Just an accidental finding.

[update]: Hmm, after some more investigation it seems, that the halfiterative of the sinh alone is not that "nugget" which I felt it were in the beginning. Still it might be a good starting point, but after some regressions with various cofactors it seems to me, that likely it cannot substantially be improved by adding terms of standardfunctions and/or further fractional iterates. So I think, Jay's very well worked function is still the way to go and this posting should only survive for historical reasons... [/update]

I use the sequence A[k]=(1,1,2,2,4,4,6,6,...) with k beginning at 0 and tried the half-iterate of the asinh()-function to translate A[k]-> k^ .

I get that nice approximation:

In reverse it looks like

Perhaps it is easy to finetune this much more with not too much effort.

[update]: a first finetuning: a^ (k) = sinh°05( (1.018 k -0.174)/2)

See the pictures with the already improved data.

Absolute values:

Ratios: [update]: upps, after looking behind index k>150 things are no more looking so nice... :-(

Technical details:

I computed the powerseries for the half-iterate from the sqrt of the carlemanmatrix for sinh to 128 terms.

The convergence-radius is zero, but for x near zero one can evaluate the first 128 terms of the powerseries and get a value, which reinserted gives indeed the one-time-iterate to, say 20 or 30 dec digits precision.

So high arguments x of the sinh°05(x) must be transferred by asinh()-iterations sufficiently near towards zero (say 0<x<0.5 or x<0.2 for k~80) , then the powerseries can be evaluated with that x, and then the result must be retransferred by appropriate sinh()-iterations.

I computed this with 400 digits internal precision and float algebra (although the square-root of the Carlemanmatrix can be determined in rational algebra).

Gottfried

(If someone needs it I can supply the Pari/GP-code)

[update]: Hmm, after some more investigation it seems, that the halfiterative of the sinh alone is not that "nugget" which I felt it were in the beginning. Still it might be a good starting point, but after some regressions with various cofactors it seems to me, that likely it cannot substantially be improved by adding terms of standardfunctions and/or further fractional iterates. So I think, Jay's very well worked function is still the way to go and this posting should only survive for historical reasons... [/update]

I use the sequence A[k]=(1,1,2,2,4,4,6,6,...) with k beginning at 0 and tried the half-iterate of the asinh()-function to translate A[k]-> k^ .

I get that nice approximation:

Code:

`. `

k | k^ = -1 | |

| +2*asinh°0.5(A[k]) | A[k] | d = k^ - k

-----+---------------------+------+-------------------

0 0.871122566717836 1 0.8711225667178365

1 0.871122566717836 1 -0.1288774332821635

2 2.333123406391677 2 0.3331234063916770

3 2.333123406391677 2 -0.6668765936083230

4 4.444125935755581 4 0.4441259357555807

5 4.444125935755581 4 -0.5558740642444193

6 5.996686817157767 6 -0.003313182842233339

7 5.996686817157767 6 -1.003313182842233

8 8.309891291850146 10 0.3098912918501464

9 8.309891291850146 10 -0.6901087081498536

10 10.06605408134559 14 0.06605408134559265

11 10.06605408134559 14 -0.9339459186544073

12 12.14359799584454 20 0.1435979958445373

13 12.14359799584454 20 -0.8564020041554627

14 13.82177298614942 26 -0.1782270138505756

15 13.82177298614942 26 -1.178227013850576

16 16.08985351232628 36 0.08985351232627991

17 16.08985351232628 36 -0.9101464876737201

18 17.94134162168168 46 -0.05865837831832136

19 17.94134162168168 46 -1.058658378318321

20 20.09383140688473 60 0.09383140688473422

21 20.09383140688473 60 -0.9061685931152658

22 21.90407738775139 74 -0.09592261224861381

23 21.90407738775139 74 -1.095922612248614

24 24.09366985432521 94 0.09366985432520982

25 24.09366985432521 94 -0.9063301456747902

26 25.95938189509372 114 -0.04061810490628340

27 25.95938189509372 114 -1.040618104906283

28 28.04805382734139 140 0.04805382734139237

29 28.04805382734139 140 -0.9519461726586076

30 29.86181858151267 166 -0.1381814184873316

31 29.86181858151267 166 -1.138181418487332

32 32.04658019188079 202 0.04658019188078527

33 32.04658019188079 202 -0.9534198081192147

34 33.95209498128483 238 -0.04790501871516933

35 33.95209498128483 238 -1.047905018715169

36 36.08894669829039 284 0.08894669829039405

37 36.08894669829039 284 -0.9110533017096059

38 37.97426693042275 330 -0.02573306957725277

39 37.97426693042275 330 -1.025733069577253

40 40.14969749681103 390 0.1496974968110288

41 40.14969749681103 390 -0.8503025031889712

42 42.07952641659918 450 0.07952641659918464

43 42.07952641659918 450 -0.9204735834008154

44 44.20137456045023 524 0.2013745604502300

45 44.20137456045023 524 -0.7986254395497700

46 46.10105039582103 598 0.1010503958210259

47 46.10105039582103 598 -0.8989496041789741

48 48.26520579699401 692 0.2652057969940090

49 48.26520579699401 692 -0.7347942030059910

50 50.20979388257698 786 0.2097938825769775

51 50.20979388257698 786 -0.7902061174230225

52 52.33643071077409 900 0.3364307107740934

53 52.33643071077409 900 -0.6635692892259066

54 54.26025209047609 1014 0.2602520904760932

55 54.26025209047609 1014 -0.7397479095239068

56 56.40159041927338 1154 0.4015904192733788

57 56.40159041927338 1154 -0.5984095807266212

58 58.34609979287966 1294 0.3460997928796596

59 58.34609979287966 1294 -0.6539002071203404

60 60.44611088173228 1460 0.4461108817322844

61 60.44611088173228 1460 -0.5538891182677156

62 62.36400319817613 1626 0.3640031981761257

63 62.36400319817613 1626 -0.6359968018238743

64 64.49766201469299 1828 0.4976620146929893

65 64.49766201469299 1828 -0.5023379853070107

66 66.45061225735103 2030 0.4506122573510340

67 66.45061225735103 2030 -0.5493877426489660

68 68.56130277140938 2268 0.5613027714093829

69 68.56130277140938 2268 -0.4386972285906171

In reverse it looks like

Code:

