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- W1596559244 abstract "In this paper we give an algorithm for finding the mth base-b digit of a positive integer n (m = 1 is the least significant digit) defined as the final number in a sequence of integers gotten by multiplying, adding, and subtracting previous numbers in the sequence (actually, the algorithm finds arbitrarily precise approximations to n/b m (mod 1), which can be used to get this mth digit whenever the lower m − 1 digits do not begin with a long run of the digit b − 1). In many cases, this algorithm will require far less memory than it takes to write down the base-b digits of n, and will have a running time (in bit operations) only slightly worse than linear in the number of these base-b digits. One easy-to-state consequence of the above result is that the mth base-10 digit of 2 t can be found using O(t 2/3 log C t) bits of memory and O(tlog C t) bit operations, where C > 0 is constant. Of course, if m = O(t 2/3 ) then one can do much better (by just computing n (mod b m )), so the result is only non-trivial when t = O(m 3/2 ). The algorithm we give is highly parallelizable, although if one uses M processors, to get an M-fold reduction in running time, the memory requirements will increase by a factor of M." @default.
- W1596559244 created "2016-06-24" @default.
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- W1596559244 date "2003-02-25" @default.
- W1596559244 modified "2023-09-27" @default.
- W1596559244 title "Memory Efficient Arithmetic" @default.
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