10.4902696854	0.5000	0.509912440917	133.502213614	0.509864028985
96	5.300	10.6308437137	0.5000	0.509913231534	133.494088430	0.509864811935
97	5.350	10.7719718467	0.5000	0.509914010622	133.486082986	0.509865583467
98	5.400	10.9136510060	0.5000	0.509914778475	133.478194216	0.509866343873
99	5.450	11.0558781590	0.5000	0.509915535375	133.470419162	0.509867093431
100	5.500	11.1986503181	0.5000	0.509916281594	133.462754976	0.509867832413

Table 2) of sequential 1/20 iterates of f(x) = x² + 1/4, beginning at x₀=1

We begin at x₀=1 and perform 31 iterations of height=1/10 . These are the 31 x_m-coordinates in the second column. In the third column each of that values was again iterated by 1/20, so we have actually 60 1/20-iterates beginning at x₀=1.0 .

The other columns are the same as in the above table.

m	x_m	x_m+0.05	h	x_m-j <=0.51	fAbel(x_m-j-0.5)	x_m-j+0.5
0	1	1.00892744	1/20	0.50997261	132.887564	0.50987992
0.1	1.01813539	1.02763611	1/20	0.50998246	132.787564	0.50988959
0.2	1.03744256	1.04756841	1/20	0.50999234	132.687564	0.50989929
0.3	1.05802812	1.06883696	1/20	0.50990414	133.587564	0.50981271
0.4	1.0800111	1.09156765	1/20	0.50991387	133.487564	0.50982226
0.5	1.10352474	1.11590156	1/20	0.50992361	133.387564	0.50983182
0.6	1.12871848	1.14199714	1/20	0.50993337	133.287564	0.5098414
0.7	1.15576047	1.17003288	1/20	0.50994315	133.187564	0.509851
0.8	1.18484031	1.20021036	1/20	0.50995295	133.087564	0.50986062
0.9	1.21617242	1.23275783	1/20	0.50996277	132.987564	0.50987026
1	1.25	1.26793458	1/20	0.50997261	132.887564	0.50987992
1.1	1.28659966	1.30603597	1/20	0.50998246	132.787564	0.50988959
1.2	1.32628706	1.34739957	1/20	0.50999234	132.687564	0.50989929
1.3	1.3694235	1.39241245	1/20	0.50990414	133.587564	0.50981271
1.4	1.41642398	1.44151994	1/20	0.50991387	133.487564	0.50982226
1.5	1.46776684	1.49523628	1/20	0.50992361	133.387564	0.50983182
1.6	1.52400542	1.55415746	1/20	0.50993337	133.287564	0.5098414
1.7	1.58578227	1.61897695	1/20	0.50994315	133.187564	0.509851
1.8	1.65384656	1.69050491	1/20	0.50995295	133.087564	0.50986062
1.9	1.72907537	1.76969188	1/20	0.50996277	132.987564	0.50987026
2	1.8125	1.8576581	1/20	0.50997261	132.887564	0.50987992
2.1	1.9053387	1.95572995	1/20	0.50998246	132.787564	0.50988959
2.2	2.00903736	2.0654856	1/20	0.50999234	132.687564	0.50989929
2.3	2.12532071	2.18881243	1/20	0.50990414	133.587564	0.50981271
2.4	2.2562569	2.32797974	1/20	0.50991387	133.487564	0.50982226
2.5	2.4043395	2.48573154	1/20	0.50992361	133.387564	0.50983182
2.6	2.57259251	2.66540542	1/20	0.50993337	133.287564	0.5098414
2.7	2.76470541	2.87108636	1/20	0.50994315	133.187564	0.509851
2.8	2.98520845	3.10780684	1/20	0.50995295	133.087564	0.50986062
2.9	3.23970162	3.38180935	1/20	0.50996277	132.987564	0.50987026
3	3.53515625	3.70089362	1/20	0.50997261	132.887564	0.50987992

Gottfried Helms, 2012-10-10