problem with the precision of numbers
<Ted.Harding <at> manchester.ac.uk> writes:
[...] I suspect this is an invented computation -- the "3456" strikes me as "unlikely" (it reminds me of my habitual illustrative use of set.seed(54321)). There is a definite problem with the development given by kayj. When k=2000 and i=k, the formula requires evaluation of 3456*(2^3000) on a log10 scale this is log10(3456) + 3000*log10(2) = 906.6286 Since R "gives up" at 10^308.25471 = 1.79767e+308 (10^308.25472 => Inf), this algorithm is going to be tricky to evaluate!
Just to finish this off: 'Rmpfr' works fine with this example case and
computes the total sum in less than 2 seconds:
library(Rmpfr)
j <- mpfr(-1, 120)
i <- mpfr(0:2000, 120)
s <- sum(j^i * 3456 * (1+i*1/2000)^3000)
s
# 1 'mpfr' number of precision 120 bits
# [1] 2.8876826480125324342665158348085465188e906
which is the same result that Maple returns. But Maple (I admit, mine is
quite old) and 'bc' (couldn't wait for the answer) are much slower at it.
It is said that 'gmp' (on which Rmpfr is based) is faster than any other
multi-precison library "on earth" --- seems to be true.
Hans Werner
I don't know how well Rmpfr copes with very large numbers (the
available documentation seems cryptic). However, I can go along
with the recommendation in the URL the Ben gives, to use 'bc'
("Berkeley Calculator"), available on unix[oid] systems since
a long time ago. That is an old friend of mine, and works well
(it can cope with exponents up to X^2147483647 in the version
I have). It can eat for breakfast the task of checking whether
Kate Bush can accurately sing pi to 117 significant figures:
http://www.absolutelyrics.com/lyrics/view/kate_bush/pi
(Try it in R).
Ted.
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Date: 19-Jan-10 Time: 18:41:27
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