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 x0.5 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 x0.5 in the second column.

 x0 x0.5 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 xm-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 xm xm+0.05 h xm-j <=0.51 fAbel(xm-j-0.5) xm-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