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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.
------------------------------------------------------------------
***** BEGIN LICENSE BLOCK *****
Version: MPL 1.1/GPL 2.0/LGPL 2.1
The contents of this file are subject to the Mozilla Public License Version
1.1 (the "License"); you may not use this file except in compliance with
the License. You may obtain a copy of the License at
http://www.mozilla.org/MPL/
Software distributed under the License is distributed on an "AS IS" basis,
WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
for the specific language governing rights and limitations under the
License.
The Original Code is the MPI Arbitrary Precision Integer Arithmetic
library.
The Initial Developer of the Original Code is
Michael J. Fromberger <sting@linguist.dartmouth.edu>
Portions created by the Initial Developer are Copyright (C) 1998, 2000
the Initial Developer. All Rights Reserved.
Contributor(s):
Alternatively, the contents of this file may be used under the terms of
either the GNU General Public License Version 2 or later (the "GPL"), or
the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
in which case the provisions of the GPL or the LGPL are applicable instead
of those above. If you wish to allow use of your version of this file only
under the terms of either the GPL or the LGPL, and not to allow others to
use your version of this file under the terms of the MPL, indicate your
decision by deleting the provisions above and replace them with the notice
and other provisions required by the GPL or the LGPL. If you do not delete
the provisions above, a recipient may use your version of this file under
the terms of any one of the MPL, the GPL or the LGPL.
***** END LICENSE BLOCK *****
$Id$
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