Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
[AIS] (alternating) Iteration series: Half-iterate using the AIS?
#16
(12/15/2012, 06:49 AM)Gottfried Wrote: Hi Sheldon -

this sounds really great ...

What about the cosine instead of the sine? You introduced the cosine in the formula: is there some optimization argument for it? I think it shifts the height-zero-definition from to the -position and must be compensated elsewhere in the formulae?

Gottfried
Hey Gottfried,

I think using cosine vs sine is completely arbitrary. has a period of 2, so to switch from cosine to sine requires a shift of -0.5. To generate the function, I needed to empirically calculate a very accurate maximum amplitude for your asum function, so I think that's when I started centering on the cosine(z) maximum amplitude of asum, as opposed to the sin(z) zero crossing of asum. I also noticed the maximum occurs approximately at the midway point between the two fixed points, so I kept working with cosine after that. Here is the equivalent series, shifted by -0.5, for the . If you want the series centered anywhere else, +/-2.35i in the complex plane, I can provide that too, with equivalent accuracy.
Code:
{asumsine=
        5.5313086758882852195782696632367
+x^ 1* -2.0430674012650642389618054781319
+x^ 2* -0.37225775792009989386677189320438
+x^ 3*  0.099658659136533363777013664824123
+x^ 4*  0.056164249815831421690267039207707
+x^ 5*  0.0024341739796340503006223611990660
+x^ 6* -0.0048821136236917176881522235140457
+x^ 7* -0.0013625033138659058581259860969190
+x^ 8*  0.00013860229935952324746169580217982
+x^ 9*  0.00016031197013218517260959597928241
+x^10*  0.000026233716766403365591594754596210
+x^11* -0.0000078397741621291476048490513062752
+x^12* -0.0000041654570665036359076289439137305
+x^13* -0.00000043429845750234876856499448438055
+x^14*  0.00000020220255943526951948556412163547
+x^15*  0.00000010469795440929394604162385515811
+x^16*  0.000000022104640668047825697063550269138
+x^17* -0.0000000050752778367022374662152498065127
+x^18* -0.0000000054268147119071794970623507258578
+x^19* -0.00000000088377028240205109643729838693901
+x^20*  0.00000000053724808383268656860483599781710
+x^21*  2.2695303649482206760147509826254 E-10
+x^22* -2.0755300318667016920518969613493 E-11
+x^23* -2.8606106192156205211842427520580 E-11
+x^24* -2.4115669661617778343439526060037 E-12
+x^25*  2.4097061226690732664291635948851 E-12
+x^26*  5.3024239886580580464040016659410 E-13
+x^27* -1.3352860389778966943126997119541 E-13
+x^28* -4.4676066586863015132548102097061 E-14
+x^29*  5.6825914165455658434390405670152 E-15
+x^30* -4.8592335192576129248775913955152 E-16
+x^31* -1.2100990011140146543285064592901 E-15
+x^32*  7.6111952883565188094575760424171 E-16
+x^33*  4.0350523000744874273361378890307 E-16
+x^34* -1.2914109236182899564695839116964 E-16
+x^35* -8.9756371000000830488253273137664 E-17
+x^36*  1.1989065889716564053858711596068 E-17
+x^37*  1.4933359276677119637947130427580 E-17
+x^38* -2.6806360741610229880740798420225 E-19
+x^39* -2.0268972881933589191981413633055 E-18
+x^40* -1.2667505083540508761525831833105 E-19
+x^41*  2.3992792073090946823641233544269 E-19
+x^42*  2.7280436906032629343367213777381 E-20
+x^43* -2.7155583649715904622682398258307 E-20
+x^44* -3.3406372025307868881770171768099 E-21
+x^45*  3.3583042746061851597680641775725 E-21
+x^46*  2.8315393198749754587922156142229 E-22
+x^47* -4.9201511971234663439574586786040 E-22
+x^48* -1.9937960921207882784670984182728 E-23
+x^49*  7.9867782170632207111309500434223 E-23
+x^50*  2.8277790598852084735133034329599 E-24
+x^51* -1.2866120668308479467449444353412 E-23
+x^52* -8.0009133087978621182694530691203 E-25
+x^53*  1.9452214442357667538533694067711 E-24
+x^54*  1.9749461280904030199079122668892 E-25
+x^55* -2.7421549570850430614725716263261 E-25
+x^56* -3.8853147981092745620598414279056 E-26
+x^57*  3.6832634896487287902937856325878 E-26
+x^58*  6.4643562945915383055244186623733 E-27
+x^59* -4.8954244803017602327236947884728 E-27
+x^60* -9.6822083644359538571851444600130 E-28
+x^61*  6.6905004127327106773176137800281 E-28
+x^62*  1.3903608890787339354290348605861 E-28
+x^63* -9.5633563001034297889480127198725 E-29
+x^64* -2.0299634580180042177140372334119 E-29
+x^65*  1.4105236281490162329783786025153 E-29
+x^66*  3.1114882040622484025316768678442 E-30
+x^67* -2.0873558246659861647640896462484 E-30
+x^68* -4.9639057430754083874748766760233 E-31
+x^69*  3.0323328554893664432512738086894 E-31
+x^70*  7.9943649080493773011295509130754 E-32
+x^71* -4.2928725325635863410096873706367 E-32
+x^72* -1.2676013386130031721900449722132 E-32
}
Reply


Messages In This Thread
RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/15/2012, 01:43 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Perhaps a new series for log^0.5(x) Gottfried 3 675 03/21/2020, 08:28 AM
Last Post: Daniel
  Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 607 09/09/2019, 10:55 PM
Last Post: tommy1729
  Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) Gottfried 1 879 09/09/2019, 10:50 PM
Last Post: tommy1729
Question Taylor series of i[x] Xorter 12 13,258 02/20/2018, 09:55 PM
Last Post: Xorter
  Does tetration take the right half plane to itself? JmsNxn 7 7,063 05/16/2017, 08:46 PM
Last Post: JmsNxn
  Half-iteration of x^(n^2) + 1 tommy1729 3 4,578 03/09/2017, 10:02 PM
Last Post: Xorter
  Dynamics as alternating waves tommy1729 2 2,685 02/13/2017, 01:01 AM
Last Post: tommy1729
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 16,583 01/09/2017, 02:41 AM
Last Post: Gottfried
  Complaining about MSE ; attitude against tetration and iteration series ! tommy1729 0 1,823 12/26/2016, 03:01 AM
Last Post: tommy1729
  2 fixpoints , 1 period --> method of iteration series tommy1729 0 1,876 12/21/2016, 01:27 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)