x↑↑x = -1

[sheldonison]

Numerically, it can't possibly give results of any reasonable precision, due to the really nasty singularity for the Abel function of 0.

Ha! Don't I know it! You can see the convergence analysis in my post. Using a 2835x2835 system, and using 11-point Newton-Cotes interpolation, I still barely managed to get convergence on the order of 10^-7, or about 20-25 bits of accuracy.

For comparison, the intuitive solution with a matrix that large would already get about 30-50 bits of accuracy (depends on where you measure it), and my accelerated version tends to get twice the accuracy (in bits) of the intuitive solution, for a given system size.

However, you mentioned that the singularity is "really nasty". I used to think so too, but when I was analyzing it last year, I discovered it's actually not that bad. I did some Riemann mapping tests (after rewriting my code), and I used iterated logarithms as a test subject. The Kneser singularities are basically an infinitely iterated logarithm.

I have no idea how you got results that are that good, for a Kneser Riemann mapping. I just generated a 2000 term Taylor series for the Riemann Circle mapping, from the theta(z) mapping, using the equations I posted. If \( \Im(z)>0.002 \), then a 2000 term series is sufficient for approximately 10^-15 decimal digits accuracy for the end result sexp(z), and I verified that. But for a 2000 term series, at the real axis, I get an error term of a little bit bigger than 10^-4 when compared to sexp(z). That's nowhere near 10^-7. Anyway, here are my error terms, followed by the first 60 terms of the Riemann mapping Taylor series. If Jay gets his code working, perhaps he can post his series.

Code:

error term at 0.002*I
1.48539974604412 E-15 -0.500000000000000 + 0.00200000000000000*I
1.47489069472385 E-15 -0.450000000000000 + 0.00200000000000000*I
1.46121503057057 E-15 -0.400000000000000 + 0.00200000000000000*I
1.44395154499657 E-15 -0.350000000000000 + 0.00200000000000000*I
1.42240825152162 E-15 -0.300000000000000 + 0.00200000000000000*I
1.39551703699474 E-15 -0.250000000000000 + 0.00200000000000000*I
1.36161960857917 E-15 -0.200000000000000 + 0.00200000000000000*I
1.31799885508938 E-15 -0.150000000000000 + 0.00200000000000000*I
1.25966680569473 E-15 -0.100000000000000 + 0.00200000000000000*I
1.17508969251408 E-15 -0.0500000000000000 + 0.00200000000000000*I
1.05484803687254 E-15 0.E-67 + 0.00200000000000000*I
1.42448041167858 E-15 0.0500000000000000 + 0.00200000000000000*I
1.57521942898691 E-15 0.100000000000000 + 0.00200000000000000*I
1.70167669559400 E-15 0.150000000000000 + 0.00200000000000000*I
1.81712691376729 E-15 0.200000000000000 + 0.00200000000000000*I
1.92663904167934 E-15 0.250000000000000 + 0.00200000000000000*I
2.03279073901222 E-15 0.300000000000000 + 0.00200000000000000*I
2.13706715820979 E-15 0.350000000000000 + 0.00200000000000000*I
2.24034990997028 E-15 0.400000000000000 + 0.00200000000000000*I
2.34311301352602 E-15 0.450000000000000 + 0.00200000000000000*I
2.44549405599774 E-15 0.500000000000000 + 0.00200000000000000*I

error term at the real axis
0.000122279589843505 -0.500000000000000 + 0.E-67*I
0.000121410233128676 -0.450000000000000 + 0.E-67*I
0.000120278572589665 -0.400000000000000 + 0.E-67*I
0.000118849428912911 -0.350000000000000 + 0.E-67*I
0.000117064967417645 -0.300000000000000 + 0.E-67*I
0.000114835709026505 -0.250000000000000 + 0.E-67*I
0.000112022145936091 -0.200000000000000 + 0.E-67*I
0.000108393990959996 -0.150000000000000 + 0.E-67*I
0.000103522306328356 -0.100000000000000 + 0.E-67*I
0.0000963767408324917 -0.0500000000000000 + 0.E-67*I
0.0000486991561415855 0.E-67 + 0.E-67*I
0.000117229100049087 0.0500000000000000 + 0.E-67*I
0.000129647505486836 0.100000000000000 + 0.E-67*I
0.000140069525671477 0.150000000000000 + 0.E-67*I
0.000149581759266669 0.200000000000000 + 0.E-67*I
0.000158602201407647 0.250000000000000 + 0.E-67*I
0.000167343698487192 0.300000000000000 + 0.E-67*I
0.000175928857373946 0.350000000000000 + 0.E-67*I
0.000184430363192777 0.400000000000000 + 0.E-67*I
0.000192887150257634 0.450000000000000 + 0.E-67*I
0.000201310257945835 0.500000000000000 + 0.E-67*I

