mpir/mpz/lucnum_ui.c

/* mpz_lucnum_ui -- calculate Lucas number.

Copyright 2001, 2003, 2005, 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 <stdio.h>
#include "mpir.h"
#include "gmp-impl.h"


/* change this to "#define TRACE(x) x" for diagnostics */
#define TRACE(x)


/* Notes:

   For the +4 in L[2k+1] when k is even, all L[4m+3] == 4, 5 or 7 mod 8, so
   there can't be an overflow applying +4 to just the low limb (since that
   would leave 0, 1, 2 or 3 mod 8).

   For the -4 in L[2k+1] when k is even, it seems (no proof) that
   L[3*2^(b-2)-3] == -4 mod 2^b, so for instance with a 32-bit limb
   L[0xBFFFFFFD] == 0xFFFFFFFC mod 2^32, and this implies a borrow from the
   low limb.  Obviously L[0xBFFFFFFD] is a huge number, but it's at least
   conceivable to calculate it, so it probably should be handled.

   For the -2 in L[2k] with k even, it seems (no proof) L[2^(b-1)] == -1 mod
   2^b, so for instance in 32-bits L[0x80000000] has a low limb of
   0xFFFFFFFF so there would have been a borrow.  Again L[0x80000000] is
   obviously huge, but probably should be made to work.  */

void
mpz_lucnum_ui (mpz_ptr ln, mpir_ui n)
{
  mp_size_t  lalloc, xalloc, lsize, xsize;
  mp_ptr     lp, xp;
  mp_limb_t  c;
  int        zeros;
  TMP_DECL;

  TRACE (printf ("mpn_lucnum_ui n=%lu\n", n));

  if (n <= FIB_TABLE_LUCNUM_LIMIT)
    {
      /* L[n] = F[n] + 2F[n-1] */
      PTR(ln)[0] = FIB_TABLE(n) + 2 * FIB_TABLE ((int) n - 1);
      SIZ(ln) = 1;
      return;
    }

  /* +1 since L[n]=F[n]+2F[n-1] might be 1 limb bigger than F[n], further +1
     since square or mul used below might need an extra limb over the true
     size */
  lalloc = MPN_FIB2_SIZE (n) + 2;
  lp = MPZ_REALLOC (ln, lalloc);

  TMP_MARK;
  xalloc = lalloc;
  xp = TMP_ALLOC_LIMBS (xalloc);

  /* Strip trailing zeros from n, until either an odd number is reached
     where the L[2k+1] formula can be used, or until n fits within the
     FIB_TABLE data.  The table is preferred of course.  */
  zeros = 0;
  for (;;)
    {
      if (n & 1)
	{
	  /* L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k */

	  mp_size_t  yalloc, ysize;
	  mp_ptr     yp;

	  TRACE (printf ("  initial odd n=%lu\n", n));

	  yalloc = MPN_FIB2_SIZE (n/2);
	  yp = TMP_ALLOC_LIMBS (yalloc);
	  ASSERT (xalloc >= yalloc);

	  xsize = mpn_fib2_ui (xp, yp, n/2);

	  /* possible high zero on F[k-1] */
	  ysize = xsize;
	  ysize -= (yp[ysize-1] == 0);
	  ASSERT (yp[ysize-1] != 0);

	  /* xp = 2*F[k] + F[k-1] */
#if HAVE_NATIVE_mpn_addlsh1_n
	  c = mpn_addlsh1_n (xp, yp, xp, xsize);
#else
	  c = mpn_lshift (xp, xp, xsize, 1);
	  c += mpn_add_n (xp, xp, yp, xsize);
#endif
	  ASSERT (xalloc >= xsize+1);
	  xp[xsize] = c;
	  xsize += (c != 0);
	  ASSERT (xp[xsize-1] != 0);

	  ASSERT (lalloc >= xsize + ysize);
	  c = mpn_mul (lp, xp, xsize, yp, ysize);
	  lsize = xsize + ysize;
	  lsize -= (c == 0);

	  /* lp = 5*lp */
#if HAVE_NATIVE_mpn_addlsh2_n
	  c = mpn_addlsh2_n (lp, lp, lp, lsize);
#else
	  /* FIXME: Is this faster than mpn_mul_1 ? */
	  c = mpn_lshift (xp, lp, lsize, 2);
	  c += mpn_add_n (lp, lp, xp, lsize);
#endif
	  ASSERT (lalloc >= lsize+1);
	  lp[lsize] = c;
	  lsize += (c != 0);

	  /* lp = lp - 4*(-1)^k */
	  if (n & 2)
	    {
	      /* no overflow, see comments above */
	      ASSERT (lp[0] <= MP_LIMB_T_MAX-4);
	      lp[0] += 4;
	    }
	  else
	    {
	      /* won't go negative */
	      MPN_DECR_U (lp, lsize, CNST_LIMB(4));
	    }

	  TRACE (mpn_trace ("  l",lp, lsize));
	  break;
	}

      MP_PTR_SWAP (xp, lp); /* balance the swaps wanted in the L[2k] below */
      zeros++;
      n /= 2;

      if (n <= FIB_TABLE_LUCNUM_LIMIT)
	{
	  /* L[n] = F[n] + 2F[n-1] */
	  lp[0] = FIB_TABLE (n) + 2 * FIB_TABLE ((int) n - 1);
	  lsize = 1;

	  TRACE (printf ("  initial small n=%lu\n", n);
		 mpn_trace ("  l",lp, lsize));
	  break;
	}
    }

  for ( ; zeros != 0; zeros--)
    {
      /* L[2k] = L[k]^2 + 2*(-1)^k */

      TRACE (printf ("  zeros=%d\n", zeros));

      ASSERT (xalloc >= 2*lsize);
      mpn_sqr (xp, lp, lsize);
      lsize *= 2;
      lsize -= (xp[lsize-1] == 0);

      /* First time around the loop k==n determines (-1)^k, after that k is
	 always even and we set n=0 to indicate that.  */
      if (n & 1)
	{
	  /* L[n]^2 == 0 or 1 mod 4, like all squares, so +2 gives no carry */
	  ASSERT (xp[0] <= MP_LIMB_T_MAX-2);
	  xp[0] += 2;
	  n = 0;
	}
      else
	{
	  /* won't go negative */
	  MPN_DECR_U (xp, lsize, CNST_LIMB(2));
	}

      MP_PTR_SWAP (xp, lp);
      ASSERT (lp[lsize-1] != 0);
    }

  /* should end up in the right spot after all the xp/lp swaps */
  ASSERT (lp == PTR(ln));
  SIZ(ln) = lsize;

  TMP_FREE;
}