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 [AIS] (alternating) Iteration series: Half-iterate using the AIS? sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 12/15/2012, 01:43 PM (This post was last modified: 12/15/2012, 02:25 PM by sheldonison.) (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 $x_0$ to the $x_{0.5}$ -position and must be compensated elsewhere in the formulae? GottfriedHey Gottfried, I think using cosine vs sine is completely arbitrary. $\text{asum}(\alpha^{-1}(z))$ has a period of 2, so to switch from cosine to sine requires a shift of -0.5. To generate the $\alpha^{-1}$ 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 $\text{asum}(\alpha^{-1}(z))=\text{amplitude}\times\sin(z)$. 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 } « Next Oldest | Next Newest »

 Messages In This Thread [AIS] (alternating) Iteration series: Half-iterate using the AIS? - by Gottfried - 12/06/2012, 12:10 AM RE: Half-iterate using the infinite iteration-series? - by tommy1729 - 12/08/2012, 06:27 PM RE: Half-iterate using the infinite iteration-series? - by Gottfried - 12/09/2012, 03:13 PM RE: Half-iterate using the infinite iteration-series? - by Gottfried - 12/10/2012, 02:23 AM RE: Half-iterate using the infinite iteration-series? - by mike3 - 12/10/2012, 03:33 AM RE: Half-iterate using the infinite iteration-series? - by Gottfried - 12/10/2012, 10:38 AM RE: Half-iterate using the infinite iteration-series? - by mike3 - 12/10/2012, 11:26 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/11/2012, 12:44 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/11/2012, 01:16 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/13/2012, 02:49 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/15/2012, 04:47 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/15/2012, 06:49 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/15/2012, 01:43 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/15/2012, 11:27 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/16/2012, 12:05 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/17/2012, 10:19 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/18/2012, 11:17 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/22/2012, 04:12 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/23/2012, 09:01 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 12/23/2012, 03:51 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Nasser - 12/14/2012, 05:06 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/14/2012, 07:16 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/15/2012, 06:37 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by andydude - 12/18/2012, 06:05 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by tommy1729 - 12/15/2012, 10:06 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/23/2012, 04:43 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 12/23/2012, 05:59 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 01/02/2013, 03:28 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 01/02/2013, 04:05 PM RE: Iteration series: Half-iterate using the infinite iteration-series? - by Gottfried - 01/03/2013, 07:07 AM RE: Iteration series: Half-iterate using the infinite iteration-series? - by sheldonison - 01/03/2013, 09:58 PM RE: [AIS] (alternating) Iteration series: Half-iterate using the AIS? - by Gottfried - 03/26/2015, 01:01 PM RE: [AIS] (alternating) Iteration series: Half-iterate using the AIS? - by tommy1729 - 03/27/2015, 11:28 PM

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