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

 

Table 1) of half-iterates of f(x) = x + 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) = x + 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