mpir/mpn/generic/hgcd.c

/* hgcd.c.

   THE FUNCTIONS IN THIS FILE ARE INTERNAL WITH MUTABLE INTERFACES.  IT IS ONLY
   SAFE TO REACH THEM THROUGH DOCUMENTED INTERFACES.  IN FACT, IT IS ALMOST
   GUARANTEED THAT THEY'LL CHANGE OR DISAPPEAR IN A FUTURE GNU MP RELEASE.

Copyright 2003, 2004, 2005, 2008, 2011, 2012 Free Software Foundation, Inc.

This file is part of the GNU MP Library.

The GNU MP Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.

The GNU MP Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
License for more details.

You should have received a copy of the GNU Lesser General Public License
along with the GNU MP Library.  If not, see http://www.gnu.org/licenses/.  */

#include "mpir.h"
#include "gmp-impl.h"
#include "longlong.h"


/* Size analysis for hgcd:

   For the recursive calls, we have n1 <= ceil(n / 2). Then the
   storage need is determined by the storage for the recursive call
   computing M1, and hgcd_matrix_adjust and hgcd_matrix_mul calls that use M1
   (after this, the storage needed for M1 can be recycled).

   Let S(r) denote the required storage. For M1 we need 4 * (ceil(n1/2) + 1)
   = 4 * (ceil(n/4) + 1), for the hgcd_matrix_adjust call, we need n + 2,
   and for the hgcd_matrix_mul, we may need 3 ceil(n/2) + 8. In total,
   4 * ceil(n/4) + 3 ceil(n/2) + 12 <= 10 ceil(n/4) + 12.

   For the recursive call, we need S(n1) = S(ceil(n/2)).

   S(n) <= 10*ceil(n/4) + 12 + S(ceil(n/2))
	<= 10*(ceil(n/4) + ... + ceil(n/2^(1+k))) + 12k + S(ceil(n/2^k))
	<= 10*(2 ceil(n/4) + k) + 12k + S(ceil(n/2^k))
	<= 20 ceil(n/4) + 22k + S(ceil(n/2^k))
*/

mp_size_t
mpn_hgcd_itch (mp_size_t n)
{
  unsigned k;
  int count;
  mp_size_t nscaled;

  if (BELOW_THRESHOLD (n, HGCD_THRESHOLD))
    return n;

  /* Get the recursion depth. */
  nscaled = (n - 1) / (HGCD_THRESHOLD - 1);
  count_leading_zeros (count, nscaled);
  k = GMP_LIMB_BITS - count;

  return 20 * ((n+3) / 4) + 22 * k + HGCD_THRESHOLD;
}

/* Reduces a,b until |a-b| fits in n/2 + 1 limbs. Constructs matrix M
   with elements of size at most (n+1)/2 - 1. Returns new size of a,
   b, or zero if no reduction is possible. */

mp_size_t
mpn_hgcd (mp_ptr ap, mp_ptr bp, mp_size_t n,
	  struct hgcd_matrix *M, mp_ptr tp)
{
  mp_size_t s = n/2 + 1;

  mp_size_t nn;
  int success = 0;

  if (n <= s)
    /* Happens when n <= 2, a fairly uninteresting case but exercised
       by the random inputs of the testsuite. */
    return 0;

  ASSERT ((ap[n-1] | bp[n-1]) > 0);

  ASSERT ((n+1)/2 - 1 < M->alloc);

  if (ABOVE_THRESHOLD (n, HGCD_THRESHOLD))
    {
      mp_size_t n2 = (3*n)/4 + 1;
      mp_size_t p = n/2;

      nn = mpn_hgcd_reduce (M, ap, bp, n, p, tp);
      if (nn)
	{
	  n = nn;
	  success = 1;
	}

      /* NOTE: It apppears this loop never runs more than once (at
	 least when not recursing to hgcd_appr). */
      while (n > n2)
	{
	  /* Needs n + 1 storage */
	  nn = mpn_hgcd_step (n, ap, bp, s, M, tp);
	  if (!nn)
	    return success ? n : 0;

	  n = nn;
	  success = 1;
	}

      if (n > s + 2)
	{
	  struct hgcd_matrix M1;
	  mp_size_t scratch;

	  p = 2*s - n + 1;
	  scratch = MPN_HGCD_MATRIX_INIT_ITCH (n-p);

	  mpn_hgcd_matrix_init(&M1, n - p, tp);

	  /* FIXME: Should use hgcd_reduce, but that may require more
	     scratch space, which requires review. */

	  nn = mpn_hgcd (ap + p, bp + p, n - p, &M1, tp + scratch);
	  if (nn > 0)
	    {
	      /* We always have max(M) > 2^{-(GMP_NUMB_BITS + 1)} max(M1) */
	      ASSERT (M->n + 2 >= M1.n);

	      /* Furthermore, assume M ends with a quotient (1, q; 0, 1),
		 then either q or q + 1 is a correct quotient, and M1 will
		 start with either (1, 0; 1, 1) or (2, 1; 1, 1). This
		 rules out the case that the size of M * M1 is much
		 smaller than the expected M->n + M1->n. */

	      ASSERT (M->n + M1.n < M->alloc);

	      /* Needs 2 (p + M->n) <= 2 (2*s - n2 + 1 + n2 - s - 1)
		 = 2*s <= 2*(floor(n/2) + 1) <= n + 2. */
	      n = mpn_hgcd_matrix_adjust (&M1, p + nn, ap, bp, p, tp + scratch);

	      /* We need a bound for of M->n + M1.n. Let n be the original
		 input size. Then

		 ceil(n/2) - 1 >= size of product >= M.n + M1.n - 2

		 and it follows that

		 M.n + M1.n <= ceil(n/2) + 1

		 Then 3*(M.n + M1.n) + 5 <= 3 * ceil(n/2) + 8 is the
		 amount of needed scratch space. */
	      mpn_hgcd_matrix_mul (M, &M1, tp + scratch);
	      success = 1;
	    }
	}
    }

  for (;;)
    {
      /* Needs s+3 < n */
      nn = mpn_hgcd_step (n, ap, bp, s, M, tp);
      if (!nn)
	return success ? n : 0;

      n = nn;
      success = 1;
    }
}