mpir/mpz/lucnum_ui.c

/* mpz_lucnum_ui -- calculate Lucas number.

Copyright 2001, 2003, 2005 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 2.1 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; see the file COPYING.LIB.  If not, write to
the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
MA 02110-1301, USA. */

#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, unsigned long 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;
  MPZ_REALLOC (ln, lalloc);
  lp = PTR (ln);

  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_lshift1 (xp, xp, xsize);
          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_addlshift
          c = mpn_addlshift (lp, lp, lsize, 2);
#else
          c = mpn_lshift2 (xp, lp, lsize);
          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_n (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;
}