broot.c revision 1.1.1.2 
1/* mpn_broot -- Compute hensel sqrt
2
3   Contributed to the GNU project by Niels M��ller
4
5   THE FUNCTIONS IN THIS FILE ARE INTERNAL WITH MUTABLE INTERFACES.  IT IS ONLY
6   SAFE TO REACH THEM THROUGH DOCUMENTED INTERFACES.  IN FACT, IT IS ALMOST
7   GUARANTEED THAT THEY WILL CHANGE OR DISAPPEAR IN A FUTURE GMP RELEASE.
8
9Copyright 2012 Free Software Foundation, Inc.
10
11This file is part of the GNU MP Library.
12
13The GNU MP Library is free software; you can redistribute it and/or modify
14it under the terms of either:
15
16  * the GNU Lesser General Public License as published by the Free
17    Software Foundation; either version 3 of the License, or (at your
18    option) any later version.
19
20or
21
22  * the GNU General Public License as published by the Free Software
23    Foundation; either version 2 of the License, or (at your option) any
24    later version.
25
26or both in parallel, as here.
27
28The GNU MP Library is distributed in the hope that it will be useful, but
29WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
30or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
31for more details.
32
33You should have received copies of the GNU General Public License and the
34GNU Lesser General Public License along with the GNU MP Library.  If not,
35see https://www.gnu.org/licenses/.  */
36
37#include "gmp.h"
38#include "gmp-impl.h"
39
40/* Computes a^e (mod B). Uses right-to-left binary algorithm, since
41   typical use will have e small. */
42static mp_limb_t
43powlimb (mp_limb_t a, mp_limb_t e)
44{
45  mp_limb_t r = 1;
46  mp_limb_t s = a;
47
48  for (r = 1, s = a; e > 0; e >>= 1, s *= s)
49    if (e & 1)
50      r *= s;
51
52  return r;
53}
54
55/* Computes a^{1/k - 1} (mod B^n). Both a and k must be odd.
56
57   Iterates
58
59     r' <-- r - r * (a^{k-1} r^k - 1) / n
60
61   If
62
63     a^{k-1} r^k = 1 (mod 2^m),
64
65   then
66
67     a^{k-1} r'^k = 1 (mod 2^{2m}),
68
69   Compute the update term as
70
71     r' = r - (a^{k-1} r^{k+1} - r) / k
72
73   where we still have cancellation of low limbs.
74
75 */
76void
77mpn_broot_invm1 (mp_ptr rp, mp_srcptr ap, mp_size_t n, mp_limb_t k)
78{
79  mp_size_t sizes[GMP_LIMB_BITS * 2];
80  mp_ptr akm1, tp, rnp, ep;
81  mp_limb_t a0, r0, km1, kp1h, kinv;
82  mp_size_t rn;
83  unsigned i;
84
85  TMP_DECL;
86
87  ASSERT (n > 0);
88  ASSERT (ap[0] & 1);
89  ASSERT (k & 1);
90  ASSERT (k >= 3);
91
92  TMP_MARK;
93
94  akm1 = TMP_ALLOC_LIMBS (4*n);
95  tp = akm1 + n;
96
97  km1 = k-1;
98  /* FIXME: Could arrange the iteration so we don't need to compute
99     this up front, computing a^{k-1} * r^k as (a r)^{k-1} * r. Note
100     that we can use wraparound also for a*r, since the low half is
101     unchanged from the previous iteration. Or possibly mulmid. Also,
102     a r = a^{1/k}, so we get that value too, for free? */
103  mpn_powlo (akm1, ap, &km1, 1, n, tp); /* 3 n scratch space */
104
105  a0 = ap[0];
106  binvert_limb (kinv, k);
107
108  /* 4 bits: a^{1/k - 1} (mod 16):
109
110	a % 8
111	1 3 5 7
112   k%4 +-------
113     1 |1 1 1 1
114     3 |1 9 9 1
115  */
116  r0 = 1 + (((k << 2) & ((a0 << 1) ^ (a0 << 2))) & 8);
117  r0 = kinv * r0 * (k+1 - akm1[0] * powlimb (r0, k & 0x7f)); /* 8 bits */
118  r0 = kinv * r0 * (k+1 - akm1[0] * powlimb (r0, k & 0x7fff)); /* 16 bits */
119  r0 = kinv * r0 * (k+1 - akm1[0] * powlimb (r0, k)); /* 32 bits */
120#if GMP_NUMB_BITS > 32
121  {
122    unsigned prec = 32;
123    do
124      {
125	r0 = kinv * r0 * (k+1 - akm1[0] * powlimb (r0, k));
126	prec *= 2;
127      }
128    while (prec < GMP_NUMB_BITS);
129  }
130#endif
131
132  rp[0] = r0;
133  if (n == 1)
134    {
135      TMP_FREE;
136      return;
137    }
138
139  /* For odd k, (k+1)/2 = k/2+1, and the latter avoids overflow. */
140  kp1h = k/2 + 1;
141
142  /* FIXME: Special case for two limb iteration. */
143  rnp = TMP_ALLOC_LIMBS (2*n + 1);
144  ep = rnp + n;
145
146  /* FIXME: Possible to this on the fly with some bit fiddling. */
147  for (i = 0; n > 1; n = (n + 1)/2)
148    sizes[i++] = n;
149
150  rn = 1;
151
152  while (i-- > 0)
153    {
154      /* Compute x^{k+1}. */
155      mpn_sqr (ep, rp, rn); /* For odd n, writes n+1 limbs in the
156			       final iteration. */
157      mpn_powlo (rnp, ep, &kp1h, 1, sizes[i], tp);
158
159      /* Multiply by a^{k-1}. Can use wraparound; low part equals r. */
160
161      mpn_mullo_n (ep, rnp, akm1, sizes[i]);
162      ASSERT (mpn_cmp (ep, rp, rn) == 0);
163
164      ASSERT (sizes[i] <= 2*rn);
165      mpn_pi1_bdiv_q_1 (rp + rn, ep + rn, sizes[i] - rn, k, kinv, 0);
166      mpn_neg (rp + rn, rp + rn, sizes[i] - rn);
167      rn = sizes[i];
168    }
169  TMP_FREE;
170}
171
172/* Computes a^{1/k} (mod B^n). Both a and k must be odd. */
173void
174mpn_broot (mp_ptr rp, mp_srcptr ap, mp_size_t n, mp_limb_t k)
175{
176  mp_ptr tp;
177  TMP_DECL;
178
179  ASSERT (n > 0);
180  ASSERT (ap[0] & 1);
181  ASSERT (k & 1);
182
183  if (k == 1)
184    {
185      MPN_COPY (rp, ap, n);
186      return;
187    }
188
189  TMP_MARK;
190  tp = TMP_ALLOC_LIMBS (n);
191
192  mpn_broot_invm1 (tp, ap, n, k);
193  mpn_mullo_n (rp, tp, ap, n);
194
195  TMP_FREE;
196}
197