Comparision: 4 different methods of interpolation

[sheldonison]

....
Because I could not yet adapt the "initial value" from the (implicite) 1 to some other value (my z_k() values) (and didn't implement a binary search to find the relative height of the z_k to 1) I've to postpone that .

But just for a quick response ...

What I tried for the beginning was the curve for the complex heights of say h=exp(2*Pi*I *k) and then sexp(h ) for k=0..1 in a small stepwidth. I found that sexp could not handle the full complex circle of h
Here is the output:
...

Hi Gottfried,

I'm glad you could try the Kneser.gp program, I need to add some comments to it, and also explain the updated algorithm.

The problem you're having with imag(z)>2*I, is that sexp(z) implements the Taylor series, so it has a convergence radius of "2", in the imaginary direction, due to the singularity at sexp(-2). sexp(z) works for other values in the real direction, by using exp(TaylorSeries(z)), and log(TaylorSeries(z)). But the Taylor series itself has a radius of convergence of 2, since there is a singularity at sexp(-2). Also, the Taylor series convergence with 100 or so terms is best within a radius of about <1.

Try the following function, which should be reasonably accurate everywhere in the complex plane. It stitches together sexp(z), with the riemzprx(z), which converges best for values of imag(z)>=idelta, which is about 0.016i after 14 iterations. My algorithm works by iterating back and fourth, generating a more accurate riemaprx(z) from sexp(z), and then generating a more accurate sexp(z) from the updated riemaprx(z). Then it updates the array sizes and makes idelta a little smaller.
- Sheldon

Code:

stitchsexp(z) = {
  print(z);
  if (imag(z)>=1, return (riemaprx(z)));
  if (imag(z)<=-1, return (conj(riemaprx(conj(z)))));
  return(sexp(z));
}
pc = vectorv(65,r,stitchsexp(2*Pi*I*(r-1)/64))

Code:

for (s=1,65,print(pc[s]));
1.0000000000000000000000000000000 + 0.E-105*I
0.99411091497356251121660334054347 + 0.12727677792455698141545403455080*I
0.97672654305998578305917759789250 + 0.25197118483864734183151398762201*I
0.94866782293218727219984591701567 + 0.37167198380072132579505764662646*I
0.91121508246980523613033586612553 + 0.48428898713827736928291690621458*I
0.86599076273649904199094947918957 + 0.58816463017016779675563269986350*I
0.81482093582936119822241434165409 + 0.68213715983980310094848042368576*I
0.75959435854105266354457468711565 + 0.76555362874190903322261122971610*I
0.70213558596085847285750652219703 + 0.83823835072842774548779489678433*I
0.64410392653404100704979636884033 + 0.90042772769655387626345342340194*I
0.58692420585720595244283156324057 + 0.95268475759286478172829599485305*I
0.53174980674745764818795178029890 + 0.99580624694707393858489014630534*I
0.47945423655143147463739109030877 + 1.0307335062742955072331330682852*I
0.43064495930562204323634954278206 + 1.0584740535001761782448191001151*I
0.38569235612456845094921072388723 + 1.0800384604495544323525243042957*I
0.34476706887833815417334909473206 + 1.0963935643563012430808924865586*I
0.30788015100843650707563101146503 + 1.1084311394694043905329177454976*I
0.27492194001038214413926826693241 + 1.1169498344674768677723986077593*I
0.24569703876602253338971340665618 + 1.1226476131982804611669541038761*I
0.21995404404932586037347813521729 + 1.1261218964845432608621383877020*I
0.19740960493527573955655725416746 + 1.1278748910342390826684105768548*I
0.17776702936849884487293749742861 + 1.1283220397701643216380955188343*I
0.16073002863623299255628073431661 + 1.1278020154970139874441116391279*I
0.14601236024543245526018230481850 + 1.1265871324641023617450540637499*I
0.13334416296472653516342485574701 + 1.1248934309410459502330586563960*I
0.12247572704559135736508544106469 + 1.1228899870416972600103884620249*I
0.11317934826520773235503638826490 + 1.1207072178116520667083477139737*I
0.10524980412933908586090548559599 + 1.1184441017971748774371045992344*I
0.098503881541559116527366610557030 + 1.1161743321749868424037832109684*I
0.092779286895306502005151548157326 + 1.1139514767511686074259337461195*I
0.087933185984612679199410547995810 + 1.1118132484652623475840711442351*I
0.083840553159988453591219696326586 + 1.1097850007567059445554175501301*I
0.080392455728788036877818724515250 + 1.1078825612100714229445599015055*I
0.077494358790625486575004343963805 + 1.1061145092323597178189207359442*I
0.075064505315839933093265775124783 + 1.1044839924824197901153148522678*I
0.073032404200808376610851753152679 + 1.1029901645356753348872249262035*I
0.071337443393641346948052091906923 + 1.1016293141457599148697176131381*I
0.069927634407947893925195927960497 + 1.1003957451968517946114151542835*I
0.068758487367060749922974475255537 + 1.0992824563833147144830256779775*I
0.067792011158244554068295700263757 + 1.0982816609240691819657330364023*I
0.066995830571445197945752119851721 + 1.0973851791936958574531273262382*I
0.066342410885254327447809150203682 + 1.0965847309297133648948693130044*I
0.065808379828507206066064649667926 + 1.0958721485197513155662294392481*I
0.065373936888691168880042862182397 + 1.0952395286376436974032454744279*I
0.065022340344101677224898321355895 + 1.0946793360427426155349636051232*I
0.064739463015669870641047317468861 + 1.0941844705532570605920106190814*I
0.064513408463790668464414968424662 + 1.0937483059389388676853763144844*I
0.064334180126366560115336601834380 + 1.0933647076538990624484740937655*I
0.064193396661816161171945934128667 + 1.0930280348650106203942826351996*I
0.064084047497062011236948413414600 + 1.0927331310573251150080966931013*I
0.064000283269284013237609881379406 + 1.0924753065593200270793805495699*I
0.063937236483214214778898014606209 + 1.0922503155819538626738985873120*I
0.063890868279932516377395753476112 + 1.0920543297693504886801868088716*I
0.063857837728869286625789537680067 + 1.0918839097854470174685462083430*I
0.063835390514486963192534150305786 + 1.0917359760858963602760903651234*I
0.063821264296554837378338419121594 + 1.0916077797284012606331404037588*I
0.063813608382306521538321775082730 + 1.0914968738417688583645880251576*I
0.063810915664542902701493063497407 + 1.0914010861916872518284773701780*I
0.063811965056336580062258923458939 + 1.0913184931393879726606969700231*I
0.063815772894633215968042430315087 + 1.0912473951798114411841652328590*I
0.063821551995675476170516382327680 + 1.0911862941620766837225532123451*I
0.063828677228417059608758993813316 + 1.0911338722316975944866139433040*I
0.063836656631231378850492509629980 + 1.0910889724868448218285695781494*I
0.063845107235203121006324848070933 + 1.0910505813066151945143721194966*I
0.063853734876755975181544200369394 + 1.0910178122849990427062282998385*I