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Revisting my accelerated slog solution using Abel matrix inversion
(01/07/2019, 01:40 AM)jaydfox Wrote: Also, here is a copy of the full 4096-term accelerated solution, at 256 bits of precision.  As I noted before, the terms start to become inaccurate after about 600-700 terms, so the remaining terms are mostly there for completeness, and to validate the solution.

Hey Jay, welcome back. I've always been fascinated by your accelerated matrix solution and I 've always wondered how stable the solution is.  I use the same accelerator to cancel out a lot of the slog singularity at L and thereby speed up computation.  However, other than the speedup, I use a completely different slog algorithm in my program.  I also included an implementation of your accelerated Matrix solution in my program, although it is limited by pari-gp's memory limitations.  I've typically generated your matrix solution for a 100x100 matrix which is only a 16-17 decimal digit solution; its been awhile since I experimented with it.

When I used your accelerated solution to improve my abel function convergence, I center my solution between the two fixed points, so the results aren't immediately comparable.  To help compare the results, I re-sampled  my solution, in an attempt to match your accelerated solution, centered at x=0.   

Here are the first 40 taylor series term of the result, printed to 78 decimal digits of precision, centered at 0.  I used to generated a solution which should match Kneser's Riemann mapping solution to about 111 decimal digits.  Then I resampled and used 1024 equally spaced points on a unit circle centered at zero so the resample should have all of the precision of the 111 decimal digit solution.  Term x^692 was 10^-111, so I could post 692 terms if interested for comparison if interested.  The results match your post at the beginning of this thread for about 25-26 decimal digits; which is interesting and way too good to be due to chance, but the differences will require more investigation... edit: perhaps something like this explains what's going on where theta is a 1-cyclic function.

Such a theta mapping might explain why Jay's computation is internally consistent to 450 decimal digits, of which Jay posted ~78 decimal digits here on the forum.  Perhaps one could calculate such a theta mapping to help explain what's going on; more work is required Smile  Monday evening I'll post some updates on how to bring Jay's slogseries into ...
+x^ 1* -0.0291847169160739298766307382174481294886258010492417221318557874529766446280271
+x^ 2*  0.00110090814344297056968002550433916758416294801484763606535323215947742312998891
+x^ 3*  0.000543879513462731464670816527227773747790273366131704879288264439117105813938343
+x^ 4* -0.000203213036542027025247546663475145270021938728061876580382634944327259507613070
+x^ 5*  3.22280652811476939965291611214222619698404045553233990135743756086579853787466 E-6
+x^ 6*  1.84668813053769065513146911482491187292420670063380010345839003346342341354811 E-5
+x^ 7* -2.55239021146659093615783219124309045163219508036433185642511568187908692740179 E-6
+x^ 8* -2.17351637826570837729804963941797089491977146333679699165249014512560057413400 E-6
+x^ 9*  4.33349226539374888910868041733179237880862838402420299817266589574554488350356 E-7
+x^10*  3.59008961404512998761419264561270511400560807428712691206276402209143971317407 E-7
+x^11* -6.43927348279012903879559091201250729509557812545956649627961806138194783640006 E-8
+x^12* -7.58187804421757561572052356032477063125979811760713129502174141007384371312194 E-8
+x^13*  7.30188108425733435064094181207366720162958780594310571075880451665411307408281 E-9
+x^14*  1.83129209024872042204766971560789608994275974103846108186858008329147982370143 E-8
+x^15*  4.01473257932598032346226007262089210881377546684696208058213238500627295459535 E-10
+x^16* -4.67885672554722345567795420931402756716458148476094002591681241111939059855612 E-9
+x^17* -7.82733541235021758641760171001880700472394347501908856378580767748054180877594 E-10
+x^18*  1.19380072264340389022833263428990612278791993921818383464545223232835517887038 E-9
+x^19*  4.22665522239028379306756941809347307667096404974011552334418245250598871959460 E-10
+x^20* -2.85665608816080013429919548538029002477867622159882429937211279129282931350602 E-10
+x^21* -1.81574577166171723785466296557378162670229177144708098538943655575248286391841 E-10
+x^22*  5.66239713718898883906312180713639755874555979636208960430143220041561095439608 E-11
+x^23*  6.96852050843926611038308842481707217583284225194934498017797384941774673015590 E-11
+x^24* -4.91620352304397459943356547431415494679778546116207934625863282799255486528870 E-12
+x^25* -2.44941717030514599798779423273779319972876052437616659420443888252926158412673 E-11
+x^26* -3.42123870391631441053377789422079448618314584584920132740140702413291174523968 E-12
+x^27*  7.81601611754706608693289451350701814359130060609483928174907844754635967785938 E-12
+x^28*  2.91220724036209763215394808304886606041356251273619264186884692847430324022217 E-12
+x^29* -2.16293011912428340286672961802220042913349279225855648441601827627055552941372 E-12
+x^30* -1.54770233704316926732425106606959328377206243330542890682121857904957483929056 E-12
+x^31*  4.42938340560344887035834315164496119841728421490728778076978332679969734227954 E-13
+x^32*  6.79973546377122236399487539221594812310631216618695038174924709985643998954290 E-13
+x^33* -6.90332802128681628839824907472046040540394965670779559600694012846196711125779 E-15
+x^34* -2.60582710142232072109007134613071747204873818687668425522691018054763266928324 E-13
+x^35* -6.01823146772145101019064836071571136694904735629484078612466275998052975606794 E-14
+x^36*  8.64299131187677287397417683498580770292146371585971227254498127557791367936589 E-14
+x^37*  4.48362256100150041731885432251130944047255099679585467364604096135948842001883 E-14
+x^38* -2.29484574790082762339099709778899822955675262384458506531727251403906417766592 E-14
+x^39* -2.35822617237345894785316721974155376877119178761358093987952559243891279892858 E-14
+x^40*  3.22266790018566996903291120532225424437815251483935826863343561533048881414813 E-15}
- Sheldon

Messages In This Thread
RE: Revisting my accelerated slog solution using Abel matrix inversion - by sheldonison - 01/07/2019, 05:14 AM
Analysis of Jay's slog vs Kneser - by sheldonison - 01/17/2019, 06:44 PM
RE: Analysis of Jay's slog vs Kneser - by jaydfox - 01/18/2019, 06:35 AM
RE: Analysis of Jay's slog vs Kneser - by jaydfox - 01/18/2019, 06:42 AM
RE: Analysis of Jay's slog vs Kneser - by jaydfox - 01/18/2019, 06:17 PM

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