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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.
------------------------------------------------------------------
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 Michael J. Fromberger are
Copyright (C) 1998, 2000 Michael J. Fromberger. All Rights Reserved.
Contributor(s):
Alternatively, the contents of this file may be used under the
terms of the GNU General Public License Version 2 or later (the
"GPL"), in which case the provisions of the GPL are applicable
instead of those above. If you wish to allow use of your
version of this file only under the terms of the GPL and not to
allow others to use your version of this file under the MPL,
indicate your decision by deleting the provisions above and
replace them with the notice and other provisions required by
the GPL. If you do not delete the provisions above, a recipient
may use your version of this file under either the MPL or the GPL.
$Id$
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