This article is about algorithms for division of integers. For the pencil-and-paper algorithm, see Long division. For the division algorithm for polynomials, see Polynomial long division.
A division algorithm is an algorithm which, given two integers N and D (respectively the numerator and the denominator), computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software.
Division algorithms fall into two main categories: slow division and fast division. Slow division algorithms produce one digit of the final quotient per iteration. Examples of slow division include restoring, non-performing restoring, non-restoring, and SRT division. Fast division methods start with a close approximation to the final quotient and produce twice as many digits of the final quotient on each iteration.[1] Newton–Raphson and Goldschmidt algorithms fall into this category.
Variants of these algorithms allow using fast multiplication algorithms. It results that, for large integers, the computer time needed for a division is the same, up to a constant factor, as the time needed for a multiplication, whichever multiplication algorithm is used.
Discussion will refer to the form , where
N = numerator (dividend)
D = denominator (divisor)
is the input, and
Q = quotient
R = remainder
is the output.
^Rodeheffer, Thomas L. (2008-08-26). Software Integer Division(PDF) (Technical report). Microsoft Research, Silicon Valley.
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