1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
/*
Copyright (C) 2023 Jean Kieffer
Copyright (C) 2026 Edgar Costa
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "fmpz.h"
#include "fmpz_vec.h"
#include "fmpz_mat.h"
/*
Compute the Smith normal form S of A and unimodular matrices U, V
such that S = U * A * V.
S is m x n, U is m x m (or NULL), V is n x n (or NULL).
Algorithm: iterative Hermite normal form.
Phase 1: alternately compute row-HNF and column-HNF (via transpose)
until X is diagonal, tracking transforms in U and V.
Phase 2: fix divisibility chain on the diagonal using xgcd,
with O(m+n) row/column operations per pair instead of
O((m+n)^3) matrix multiplications.
Based on the implementation by Jean Kieffer in acb_theta, generalized
to non-square matrices.
*/
void
fmpz_mat_snf_transform(fmpz_mat_t S, fmpz_mat_t U, fmpz_mat_t V,
const fmpz_mat_t A)
{
slong m = fmpz_mat_nrows(A);
slong n = fmpz_mat_ncols(A);
slong d = FLINT_MIN(m, n);
slong max_iters, iter;
fmpz_mat_t X, Xt, Mr, Mc;
slong j, k, i;
if (d == 0)
{
fmpz_mat_zero(S);
if (U != NULL)
fmpz_mat_one(U);
if (V != NULL)
fmpz_mat_one(V);
return;
}
if (U == NULL && V == NULL)
{
fmpz_mat_snf(S, A);
return;
}
fmpz_mat_init(X, m, n);
fmpz_mat_init(Xt, n, m);
fmpz_mat_init(Mr, m, m);
fmpz_mat_init(Mc, n, n);
fmpz_mat_set(X, A);
if (U != NULL)
fmpz_mat_one(U);
if (V != NULL)
fmpz_mat_one(V);
max_iters = _fmpz_mat_snf_iter_bound(A);
for (iter = 0; !fmpz_mat_is_diagonal(X); iter++)
{
if (iter >= max_iters)
flint_throw(FLINT_ERROR,
"(fmpz_mat_snf_transform): Phase 1 exceeded iteration "
"bound (%wd). Likely bug in fmpz_mat_hnf_transform or "
"unexpected input; please report.\n", max_iters);
/* Row HNF: X -> Mr*X, U -> Mr*U (Mr is m x m) */
fmpz_mat_hnf_transform(X, Mr, X);
if (U != NULL)
fmpz_mat_mul(U, Mr, U);
if (fmpz_mat_is_diagonal(X))
break;
/* Column HNF via transpose:
Xt = X^T, compute Xt -> Mc*Xt (row HNF of transpose),
then X = Xt^T, V -> V * Mc^T */
fmpz_mat_transpose(Xt, X);
fmpz_mat_hnf_transform(Xt, Mc, Xt);
fmpz_mat_transpose(X, Xt);
fmpz_mat_transpose(Mc, Mc);
if (V != NULL)
fmpz_mat_mul(V, V, Mc);
}
/*
Phase 2: fix divisibility chain on diagonal.
For each pair (j, k) with j < k, if X[j,j] does not divide X[k,k],
apply xgcd-based row/column operations.
Let a = X[j,j], b = X[k,k], d = gcd(a,b), a = d*p, b = d*q,
u*a + v*b = d (so u*p + v*q = 1).
Left transform L (rows j,k):
L[j,j] = 1, L[j,k] = v, L[k,j] = q, L[k,k] = vq-1
det(L) = vq-1 - vq = -1
Right transform R (cols j,k):
R[j,j] = u, R[j,k] = 1-up, R[k,j] = 1, R[k,k] = -p
det(R) = -up - (1-up) = -1
Result: L * diag(a,b) * R = diag(d, pqd) = diag(gcd, lcm).
