Using MPFR And Adding - How many Digits are Correct?

Question

I have a pretty easy question (I think). As much as I've tried, I can not find an answer to this question.

I am creating a function, for which I want the user to enter two numbers. The first is the the number of terms of a certain infinite series to add together. The second is the number of digits the user would like the truncated sum to be accurate to.

Say the terms of the sequence are a_i. How much precision n, would be required in mpfr to ensure the result of adding these a_i from i=0 up to the user's entered value would be needed to guarantee the number of digits the user needs?

By the way, I'm adding the a_i in a naive way.

Any help will be much appreciated.

Thanks,

Rick

Answer 1

You can convert between decimal digits of precision, d, and binary digits of precision, b, with logarithms

b = d × log(10) / log(2)

A little rearranging shows why

b × log(2) = d × log(10)
log(2^b) = log(10^d)
2^b = 10^d

Each term of the series (and each addition) will introduce a rounding error at the least significant digit so, assuming each of the t terms involves n (two argument) arithmetic operations, you will want to add an extra

log(t * (n+2))/log(2)

bits.

You'll need to round the number of bits of precision up to be sure that you have enough room for your decimal digits of precision

b = ceil((d*log(10.0) + log(t*(n+2)))/log(2.0));

Finally, you should be aware that the terms may introduce cancellation errors, in which case this simple calculation will dramatically underestimate the required number of bits, even assuming I've got it right in the first place ;-)