`. `

| a^ = | |

k | sinh°05(k/2+1/2) | A[k] | r = a^ /A[k] - 1

-----+---------------------+------+-------------------

0 0.5102310310942976 1 -0.4897689689057024

1 1.078138996064359 1 0.0781389960643593

2 1.748304390930525 2 -0.1258478045347374

3 2.552519461332870 2 0.2762597306664349

4 3.515579526352944 4 -0.1211051184117640

5 4.659355506757197 4 0.1648388766892992

6 6.004813731822326 6 0.0008022886370542993

7 7.572966044912433 6 0.2621610074854055

8 9.385336346538606 10 -0.06146636534613942

9 11.46420789661713 10 0.1464207896617127

10 13.83276703665167 14 -0.01194521166773818

11 16.51519557483509 14 0.1796568267739347

12 19.53673656870888 20 -0.02316317156455596

13 22.92374579523910 20 0.1461872897619548

14 26.70373529679550 26 0.02706674218444246

15 30.90541246272600 26 0.1886697101048460

16 35.55871659041376 36 -0.01225787248850666

17 40.69485405483909 36 0.1304126126344191

18 46.34633276239224 46 0.007528973095483409

19 52.54699630468353 46 0.1423260066235550

20 59.33205807477817 60 -0.01113236542036384

21 66.73813551558644 60 0.1123022585931074

22 74.80328461281173 74 0.01085519747042881

23 83.56703470866675 74 0.1292842528198210

24 93.07042368928640 94 -0.009889109688442508

25 103.3560335835290 94 0.09953227216520219

26 114.4680266007278 114 0.004105496497612463

27 126.4521816281218 114 0.1092296634045772

28 139.3559312040241 140 -0.004600491399828113

29 153.2283989795549 140 0.09448856413967771

30 168.1204376795053 166 0.01277372096087535

31 184.0846675712996 166 0.1089437805499975

32 201.1755154498853 202 -0.004081606683735970

33 219.4492541455617 202 0.08638244626515684

34 238.9640425611633 238 0.004050598996484624

35 259.7799662445876 238 0.09151246321255298

36 281.9590785023372 284 -0.007186343301629582

37 305.5654420595219 284 0.07593465513916145

38 330.6651712715941 330 0.002015670519982010

39 357.3264748929723 330 0.08280749967567349

40 385.6196994076169 390 -0.01123153998046941

41 415.6173729265633 390 0.06568557160657261

42 447.3942496573710 450 -0.005790556316953349

43 481.0273549504231 450 0.06894967766760693

44 516.5960309269911 524 -0.01412971197139100

45 554.1819826939725 524 0.05759920361445136

46 593.8693251502086 598 -0.006907483026407088

47 635.7446303892922 598 0.06311811101888332

48 679.8969757037854 692 -0.01748991950320035

49 726.4179921957754 692 0.04973698294187201

50 775.4019139987164 786 -0.01348356997618778

51 826.9456281155144 786 0.05209367444721939

52 881.1487248778374 900 -0.02094586124684738

53 938.1135490316460 900 0.04234838781293998

54 997.9452514539660 1014 -0.01583308535111831

55 1060.751841505941 1014 0.04610635256996131

56 1126.644240027227 1154 -0.02370516462112075

57 1195.736332976817 1154 0.03616666635772742

58 1268.145025725405 1294 -0.01998066018129422

59 1343.990298004411 1294 0.03863237867419740

60 1423.395259516844 1460 -0.02507174005695639

61 1506.486206215162 1460 0.03183986727065867

62 1593.392677251357 1626 -0.02005370402745561

63 1684.247512604481 1626 0.03582257847754043

64 1779.186911390872 1828 -0.02670300252140491

65 1878.350490862376 1828 0.02754403220042451

66 1981.881346097857 2030 -0.02370377039514422

67 2089.926110393341 2030 0.02952025142529100

68 2202.635016356147 2268 -0.02882053952550826

69 2320.161957708406 2268 0.02299909951869768

Perhaps it is easy to finetune this much more with not too much effort.

[update]: a first finetuning: a^ (k) = sinh°05( (1.018 k -0.174)/2)

See the pictures with the already improved data.

Absolute values:

Ratios: [update]: upps, after looking behind index k>150 things are no more looking so nice... :-(

Technical details:

I computed the powerseries for the half-iterate from the sqrt of the carlemanmatrix for sinh to 128 terms.

The convergence-radius is zero, but for x near zero one can evaluate the first 128 terms of the powerseries and get a value, which reinserted gives indeed the one-time-iterate to, say 20 or 30 dec digits precision.

So high arguments x of the sinh°05(x) must be transferred by asinh()-iterations sufficiently near towards zero (say 0<x<0.5 or x<0.2 for k~80) , then the powerseries can be evaluated with that x, and then the result must be retransferred by appropriate sinh()-iterations.

I computed this with 400 digits internal precision and float algebra (although the square-root of the Carlemanmatrix can be determined in rational algebra).

Gottfried

(If someone needs it I can supply the Pari/GP-code)

Gottfried Helms, Kassel