Wikipedia:Reference desk/Archives/Mathematics/2013 February 15

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February 15

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How do you solve   for n?

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How do you solve   for  , given   and  ? The parameter values of interest are of the order   and  . Can someone give a good approximation formula that can be evaluated without overflows or gross loss of precision during computation? Thanks. --173.49.13.216 (talk) 03:56, 15 February 2013 (UTC)[reply]

I'm not a specialist of this kind of thing, but this can be rewritten (exactly) as

 

 

In approximate terms, we can write this as

 

 

 

In linearizing the logarithms, the error in the terms on the left is at most about   times their value. Given the approximate value of n, the error on the left is, in proportion, about  . In other words, in using the formula   you commit an error of at most about   times the vaue of n. (In fact, a more careful calculation shows that the error is about 2/3 of this, and that it results in an underestimation of the minimum possible n.) If you replace r - 1 with r, this rises to about 1/r, or 10-9. If you replace   with  , the error rises to about   times the value of n. 64.140.122.50 (talk) 05:58, 15 February 2013 (UTC)[reply]

I forgot to mention this - I'm using natural logarithms here. 64.140.122.50 (talk) 06:08, 15 February 2013 (UTC)[reply]
Thanks. Your formula is more than good enough. --173.49.13.216 (talk) 12:15, 15 February 2013 (UTC)[reply]