Michael Grottke <Michael.Grottke <at> duke.edu> writes:
: I am currently using R for fitting a model to various data sets : (minimizing the negative log-likelihood) and calculating a number of : metrics based on the parameter estimates. Within these calculations, I : have steps of the form
: : log(log(1+x)),
: : where x can be very small (e.g., around 1e-16). Unfortunately, the : precision of doubles does not suffice for coping with the difference in : the orders of magnitude of 1 and x: 1+x is rounded to 1.
: : One way for solving this problem seems to be to use an arbitrary : precision library implemented in C and call the respective routines for : calculating the logarithm(s) from within R.
: : My questions are as follows:
: 1. Is there any better/more direct way to solve the problem?
: 2. Is there any arbitrary precision library you can suggest in particular?
:
The approximation log(1+x) = x would be accuate to several decimal
places in your case so your expression would reduce to log(log(1+x)) = log(x).
Another possibility is to use the log1p function to evaluate log1p(x) = log(1+x). The two expressions give similar answers in this case
> options(digits=12)
> log(log1p(1e-16))
[1] -36.8413614879
> log(1e-16)
[1] -36.8413614879
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