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Division
This describes the division algorithm used by the MPI library.
Input: a, b; a > b
Compute: Q, R; a = Qb + R
The input numbers are normalized so that the high-order digit of b is
at least half the radix. This guarantees that we have a reasonable
way to guess at the digits of the quotient (this method was taken from
Knuth, vol. 2, with adaptations).
To normalize, test the high-order digit of b. If it is less than half
the radix, multiply both a and b by d, where:
radix - 1
d = -----------
bmax + 1
...where bmax is the high-order digit of b. Otherwise, set d = 1.
Given normalize values for a and b, let the notation a[n] denote the
nth digit of a. Let #a be the number of significant figures of a (not
including any leading zeroes).
Let R = 0
Let p = #a - 1
while(p >= 0)
do
R = (R * radix) + a[p]
p = p - 1
while(R < b and p >= 0)
if(R < b)
break
q = (R[#R - 1] * radix) + R[#R - 2]
q = q / b[#b - 1]
T = b * q
while(T > L)
q = q - 1
T = T - b
endwhile
L = L - T
Q = (Q * radix) + q
endwhile
At this point, Q is the quotient, and R is the normalized remainder.
To denormalize R, compute:
R = (R / d)
At this point, you are finished.
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