*/
{
fmpz_t dd, uu, vv, pp, qq, vq_m1, one_m_up;
fmpz * save_j, * save_k;
fmpz_init(dd);
fmpz_init(uu);
fmpz_init(vv);
fmpz_init(pp);
fmpz_init(qq);
fmpz_init(vq_m1);
fmpz_init(one_m_up);
save_j = _fmpz_vec_init(FLINT_MAX(m, n));
save_k = _fmpz_vec_init(FLINT_MAX(m, n));
for (j = 0; j < d; j++)
{
if (fmpz_is_one(fmpz_mat_entry(X, j, j)))
continue;
for (k = j + 1; k < d; k++)
{
if (fmpz_is_zero(fmpz_mat_entry(X, k, k)))
continue;
if (!fmpz_is_zero(fmpz_mat_entry(X, j, j))
&& fmpz_divisible(
fmpz_mat_entry(X, k, k),
fmpz_mat_entry(X, j, j)))
continue;
fmpz_xgcd_canonical_bezout(dd, uu, vv,
fmpz_mat_entry(X, j, j),
fmpz_mat_entry(X, k, k));
fmpz_divexact(pp,
fmpz_mat_entry(X, j, j), dd);
fmpz_divexact(qq,
fmpz_mat_entry(X, k, k), dd);
/* vq_m1 = v*q - 1 */
fmpz_mul(vq_m1, vv, qq);
fmpz_sub_ui(vq_m1, vq_m1, 1);
/* one_m_up = 1 - u*p */
fmpz_mul(one_m_up, uu, pp);
fmpz_neg(one_m_up, one_m_up);
fmpz_add_ui(one_m_up, one_m_up, 1);
/*
Left transform on rows j,k of U:
new row j = old row j + v * old row k
new row k = q * old row j
+ (vq-1) * old row k
*/
if (U != NULL)
{
fmpz * row_j = fmpz_mat_row(U, j);
fmpz * row_k = fmpz_mat_row(U, k);
_fmpz_vec_set(save_j, row_j, m);
_fmpz_vec_set(save_k, row_k, m);
_fmpz_vec_scalar_addmul_fmpz(row_j, save_k, m, vv);
_fmpz_vec_scalar_mul_fmpz(row_k, save_j, m, qq);
_fmpz_vec_scalar_addmul_fmpz(row_k, save_k, m, vq_m1);
}
/*
Right transform on cols j,k of V:
new col j = u * old col j + old col k
new col k = (1-up) * old col j
+ (-p) * old col k
*/
if (V != NULL)
{
for (i = 0; i < n; i++)
fmpz_set(&save_j[i],
fmpz_mat_entry(V, i, j));
for (i = 0; i < n; i++)
fmpz_set(&save_k[i],
fmpz_mat_entry(V, i, k));
for (i = 0; i < n; i++)
{
fmpz_mul(
fmpz_mat_entry(V, i, j),
uu, &save_j[i]);
fmpz_add(
fmpz_mat_entry(V, i, j),
fmpz_mat_entry(V, i, j),
&save_k[i]);
fmpz_mul(
fmpz_mat_entry(V, i, k),
one_m_up, &save_j[i]);
fmpz_submul(
fmpz_mat_entry(V, i, k),
pp, &save_k[i]);
}
}
/* X[j,j] = gcd, X[k,k] = lcm */
fmpz_set(fmpz_mat_entry(X, j, j), dd);
fmpz_mul(fmpz_mat_entry(X, k, k),
pp, qq);
fmpz_mul(fmpz_mat_entry(X, k, k),
fmpz_mat_entry(X, k, k), dd);
}
}
_fmpz_vec_clear(save_k, FLINT_MAX(m, n));
_fmpz_vec_clear(save_j, FLINT_MAX(m, n));
fmpz_clear(dd);
fmpz_clear(uu);
fmpz_clear(vv);
fmpz_clear(pp);
fmpz_clear(qq);
fmpz_clear(vq_m1);
fmpz_clear(one_m_up);
}
/*
Phase 3: fix signs. SNF requires non-negative diagonal entries.
If X[j,j] < 0, negate row j of U to absorb the sign.
*/
for (j = 0; j < d; j++)
{
if (fmpz_sgn(fmpz_mat_entry(X, j, j)) < 0)
{
fmpz_neg(fmpz_mat_entry(X, j, j),
fmpz_mat_entry(X, j, j));
if (U != NULL)
{
fmpz * row_j = fmpz_mat_row(U, j);
_fmpz_vec_neg(row_j, row_j, m);
}
}
}
fmpz_mat_set(S, X);
fmpz_mat_clear(Mc);
fmpz_mat_clear(Xt);
fmpz_mat_clear(X);
fmpz_mat_clear(Mr);
}