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 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

2×37×127 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

11100001101012 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

3·19 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

2·37·127 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 2·37·127 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.

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