Comparision: 4 different methods of interpolation

[Gottfried]

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

Hi Sheldon -
I've tried your kneser.gp - many thanks: I'll have to study this in more detail later. I reproduced the coefficients for the powerseries for sexp successfully.


\r kneser.gp
init(4)
loop(14)
dumparray


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:

pc = vectorv(65,r,sexp(2*Pi*I*(r-1)/64))
-----------
1.00000000000+0.E-105*I
0.994110914974+0.127276777925*I
0.976726543060+0.251971184839*I
0.948667822932+0.371671983801*I
0.911215082470+0.484288987138*I
0.865990762736+0.588164630170*I
0.814820935829+0.682137159840*I
0.759594358541+0.765553628742*I
0.702135585961+0.838238350728*I
0.644103926534+0.900427727697*I
0.586924205857+0.952684757593*I
0.531749806747+0.995806246947*I
0.479454236551+1.03073350627*I
0.430644959306+1.05847405350*I
0.385692356125+1.08003846045*I
0.344767068878+1.09639356436*I
0.307880151008+1.10843113947*I
0.274921939972+1.11694983442*I
0.245697026317+1.12264759756*I
0.219951093578+1.12611839242*I
0.196882860140+1.12728172854*I
0.104936081109+1.05034222516*I
-7.82699075484-7.02415700313*I
-709.298724923-690.557826748*I
-51957.5575671-48491.8027706*I
-3187869.14771-2853512.97028*I
-166166092.408-142895081.478*I
-7450029334.05-6164718982.34*I
-290496242538.-231637469246.*I
-9.94894155289E12-7.65497982943E12*I
-3.01936911598E14-2.24453542988E14*I
-8.18516413940E15-5.88555696918E15*I
-1.99639929780E17-1.39005088025E17*I
-4.40983394463E18-2.97626400168E18*I
-8.87445465596E19-5.81128395357E19*I
-1.63594655086E21-1.04032968135E21*I
-2.77630447925E22-1.71596276853E22*I
-4.35731956614E23-2.61966426185E23*I
-6.35111854280E24-3.71697245818E24*I
-8.63059373889E25-4.92042449645E25*I
-1.09734573161E27-6.09848329535E26*I
-1.30977172945E28-7.10017710659E27*I
-1.47207362265E29-7.78866883271E28*I
-1.56236696609E30-8.07288957626E29*I
-1.57003044882E31-7.92695731129E30*I
-1.49753982260E32-7.39194047837E31*I
-1.35892812751E33-6.56110592764E32*I
-1.17570899338E34-5.55507719216E33*I
-9.71776298482E34-4.49538064973E34*I
-7.68810138740E35-3.48354134756E35*I
-5.83218466008E36-2.58951669954E36*I
-4.24941962635E37-1.84960552474E37*I
-2.97850538389E38-1.27139182283E38*I
-2.01132601530E39-8.42282817727E38*I
-1.31035925744E40-5.38538829227E39*I
-8.24701803610E40-3.32755801839E40*I
-5.02048567638E41-1.98939873261E41*I
-2.95972451395E42-1.15216571225E42*I
-1.69161357840E43-6.47124093462E42*I
-9.38334654197E43-3.52855934257E43*I
-5.05661665865E44-1.86973058846E44*I
-2.64987303104E45-9.63707523115E44*I
-1.35160403892E46-4.83602667122E45*I
-6.71600116592E46-2.36473793281E46*I
-3.25363228864E47-1.12767556062E47*I


I'll give it more time later. Thanks so far!

Gottfried