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 Tetra-series Gottfried Ultimate Fellow Posts: 765 Threads: 119 Joined: Aug 2007 07/02/2008, 11:01 AM A new result, which I just posted in sci.math; I'll improve the formatting later (a bit lazy...) Gottfried Code:```Am 30.06.2008 17:00 schrieb alainverghote@gmail.com: >> >> Gottfried > > > > Well, inverse function of tf(x) using lambertW() > > is  -Lambert(-1/2exp(1/2e^x - 1)) +1/2e^x - 1 > > series near 0 , x +x^2 +4/3x^3 +29/12x^4 +51/10x^5 ... > > > > Yepp, I got the same series - good! But now - iterations and especially the sum of iterations of this functions in the Lambert-representation should be intractable, so I don't assume this can be helpful to get more insight in the source of the inconsisteny-problem; remember: by the formal application of the matrix-approach       asn(x)+asp(x)-x = 0  // expected or       asn(x,f(x)) = x - asp(x,f(x)) // expected or       asp(x,f°(-1)(x)) = x - asp(x,f(x)) // expected is not true, at least for the function f(x) = exp(x)-1 Well - it was a try... -------------------------------------- Meanwhile I refined my computation-process, so I've now even the function fz(x) with the condition   e^x - 1 = fz(x) + fz(fz(x)) + fz(fz(fz(x))) + ..           = sum{h=1..inf} fz°h(x) I got fz(x) = 2*(x/4)/1! + 6*(x/4)^2/2! + 10*(x/4)^3/3! - 46*(x/4)^4/4! - 554*(x/4)^5/5!        - 1690*(x/4)^6/6! + 27882*(x/4)^7/7! + 505986*(x/4)^8/8! + 2529590*(x/4)^9/9!        - 61918794*(x/4)^10/10! - 1726391798*(x/4)^11/11! - 14268435022*(x/4)^12/12!        + 352044609814*(x/4)^13/13! + O(x^14) where the integer parts of the coefficients are [2] [6] [10] [-46] [-554] [-1690] [27882] [505986] [2529590] [-61918794] [-1726391798] [-14268435022] [352044609814] which have to be divided py powers of 4 and by factorials to give the coefficients of the function. The float representation of this function is fz(x) =   0.500000000000*x^1 + 0.187500000000*x^2 + 0.0260416666667*x^3 - 0.00748697916667*x^4          - 0.00450846354167*x^5 - 0.000573052300347*x^6 + 0.000337655203683*x^7 + 0.000191486449469*x^8          + 0.0000265917660278*x^9 - 0.0000162726971838*x^10 - 0.0000103115449999*x^11          - 0.00000177549511523*x^12 + 0.000000842437121499*x^13 + 0.000000632647393830*x^14          + O(x^15) Can we give a range for x where this converges ? The quotients of subsequent coefficients give the following sequence 0.375000000000,0.138888888889,-0.287500000000, 0.602173913043,0.127105896510,-0.589222316145, 0.567106466538,0.138870223462,-0.611944959460, 0.633671534805,0.172185168687,-0.474480112208, 0.750972835462,0.212965249501,-0.333234112499, 0.928624638217,0.256835516535,-0.218432174828, 1.22327362079,0.303250445581,-0.126531683586, 1.82969444504,0.353026162019,-0.0509774937397, 3.94499797418,0.407780103462,0.0133041813614,-13.1183083394, 0.470026310618,0.0698970628404,-2.15481130004, 0.543690120454,... ----------------------------------------------------------- Using 32 coefficients for the function and 60 iterates for the sum I could approximate e^1 -1 relatively well. I got     sum(h=1,60,fz°h(1.0)) - ( exp(1)-1 ) =  -3.24385306514 E-13 where the quality of approximation increased when terms of the function and numbers of iterates are increased. Fun... :-) Gottfried Helms``` Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Tetra-series - by Gottfried - 11/20/2007, 12:47 PM RE: Tetra-series - by andydude - 11/21/2007, 07:14 AM RE: Tetra-series - by Gottfried - 11/22/2007, 07:04 AM RE: Tetra-series - by andydude - 11/21/2007, 07:51 AM RE: Tetra-series - by Gottfried - 11/21/2007, 09:41 AM RE: Tetra-series - by Ivars - 11/21/2007, 03:58 PM RE: Tetra-series - by Gottfried - 11/21/2007, 04:37 PM RE: Tetra-series - by Gottfried - 11/21/2007, 06:59 PM RE: Tetra-series - by andydude - 11/21/2007, 07:24 PM RE: Tetra-series - by Gottfried - 11/21/2007, 07:49 PM RE: Tetra-series - by andydude - 11/21/2007, 08:39 PM RE: Tetra-series - by Gottfried - 11/23/2007, 10:47 AM RE: Tetra-series - by Gottfried - 12/26/2007, 07:39 PM RE: Tetra-series - by Gottfried - 02/18/2008, 07:19 PM RE: Tetra-series - by Gottfried - 06/13/2008, 07:15 AM RE: Tetra-series - by Gottfried - 06/22/2008, 05:25 PM Tetra-series / Inverse - by Gottfried - 06/29/2008, 09:41 PM RE: Tetra-series / Inverse - by Gottfried - 06/30/2008, 12:11 PM RE: Tetra-series / Inverse - by Gottfried - 07/02/2008, 11:01 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 10:38 AM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 11:01 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 01:25 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 02:40 PM RE: Tetra-series / Inverse - by andydude - 10/31/2009, 09:37 PM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 10:33 PM RE: Tetra-series / Inverse - by Gottfried - 11/01/2009, 07:45 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 03:56 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 04:12 AM RE: Tetra-series / Inverse - by andydude - 11/03/2009, 05:04 AM RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 12:58 PM

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