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 Arguments for the beta method not being Kneser's method sheldonison Long Time Fellow Posts: 684 Threads: 24 Joined: Oct 2008 10/23/2021, 03:13 AM (This post was last modified: 10/23/2021, 10:33 PM by sheldonison.) (10/22/2021, 03:54 AM)JmsNxn Wrote: ...  fascinating, Sheldon.... I'm a little dumbfounded by how you are calculating logrho so fast about the singularity--but it makes sense for the most part. Hey James, Now lets define a function $$\text{logrho}(z)=\ln(-\rho(z))$$ where I'll use the shorthand notation $$l\rho(z)$$ for the remainder of this post.  Lets start with the following from my previous post, again this is for the 2pii periodic beta(z,1). \begin{align} f_0(z)=\beta(z)-\ln(1+\exp(-z));\;\;\; f_n(z) = \ln^{\circ n}f(z+n)\\ \rho_0(z)=-\ln(1+\exp(-z))\\ \rho_n(z)=\ln\left(1+\frac{\rho_{n-1}(z+1)}{f_{n-2}(z+1)}\right)\\ \end{align} Now lets change the recursive equation for $$\rho$$ to a recursive equation for $$l\rho$$ \begin{align} l\rho_0(z)=\ln\Big(\ln\big(1+\exp(-z)\big)\Big)\\ l\rho_n(z)=\ln\left(-\ln\left(1+\frac{\rho_{n-1}(z+1)}{f_{n-2}(z+1)}\right)\right)\\ l\rho_n(z)=\ln\left(-\ln\left(1+\frac{-\exp(l\rho_{n-1}(z+1))}{\exp(f_{n-1}(z))}\right)\right)\\ l\rho_n(z)=\ln\bigg(-\ln\Big(1-\exp\big( l\rho_{n-1}(z+1) - f_{n-1}(z) \big) \Big) \bigg)\\ \end{align} Next I implemented in pari-gp a routine I called loglogmexp(z) which implements the following: \begin{align} \text{loglogmexp}(y)=\ln\Big(-\ln\big(1-\exp(y)\big)\Big)\\ l\rho_n(z)=\text{loglogmexp}\big( \rho_{n-1}(z+1) - f_{n-1}(z)\big);\;\;\; y=\rho_{n-1}(z+1)-f_{n-1}(z)\\ \end{align} Now, often times $$\Re(y)$$ is large enough negative, that we can replace the inner most $$-\ln\big(1-\exp(y) \big)$$ with the approximation of: $$\exp(y)$$!!  If we are closer to the singularity then I implemented either a more exact series, or else directly implemented the exponents and logarithms.  But for n=4, for most cases this is an extremely accurate approximation.  This approximation is accurate to >=~60 decimal digits at a radius of less than 99.998% of the radius of convergence!  \begin{align} l\rho_n(z) \approx l\rho_{n-1}(z+1) - f_{n-1}(z) \\ l\rho_n(z) \approx \ln\Big(\ln\big(1+\exp(-z-n)\big)\Big)-\sum_{i=1}^{n}f_{i-1}(z+n-i)\\ \end{align} edit and update: The equation above is dominated by $$f_0(z+n-1)$$ or if centering at Tet(0), $$e\uparrow\uparrow(z+n-1)$$.  In my program, I call f(z,n), beta_tau(z,n).  You can see the individual contributions, by running "logrho_n(rr,4)" instead of logrho(rr,4).   Code:z=logrho_n(rr,4);  -5.74639913386489   log(log(1+exp(-z-4)))  -3814279.10476022  -beta_tau(z+3,0)  -15.1542622414793  -beta_tau(z+2,1)  -2.71828182845905  -beta_tau(z+1,2)  -1.00000000000000  -beta_tau(z+0,3) z=-3814303.72370342;   beta_tau.gp (Size: 8.79 KB / Downloads: 110) - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Arguments for the beta method not being Kneser's method - by JmsNxn - 06/07/2021, 10:45 PM RE: Arguments for the beta method not being Kneser's method - by MphLee - 06/10/2021, 12:41 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 06/11/2021, 02:05 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 06/20/2021, 12:36 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 07/07/2021, 11:00 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 07/08/2021, 05:11 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 09/24/2021, 04:42 PM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 07/21/2021, 07:13 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 07/22/2021, 03:47 AM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 07/22/2021, 12:21 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 07/23/2021, 04:05 PM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 09/21/2021, 07:22 PM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 09/24/2021, 04:52 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/25/2021, 03:00 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 09/28/2021, 04:41 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/29/2021, 12:33 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 09/29/2021, 11:55 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/01/2021, 01:23 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 08/17/2021, 03:12 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 09/15/2021, 08:15 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/16/2021, 01:41 AM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 09/16/2021, 10:54 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/17/2021, 03:20 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 09/16/2021, 07:23 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/17/2021, 04:23 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/17/2021, 06:00 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/22/2021, 03:11 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/23/2021, 08:42 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/01/2021, 03:50 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/01/2021, 11:38 PM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/02/2021, 11:36 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/03/2021, 05:59 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/05/2021, 03:27 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/05/2021, 02:31 PM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 10/02/2021, 01:40 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/03/2021, 04:50 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/06/2021, 10:22 PM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/07/2021, 02:29 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/07/2021, 03:18 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 10/07/2021, 07:02 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/07/2021, 05:20 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 10/07/2021, 07:11 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/07/2021, 04:12 PM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 10/09/2021, 12:27 PM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 10/09/2021, 08:02 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/11/2021, 02:11 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/11/2021, 03:51 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/11/2021, 12:48 AM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 10/12/2021, 12:23 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/13/2021, 05:01 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/19/2021, 02:43 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/23/2021, 03:13 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/20/2021, 05:13 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/21/2021, 03:33 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/22/2021, 03:54 AM

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