c++ - Efficiently summing log quantities -

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working in c++, i'd find sum of quantities, , take log of sum:

log(a_1 + a_2 + a_3 + ... + a_n)

however, not have quantities themselves, have log'd values:

l_1 = log(a_1), l_2 = log(a_2), ... , l_n = log(a_n)

is there efficient way log sum of a_i's? i'd avoid

log(s) = log(exp(l_1) + exp(l_2) + ... + exp(l_n))

if possible - exp becomes bottleneck calculation done many times.

how large n?

this quantity known log-sum-exp , lieven vandenberghe talks on page 72 of book. has optimization package uses operation, , brief look, seems if doesn't special there, exponentiates , adds. perhaps exponentiation not serious bottleneck when n small enough vector fit memory.

this operation comes in modeling , bottleneck there sheer number of terms. magnitude of n=2^100 common, terms implicitly represented. in such cases, there various tricks approximating quantity relying on convexity of log-sum-exp. simplest trick -- approximate log(s) max(l1,l2,....,ln)


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