Multiplication

This describes the multiplication algorithm used by the MPI library.

This is basically a standard "schoolbook" algorithm.  It is slow --
O(mn) for m = #a, n = #b -- but easy to implement and verify.
Basically, we run two nested loops, as illustrated here (R is the
radix):

k = 0
for j <- 0 to (#b - 1)
  for i <- 0 to (#a - 1)
    w = (a[j] * b[i]) + k + c[i+j]
    c[i+j] = w mod R
    k = w div R
  endfor
  c[i+j] = k;
  k = 0;
endfor

It is necessary that 'w' have room for at least two radix R digits.
The product of any two digits in radix R is at most:

	(R - 1)(R - 1) = R^2 - 2R + 1

Since a two-digit radix-R number can hold R^2 - 1 distinct values,
this insures that the product will fit into the two-digit register.

To insure that two digits is enough for w, we must also show that
there is room for the carry-in from the previous multiplication, and
the current value of the product digit that is being recomputed.
Assuming each of these may be as big as R - 1 (and no larger,
certainly), two digits will be enough if and only if:

	(R^2 - 2R + 1) + 2(R - 1) <= R^2 - 1

Solving this equation shows that, indeed, this is the case:

	R^2 - 2R + 1 + 2R - 2 <= R^2 - 1

	R^2 - 1 <= R^2 - 1

This suggests that a good radix would be one more than the largest
value that can be held in half a machine word -- so, for example, as
in this implementation, where we used a radix of 65536 on a machine
with 4-byte words.  Another advantage of a radix of this sort is that
binary-level operations are easy on numbers in this representation.

Here's an example multiplication worked out longhand in radix-10,
using the above algorithm:

   a =     999
   b =   x 999
  -------------
   p =   98001

w = (a[jx] * b[ix]) + kin + c[ix + jx]
c[ix+jx] = w % RADIX
k = w / RADIX
                                                               product
ix	jx	a[jx]	b[ix]	kin	w	c[i+j]	kout	000000
0	0	9	9	0	81+0+0	1	8	000001
0	1	9	9	8	81+8+0	9	8	000091
0	2	9	9	8	81+8+0	9	8	000991
				8			0	008991
1	0	9	9	0	81+0+9	0	9	008901
1	1	9	9	9	81+9+9	9	9	008901
1	2	9	9	9	81+9+8	8	9	008901
				9			0	098901
2	0	9	9	0	81+0+9	0	9	098001
2	1	9	9	9	81+9+8	8	9	098001
2	2	9	9	9	81+9+9	9	9	098001

------------------------------------------------------------------
***** 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: mul.txt,v 1.2 2005/02/02 22:28:22 gerv%gerv.net Exp $


