In mathematics, binary splitting is a technique for speeding up numerical evaluation of many types of series with rational terms. In particular, it can be used to evaluate hypergeometric series at rational points.
Given a series
where pn and qn are integers, the goal of binary splitting is to compute integers P(a, b) and Q(a, b) such that
The splitting consists of setting m = [(a + b)/2] and recursively computing P(a, b) and Q(a, b) from P(a, m), P(m, b), Q(a, m), and Q(m, b). When a and b are sufficiently close, P(a, b) and Q(a, b) can be computed directly from pa...pb and qa...qb.
Comparison with other methods
Binary splitting requires more memory than direct term-by-term summation, but is asymptotically faster since the sizes of all occurring subproducts are reduced. Additionally, whereas the most naive evaluation scheme for a rational series uses a full-precision division for each term in the series, binary splitting requires only one final division at the target precision; this is not only faster, but conveniently eliminates rounding errors. To take full advantage of the scheme, fast multiplication algorithms such as Toom–Cook and Schönhage–Strassen must be used; with ordinary O(n2) multiplication, binary splitting may render no speedup at all or be slower.
In a less specific sense, binary splitting may also refer to any divide and conquer algorithm that always divides the problem in two halves.
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