Divide-and-correct methods for multiple precision division

Abstract

A division problem is defined and notation to relate it to the problem of multiple precision operation in a digital computer is introduced. A basic divide-and-correct method for multiple precision division is formulated and its known properties briefly reviewed. Of particular interest is the fact that the method produces at each step a set of precisely three estimates for the desired result, one of which is exact.

A multiple-precision division algorithm

“…It is more difficult to skip normalization during division, because the quotient digit q used in each "multiply and subtract" step is a guess. Early routines ( [3,4,5,6,8]) used integer arithmetic to estimate q. Bailey used double precision to estimate q, which made the probability of an incorrect q much smaller. Brent used Newton iteration and multiplication to avoid the long-division algorithm, but this made division much slower than multiplication.…”

Selected topics in computer generation of pseudorandom numbers

“…In other words, for B > 6 and b^^ sufficiently, large, at most two corrections of the trial quotient digit may be necessary. Stein (1964), having noted that overestimation in the basic method is the general case, has tried to make improvements by increasing the divisor by one. However, the modified method has not increased the efficiency.…”

Selected topics in computer generation of pseudorandom numbers

“…In other words, for B > 6 and b^^ sufficiently, large, at most two corrections of the trial quotient digit may be necessary. Stein (1964), having noted that overestimation in the basic method is the general case, has tried to make improvements by increasing the divisor by one. However, the modified method has not increased the efficiency.…”