 mersenneforum.org > Math Inverse of Smoothness Probability
 Register FAQ Search Today's Posts Mark Forums Read 2017-07-24, 12:24 #1 paul0   Sep 2011 3×19 Posts Inverse of Smoothness Probability The probability that a random number below X is B-smooth is given by u-u, where u=ln(X)/ln(B). However, I would like the do the inverse, that is, given the smoothness probability and B, how do I solve for X? I have a solution via Newton method. Is there a closed form?   2017-07-24, 19:53   #2
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

100100110010012 Posts Quote:
 Originally Posted by paul0 The estimate of the probability that a random number below X is B-smooth is given by u-u, where u=ln(X)/ln(B).
No need for Newton method, because it will immediately exceed the precision of the estimate that you started with. There are better estimates (and then see Dickman's and/or Buchstab's).   2017-07-25, 09:10   #3
paul0

Sep 2011

3×19 Posts Quote:
 Originally Posted by Batalov No need for Newton method, because it will immediately exceed the precision of the estimate that you started with. There are better estimates (and then see Dickman's and/or Buchstab's).
I think you misunderstood. Instead of solving the probability of smoothness, I want to solve for X given B and the probability.   2017-07-25, 10:07   #4
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3×29×83 Posts Quote:
 Originally Posted by paul0 I think you misunderstood. Instead of solving the probability of smoothness, I want to solve for X given B and the probability.
He's saying that going in reverse on that equation (p = some_function_of(X, B), as you're trying to do, isn't very useful because it's not an equation -- only an approximation, and not a very good one. Better to write p ~ some_function_of(X, B), and then note that the "~" is vague enough that undoing "some_function_of" isn't worth the effort.

The links he provides are sources for p ≈ some_better_function_of(X, B). Find those better functions before trying to invert them.   2017-07-25, 10:21   #5
paul0

Sep 2011

5710 Posts Quote:
 Originally Posted by Dubslow He's saying that going in reverse on that equation (p = some_function_of(X, B), as you're trying to do, isn't very useful because it's not an equation -- only an approximation, and not a very good one. Better to write p ~ some_function_of(X, B), and then note that the "~" is vague enough that undoing "some_function_of" isn't worth the effort. The links he provides are sources for p ≈ some_better_function_of(X, B). Find those better functions before trying to invert them.
I stand corrected. Thanks for clarifying :)   2017-07-25, 16:33   #6
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

24C916 Posts Incidentally, I have just heard from much more knowledgeable people, and I will simply quote:

Quote:
 Originally Posted by Robert Note that the question (u-u) can be answered using the Lambert W function. But, of course, one still must compute the answer numerically via series/Pade approximant, etc. Note that the W function is not real-analytic. Bob   2017-07-25, 16:41 #7 Batalov   "Serge" Mar 2008 Phi(4,2^7658614+1)/2 941710 Posts And for the Inverse of Smoothness Probability question, you want a reasonable numerical estimate the inverse of Dickman function. So you want to take Newton method on the inverse of Dickman ρ - because if you know the derivative of Dickman ρ by definition, then you know the derivative of the inverse of Dickman ρ.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post wreck NFS@Home 1 2016-05-08 15:44 Sam Kennedy Programming 18 2013-01-21 14:23 only_human Miscellaneous Math 26 2012-08-10 02:47 Raman Math 5 2011-04-13 23:29 Washuu Math 12 2005-06-27 12:19

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