• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 tetration base conversion, and sexp/slog limit equations sheldonison Long Time Fellow Posts: 626 Threads: 22 Joined: Oct 2008 02/24/2009, 08:24 PM (This post was last modified: 02/25/2009, 12:17 AM by sheldonison.) sheldonison Wrote:.... With a little additional algebra, this allows defining an sexp/slog extension to real numbers, for base e, by iterating the ln function "n" times. In practice, for base e, using n=5 will give approximately a million digits of precision. This also works for any other arbitrary base. $\text{sexp}_e(x) = \lim_{b \to \eta^+}\text{ } \lim_{n \to \infty} \text{ln(ln(ln(}\cdots \text{sexp}_b (x + \text{slog}_b(\text{sexp}_e(n))))))$In this post, I will use this equation to derive the $\text{sexp}_e(-0.5)$, accurate to five significant digits, using a linear approximation of sexp/slog. I haven't made any progress in proving the limit converges, but assuming it does, earlier in this thread I derived the error term of a linear approximation as the base approaches $\eta^+$. error term =~ $m^2*(1/(72*e))*sqrt(1/3)$ relative error term =~ $m*(1/(72*e))*sqrt(1/3)$ with m=$\text{slog}_b(e)-\text{slog}_b(\text{slog}_b(e))$, as b approaches $\eta^+$, $m=e-\text{slog}_b(e)$ The error term uses the difference between a linear approximation, and a 3rd order approximation for the critical section. A linear approximation has a continuous 1st derivative, and a 3rd order approximation has a continuous 2nd derivative. For this example, use a linear approximation to calculate sexp_e(-0.5), accurate to five significant digits. The relative error term required would be $10^{-5}$, which leads to m=~1/295. Use b=$\eta$+(1/1500). As a comparison, the other results I used were $\eta$+(1/25), but using a third order approximation. The base being used here is much closer to $\eta$, and there are a lot more iterations. Starting with the limit equation, I picked a value of n=5, for sexp_e(5). My excel spreadsheet can't handle sexp_e(5), but it can handle x=slog_b(slog_b(sexp_e(5)), which is x=10355300. Then, keep iterating x=log_b(x), another 65 times, for a total of 67 log_b iterations, until the value of x is less then log_b(e). I stopped at x=2.71365. Now that x is in the linear region of the slog_b(x) curve, let x= sexp_b(slog_b(x)-0.5). For the slog_b function, I got a remainder term of 0.638016. Subtract 0.5, and use the linear approximation to begin the conversion back to the sexp_b, x=2.71194. The error term doubles to +/-0.000002 since two different linear approximations are required, one for the slog_b and another for the sexp_b. Now iterate, taking x=b^x 66 times. I got x=5.36*10^77. One last exponentiation will be too large for the spread sheet, so I pair it up with the first ln. Let x= ln(ln(ln(ln(ln(b^x))))). The result I got was sexp(-0.5)=0.497818+/-0.00002. Using the more accurate 3rd order approximation for the critical section, I got sexp(-0.5)=0.49783297. The delta between the two values was 1.5*10^-5 which is less than 2*10^-5. The next thing I plan to do is to convert the base=e sexp function posted by Kouznetsov to another base, probably b=1.45, and graph the equations and derivatives to show that Kouznetsov's sexp function requires a 1-cyclic definition of sexp/slog base conversions, as the limit cannot converge to a constant, but instead wobbles. Jay noticed the same thing for Andy's sexp/slog extension to real numbers. I will also graph the sexp_e function converting from base=1.45. Any other suggestions? « Next Oldest | Next Newest »

 Messages In This Thread tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/18/2009, 07:01 AM RE: tetration base conversion, questions and results - by sheldonison - 02/19/2009, 12:10 AM tetration base conversion, uniqueness criterion? - by bo198214 - 02/19/2009, 04:24 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/20/2009, 10:54 AM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/20/2009, 01:07 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/20/2009, 02:51 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/21/2009, 12:18 AM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/21/2009, 12:39 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/21/2009, 02:59 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/21/2009, 06:36 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/22/2009, 04:41 AM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/22/2009, 04:04 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/24/2009, 08:24 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/24/2009, 09:57 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/24/2009, 10:21 PM RE: tetration base conversion, uniqueness criterion? - by sheldonison - 02/24/2009, 10:54 PM RE: tetration base conversion, uniqueness criterion? - by bo198214 - 02/24/2009, 11:06 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/26/2009, 11:04 AM RE: tetration base conversion, and sexp/slog limit equations - by bo198214 - 02/26/2009, 12:16 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/26/2009, 02:36 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/28/2009, 05:56 AM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/28/2009, 10:01 AM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/01/2009, 12:18 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/03/2009, 06:15 PM RE: tetration base conversion, and sexp/slog limit equations - by bo198214 - 03/03/2009, 06:46 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/03/2009, 07:27 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 03/09/2009, 06:34 PM Summay tetration base conversion, and sexp/slog limit equations - by sheldonison - 07/31/2009, 06:55 PM RE: Summay tetration base conversion, and sexp/slog limit equations - by sheldonison - 08/01/2009, 10:32 AM Is it analytic? - by sheldonison - 12/22/2009, 11:39 PM RE: tetration base conversion, and sexp/slog limit equations - by mike3 - 12/25/2009, 08:51 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 12/26/2009, 01:44 AM RE: tetration base conversion, and sexp/slog limit equations - by mike3 - 12/26/2009, 01:54 AM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 12/27/2009, 06:53 AM RE: tetration base conversion, and sexp/slog limit equations - by mike3 - 12/31/2009, 11:45 PM Inherent ringing in tetration, re: base conversion - by sheldonison - 01/02/2010, 05:31 AM RE: Inherent ringing in tetration, base conversion - by mike3 - 01/04/2010, 03:51 AM RE: Inherent ringing in tetration, base conversion - by sheldonison - 01/04/2010, 06:08 AM RE: tetration base conversion, and sexp/slog limit equations - by tommy1729 - 02/26/2013, 10:47 PM RE: tetration base conversion, and sexp/slog limit equations - by sheldonison - 02/27/2013, 07:05 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Complex Tetration, to base exp(1/e) Ember Edison 7 1,015 08/14/2019, 09:15 AM Last Post: sheldonison Is bounded tetration is analytic in the base argument? JmsNxn 0 1,162 01/02/2017, 06:38 AM Last Post: JmsNxn Sexp redefined ? Exp^[a]( - 00 ). + question ( TPID 19 ??) tommy1729 0 1,403 09/06/2016, 04:23 PM Last Post: tommy1729 Taylor polynomial. System of equations for the coefficients. marraco 17 14,926 08/23/2016, 11:25 AM Last Post: Gottfried Dangerous limits ... Tommy's limit paradox tommy1729 0 1,722 11/27/2015, 12:36 AM Last Post: tommy1729 tetration limit ?? tommy1729 40 43,275 06/15/2015, 01:00 AM Last Post: sheldonison Some slog stuff tommy1729 15 10,948 05/14/2015, 09:25 PM Last Post: tommy1729 Totient equations tommy1729 0 1,607 05/08/2015, 11:20 PM Last Post: tommy1729 Bundle equations for bases > 2 tommy1729 0 1,602 04/18/2015, 12:24 PM Last Post: tommy1729 Limit of mean of Iterations of f(x)=(ln(x);x>0,ln(-x) x<0) =-Omega constant for all x Ivars 10 13,739 03/29/2015, 08:02 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)