1 /* mpz_lucnum_ui -- calculate Lucas number.
3 Copyright 2001, 2003, 2005 Free Software Foundation, Inc.
5 This file is part of the GNU MP Library.
7 The GNU MP Library is free software; you can redistribute it and/or modify
8 it under the terms of the GNU Lesser General Public License as published by
9 the Free Software Foundation; either version 3 of the License, or (at your
10 option) any later version.
12 The GNU MP Library is distributed in the hope that it will be useful, but
13 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
14 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
15 License for more details.
17 You should have received a copy of the GNU Lesser General Public License
18 along with the GNU MP Library. If not, see http://www.gnu.org/licenses/. */
25 /* change this to "#define TRACE(x) x" for diagnostics */
31 For the +4 in L[2k+1] when k is even, all L[4m+3] == 4, 5 or 7 mod 8, so
32 there can't be an overflow applying +4 to just the low limb (since that
33 would leave 0, 1, 2 or 3 mod 8).
35 For the -4 in L[2k+1] when k is even, it seems (no proof) that
36 L[3*2^(b-2)-3] == -4 mod 2^b, so for instance with a 32-bit limb
37 L[0xBFFFFFFD] == 0xFFFFFFFC mod 2^32, and this implies a borrow from the
38 low limb. Obviously L[0xBFFFFFFD] is a huge number, but it's at least
39 conceivable to calculate it, so it probably should be handled.
41 For the -2 in L[2k] with k even, it seems (no proof) L[2^(b-1)] == -1 mod
42 2^b, so for instance in 32-bits L[0x80000000] has a low limb of
43 0xFFFFFFFF so there would have been a borrow. Again L[0x80000000] is
44 obviously huge, but probably should be made to work. */
47 mpz_lucnum_ui (mpz_ptr ln, unsigned long n)
49 mp_size_t lalloc, xalloc, lsize, xsize;
55 TRACE (printf ("mpn_lucnum_ui n=%lu\n", n));
57 if (n <= FIB_TABLE_LUCNUM_LIMIT)
59 /* L[n] = F[n] + 2F[n-1] */
60 PTR(ln)[0] = FIB_TABLE(n) + 2 * FIB_TABLE ((int) n - 1);
65 /* +1 since L[n]=F[n]+2F[n-1] might be 1 limb bigger than F[n], further +1
66 since square or mul used below might need an extra limb over the true
68 lalloc = MPN_FIB2_SIZE (n) + 2;
69 MPZ_REALLOC (ln, lalloc);
74 xp = TMP_ALLOC_LIMBS (xalloc);
76 /* Strip trailing zeros from n, until either an odd number is reached
77 where the L[2k+1] formula can be used, or until n fits within the
78 FIB_TABLE data. The table is preferred of course. */
84 /* L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k */
86 mp_size_t yalloc, ysize;
89 TRACE (printf (" initial odd n=%lu\n", n));
91 yalloc = MPN_FIB2_SIZE (n/2);
92 yp = TMP_ALLOC_LIMBS (yalloc);
93 ASSERT (xalloc >= yalloc);
95 xsize = mpn_fib2_ui (xp, yp, n/2);
97 /* possible high zero on F[k-1] */
99 ysize -= (yp[ysize-1] == 0);
100 ASSERT (yp[ysize-1] != 0);
102 /* xp = 2*F[k] + F[k-1] */
103 #if HAVE_NATIVE_mpn_addlsh1_n
104 c = mpn_addlsh1_n (xp, yp, xp, xsize);
106 c = mpn_lshift (xp, xp, xsize, 1);
107 c += mpn_add_n (xp, xp, yp, xsize);
109 ASSERT (xalloc >= xsize+1);
112 ASSERT (xp[xsize-1] != 0);
114 ASSERT (lalloc >= xsize + ysize);
115 c = mpn_mul (lp, xp, xsize, yp, ysize);
116 lsize = xsize + ysize;
120 #if HAVE_NATIVE_mpn_addlshift
121 c = mpn_addlshift (lp, lp, lsize, 2);
123 c = mpn_lshift (xp, lp, lsize, 2);
124 c += mpn_add_n (lp, lp, xp, lsize);
126 ASSERT (lalloc >= lsize+1);
130 /* lp = lp - 4*(-1)^k */
133 /* no overflow, see comments above */
134 ASSERT (lp[0] <= MP_LIMB_T_MAX-4);
139 /* won't go negative */
140 MPN_DECR_U (lp, lsize, CNST_LIMB(4));
143 TRACE (mpn_trace (" l",lp, lsize));
147 MP_PTR_SWAP (xp, lp); /* balance the swaps wanted in the L[2k] below */
151 if (n <= FIB_TABLE_LUCNUM_LIMIT)
153 /* L[n] = F[n] + 2F[n-1] */
154 lp[0] = FIB_TABLE (n) + 2 * FIB_TABLE ((int) n - 1);
157 TRACE (printf (" initial small n=%lu\n", n);
158 mpn_trace (" l",lp, lsize));
163 for ( ; zeros != 0; zeros--)
165 /* L[2k] = L[k]^2 + 2*(-1)^k */
167 TRACE (printf (" zeros=%d\n", zeros));
169 ASSERT (xalloc >= 2*lsize);
170 mpn_sqr (xp, lp, lsize);
172 lsize -= (xp[lsize-1] == 0);
174 /* First time around the loop k==n determines (-1)^k, after that k is
175 always even and we set n=0 to indicate that. */
178 /* L[n]^2 == 0 or 1 mod 4, like all squares, so +2 gives no carry */
179 ASSERT (xp[0] <= MP_LIMB_T_MAX-2);
185 /* won't go negative */
186 MPN_DECR_U (xp, lsize, CNST_LIMB(2));
189 MP_PTR_SWAP (xp, lp);
190 ASSERT (lp[lsize-1] != 0);
193 /* should end up in the right spot after all the xp/lp swaps */
194 ASSERT (lp == PTR(ln));