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 [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried Ultimate Fellow Posts: 765 Threads: 119 Joined: Aug 2007 12/15/2012, 06:37 AM (This post was last modified: 12/15/2012, 07:11 AM by Gottfried.) (12/14/2012, 07:16 PM)Gottfried Wrote: Here is some explanation in terms of Pari/GP-code, how the serial alternating iteration sums asum(x) can be expressed/computed with the help of power series.Just a short, but useful addendum: if we assume identity of the serial comnputation of asum(x) via the Pari/GP-sumalt-procedure and that via the Neumann-series-matrices asum_mat(x) and its power series then one should consider to use that second method as its standard basis. I toyed a bit around with differentating and integrating using the asum(x) and found, that the asum_mat(x) needs only about 1/20 of the computation time, so my example integral needed 80 000 msec with the serial implementation of the asum(x) but only 4 000 msec using the power-series implementation. This should also be useful for the computation of the inverse of asum(x) as long as we need to interpolate it by binary search/Newton-method, where many function calls are needed. I think moreover, that this shall prove useful, once we shall step further to analytically continue the range for the x and for the base b outside the "safe intervals" and enter the realms of truly divergent series for the asum(x). Gottfried Additional readings: An early(2008 ) discussion of this method and some of the problems, which we seemingly can resolve now, but also a (very natural) view into regions of bases outside the Euler-summable range for the serial computation of the asum(x) is here http://go.helms-net.de/math/tetdocs/Tetr...roblem.pdf An involved discussion (2007) about the ability of the Neumann-type matrix for the asum(x) to represent an analytical continuation for the divergent cases - the matrix-ansatz was crosschecked against a shanks-summation in the range, where the shanks-summation was computable: http://go.helms-net.de/math/tetdocs/Iter...tion_1.htm Gottfried Helms, Kassel « 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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