Comparision: 4 different methods of interpolation

[sheldonison]

...
What I'm missing are the computations of that example interval for the Cauchy-integral-method (could someone do this for me?) and the method derived from the slog-ansatz of Walker and Andy Robbins.
Here is the link....
Gottfried

Hi Gottfried,

Your post is very interesting. The algorithm I have for generating the sexp(z) is also based on the idea of iterating logarithms, at different values for imag(z)=constant idelta, where the constant slowly moves closer to the real axis from one iteration to the next. This compares to your \( z_0 = 0 + 0.1*i \), \( z_1 = 0 + 0.05*i \)...

If you want a challenge, one thing you could do is generate the limiting behavior of iterating the logarithm, on the path starting with \( z_0 \), and ending at \( 4^{z_0} \). Count the number of times you iterate the logarithm. Then, subtract the fixed point "L". Then multiply by c^n. Then take the logarithm, base c. This will converge to a result, as the number of iterations of logarithms increases arbitrarily large. This resulting function can be analytic, if wrapped around the unit circle, or, equivalently, if represented as a 1-cyclic Fourier series. If it is analytic inside the unit circle, or the Fourier series is analytic for imag(z)>0, then the interpolation you were using will converges to the fixed point at +I*infinity. Also, you can use that 1-cyclic Fourier series correlating to your interpolation, and use that to extend your interpolation to other points that aren't on the initial interpolation line, whose imag(z) values are bigger than your interpolations \( \Im(z_0) \), which with a little bit of trickery, can even be used to see how your interpolation behaves at values approaching the real axis, although the Fourier series has a singularity corresponding to log(z=0), and the Fourier series is not defined for imag(z)<0.

Here is the equation to use.

\( c=L\times\ln(B) \)
\( \text{isuperf}_{B}(z) = \lim_{n \to \infty}
\log_{c} (c^n \times ((\log_{B}^{[n]}(z))-L)) \)

I'm also copying a 100 term Taylor series for \( \text{sexp}_4(z) \), generated with my kneser.gp code, updated today in this post. Convergence for this series is best within a unit circle of radius of about "1", and it is accurate to about 32 decimal digits. This should be equivalent to the results for the Cauchy integral, in that the Cauchy integral also converges to the fixed point of L, L* at +/- infinity.
- Sheldon

Code:

sie sexp Taylor series approximation terms
0.9999999999999999999999999999999995070247187929191800883791929298
1.300874303182213070881446827688072813061714556754230656713299112
0.6134882416397192582109374555554626084484838898219689840815017709
0.4626136939158324624570419449320859356621508257631076067661781738
0.2580271665381400290406176995439707729428027624801088785511732678
0.1631972654995318557720687010383484813350480755688599657964371828
0.08962419078985280875791908367441636870291992358851208951306242480
0.05210943962785077456922545881576836319307194597998413171358219472
0.02787344355922481828308550101481237499554959951610037275785833015
0.01539917994225308398452361886392528717634511053121181516280489767
0.008038140872697811905389166736775937873687184851439611721620856142
0.004285814333364039540175648057057942915900950248681577696568660482
0.002190017732657062431267753342735334682355213685867116789645253986
0.001136799557106834835735373410210317366737445255609961738959439943
0.0005702933580844330487522396703324708581367111745349736234225003887
0.0002897924373405595552639524833821840895243704307010392362494910403
0.0001430633962592935882349730645760417255725999434193038877463683821
0.00007143535241514501365950250514888103430851484606325840796413359310
0.00003477052612699922375452114292328458686494995664991541145966486853
0.00001710763661062027438435026942770620065276202889441606559780515340
0.000008222668464823314272610019073460323839550881158819005935582905370
0.000003994852780504333014833969574749273443623562826957028502622836667
0.000001898398063874020939950524705115380873628494076288510587587700622
0.0000009122438146518910571971764421537032161885169426812387017113281077
0.0000004290385948155930958122182972265362537957940313092525320328638684
0.0000002041983306345472171355252468820480355653099911657003733656774073
0.00000009512318872976968042268016824201584840722705732172328549896514731
0.00000004489317526475872755381079755929962866192623484738624738947818941
0.00000002072720854761742214919996912392356563331344531270627679611102753
0.000000009709928580376567409288540894217012554522585199358692221947344054
0.000000004445457635977581093298134874172196352623391145821751830692478538
0.000000002069067831997830355680147212366306199858197057754070520301979538
0.0000000009396576972415515090385897649635544514935693430568984322768929411
0.0000000004348993806810866648462761422110304400679663399357988409106276443
1.959634169379625824533975324107307643182292461794426937648214767 E-10
9.026610708595697059589186571004657226483426982828780751255398719 E-11
4.035887940589495505962961813454430792050597579829137552712606253 E-11
1.851811158741082264579893345045055577900185993420077817637967014 E-11
8.215057781560342499234972804886894570184731791115949762884833338 E-12
3.758182099737354800100972442189710575470534180644706040162342618 E-12
1.653816116453905095315919495158619674267242509776579770628140753 E-12
7.551090292313603512182414144871894949960949212829628314979456250 E-13
3.294730854704294889645798366290620163251202126044696613532797910 E-13
1.503180202578246284601955177533010172432124301605887936254906187 E-13
6.498532414596416354625574128693528367693104525543242144376365141 E-14
2.966787694463398260101586711895208612540948702274735616760 E-14
1.269506066394563995980523451111737527416389957745425996754 E-14
5.809447211675512658406406010859664050117453892558603002430 E-15
2.456923887981265571022295372485400661348078556275414499022 E-15
1.129428858171867988405297967343673929768801862164662346633 E-15
4.711338066637788073516155078449022678560203083166264723447 E-16
2.181606517380487522129598303657716325016581211042840570040 E-16
8.951254371028589263572483822751652023271244878936893432056 E-17
4.190185379602400264475431039792092048187259950236125497032 E-17
1.684676325260767467952730786600274324353305534443332916510 E-17
8.00975310599499114034331211081169770466361095671141938590 E-18
3.139300680337191659721455825117368158842562177220748216374 E-18
1.525412448387190895963160857882397823035180144715226963088 E-18
5.787124927518397255730760899085273078103665520972826591387 E-19
2.897860085139928565881070500794655861108139200741166180865 E-19
1.0539017924966478755296435787405525807942497018701059912826 E-19
5.499741182310150685235453747877087354820770099024176682661 E-20
1.891815428233224354088005687028096685568736250862778647283 E-20
1.044661806434266648611246719755393417569252110062999131835 E-20
3.335460109848631173403946935187917579211177770612486976098 E-21
1.990391550180344368519802729546144052355230903443214604775 E-21
5.742544667618120422282442998171936954247274848707625156635 E-22
3.813938765209951555411152348110770741798315126591233113213 E-22
9.558229147437373582485505316131990712732375060961782289688 E-23
7.372258718528940476709714144299012707345703932341819279402 E-23
1.509540081900308541844775926021439334974435152344193470196 E-23
1.442355372479914264007309507475705715797347663296257830783 E-23
2.17261402936860554290691819576399424158155685466 E-24
2.86605801478900889251802391601744148442882364181 E-24
2.54262312111658143996213690853259003171756198089 E-25
5.80277732661624708084917829800408734781659878345 E-25
1.193043637496520775356974419572489627071440054140 E-26
1.200187232589791625741176000227785330653724342087 E-25
-6.14763319790035079365261034143156019979249851829 E-27
2.539858233552638308377817435553872085374218519626 E-26
-3.13057013130477445048407475360820250796497191757 E-27
5.50123125279568297609245444405153219242798923736 E-27
-1.056288294171983348089454164293866035364041344394 E-27
1.218417104876116850071141113015421401872978232018 E-27
-3.10037140662822825214492277765472433954612767950 E-28
2.75399762285912058468172127529495128841833289673 E-28
-8.49903533151774002213304461649241351169879536428 E-29
6.33607669365821705179359970435612411142282899585 E-29
-2.240662470687793097269648735303213434107446613212 E-29
1.479495589621324521501992703428978451151211123986 E-29
-5.76622626194605031140543603101348276541245427194 E-30
3.49645153513877337162203231356324697384970684598 E-30
-1.460807401340342751076050916111909985924317827766 E-30
8.34263457663592562247448858165024513769708270210 E-31
-3.66155034725519324324462302612346068105890371888 E-31
2.00634211056505813212940132182242432106754109740 E-31
-9.10436035774224627877942444074438826036859283602 E-32
4.86540080643117711358128043740583276363662229656 E-32
-2.243492597703563039982058352212732980162096265650 E-32
1.197927046648240273800047573989178344238550676938 E-32