Here is the Riemann mapping series
{RiemannMap=
x^ 1* (-327.7900737499296362245223 - 23.90211663662640951715732*I)
+x^ 2* (-185.9456815203626111385993 - 21.53372742928080246263683*I)
+x^ 3* (-130.5021375740268760564963 - 18.40398494742714600499322*I)
+x^ 4* (-100.7554518126267253224986 - 16.01165962890605241739091*I)
+x^ 5* (-82.14559469605831107094789 - 14.19251002958850769839935*I)
+x^ 6* (-69.38293686758304392583564 - 12.77142709975804454542933*I)
+x^ 7* (-60.07614697028410621039645 - 11.63101188033707572674404*I)
+x^ 8* (-52.98403795063564504697256 - 10.69456145922465454160790*I)
+x^ 9* (-47.39745732289376085443843 - 9.910756323465195981465028*I)
+x^10* (-42.88143943638113746945703 - 9.244178237858093337857055*I)
+x^11* (-39.15419551272454233078916 - 8.669646366305387333487256*I)
+x^12* (-36.02505166964368981700397 - 8.168776352931145519089286*I)
+x^13* (-33.36032741603822724385750 - 7.727829338414107315386952*I)
+x^14* (-31.06349373678005276058250 - 7.336326554807253921625497*I)
+x^15* (-29.06308871264384926429700 - 6.986131482096766468044178*I)
+x^16* (-27.30506627723899429124894 - 6.670825910226712052166535*I)
+x^17* (-25.74778878750429139698420 - 6.385275832487105490648721*I)
+x^18* (-24.35865491006778095837144 - 6.125323046096789509916112*I)
+x^19* (-23.11177149208846981595736 - 5.887561907396471181236202*I)
+x^20* (-21.98631058341126369713516 - 5.669174971195849915165872*I)
+x^21* (-20.96532720981577588630165 - 5.467810116389373439472789*I)
+x^22* (-20.03489378164942227447907 - 5.281487401183351085916574*I)
+x^23* (-19.18345636223462436180240 - 5.108527555489849135347319*I)
+x^24* (-18.40134912867140525038615 - 4.947496445373594090916479*I)
+x^25* (-17.68042342683887284539206 - 4.797161481768521336801208*I)
+x^26* (-17.01376104147164494207010 - 4.656457068689557924111830*I)
+x^27* (-16.39545017500698181535013 - 4.524456968352042340209790*I)
+x^28* (-15.82040868780648458805033 - 4.400352013216143946839389*I)
+x^29* (-15.28424335548893485935653 - 4.283431990561650601367138*I)
+x^30* (-14.78313685745141748617509 - 4.173070811871996871871108*I)
+x^31* (-14.31375632090695905289909 - 4.068714289432567183874581*I)
+x^32* (-13.87317876882008282248580 - 3.969869998212339262685917*I)
+x^33* (-13.45882993354894505984508 - 3.876098817567044447093479*I)
+x^34* (-13.06843372019828813662486 - 3.787007835259772100614620*I)
+x^35* (-12.69997021692735288669247 - 3.702244363303377394209595*I)
+x^36* (-12.35164061114453141461855 - 3.621490866598529579255885*I)
+x^37* (-12.02183772117189722174859 - 3.544460645181984030579682*I)
+x^38* (-11.70912112148618838671693 - 3.470894141964731753824588*I)
+x^39* (-11.41219604687566690125555 - 3.400555772229562073761104*I)
+x^40* (-11.12989542195876179277337 - 3.333231190431880029342656*I)
+x^41* (-10.86116448862628774538233 - 3.268724925173703485610012*I)
+x^42* (-10.60504760334181465513582 - 3.206858325479057669041820*I)
+x^43* (-10.36067685502226449503764 - 3.147467771358334465163649*I)
+x^44* (-10.12726221705588583477331 - 3.090403109621203160400691*I)
+x^45* (-9.904082997408298909140751 - 3.035526282376254789350546*I)
+x^46* (-9.690480391398340144605118 - 2.982710120945907913648327*I)
+x^47* (-9.485850974651652189477772 - 2.931837282265036591517706*I)
+x^48* (-9.289641000551706203919960 - 2.882799308407712700307342*I)
+x^49* (-9.101341388441679990739435 - 2.835495792845282582443991*I)
+x^50* (-8.920483306853164130678297 - 2.789833639497083468923510*I)
+x^51* (-8.746634270909107793359390 - 2.745726402685078925015440*I)
+x^52* (-8.579394685369404944291408 - 2.703093697819645491229504*I)
+x^53* (-8.418394775035100065267641 - 2.661860674085207203683352*I)
+x^54* (-8.263291852782031300976388 - 2.621957541609435214461432*I)
+x^55* (-8.113767882661989264022255 - 2.583319146627242910994307*I)
+x^56* (-7.969527301534856323984202 - 2.545884589022495921095650*I)
+x^57* (-7.830295067777372297714664 - 2.509596877372102599887321*I)
+x^58* (-7.695814909914436622394283 - 2.474402617250175412680592*I)
+x^59* (-7.565847751668658066339278 - 2.440251729091720368616231*I)
+x^60* (-7.440170293030741768660035 - 2.407097192380241168765835*I)
}
skipping to the 2000th term:
x^2000*(-0.2110504350274731041729573 - 0.1103488141396473954206754*I)
I went back and generated a 4000 term series too:
x^4000*(-0.1033386085272496424048194 - 0.05795348899039744714402679*I)

I would estimate that each doubling of the number of terms would give you one binary bit additional accuracy. So if a 2000 term series is 10^-4, then a 2,000,000 term series might be 10^-7. Interestingly, you can calculate sexp(z) along a unit circle in the upper half of the complex plane very accurately with this series. If \( \Im(z)>0.004i \), this series is accurate to 25 decimal digits; and if you use it along with the conj(f(conj(z))) in the lower half of the complex plane, you can recover an sexp(z) unit circle Taylor series accurate to 25 decimal digits....