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
/*
Copyright (C) 2023 Jean Kieffer
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 "arb.h"
#include "arb_mat.h"
#include "acb_mat.h"
#include "acb_theta.h"
/* This is the all-important function to increase performance. */
int
acb_theta_ql_nb_steps(slong * pattern, const acb_mat_t tau, int cst, slong prec)
{
slong g = acb_mat_nrows(tau);
slong lp = ACB_THETA_LOW_PREC;
arb_mat_t cho, yinv;
arb_t x, t;
slong s, nb, dupl;
slong * rough;
arb_init(x);
arb_init(t);
arb_mat_init(cho, g, g);
arb_mat_init(yinv, g, g);
rough = flint_malloc(g * sizeof(slong));
acb_siegel_cho_yinv(cho, yinv, tau, lp);
/* Compute rough pattern (could be negative) */
/* Note: we could be more precise by scaling x by something else than
powers of 2 */
for (s = 0; s < g; s++)
{
arb_sqr(x, arb_mat_entry(cho, s, s), lp);
arb_const_log2(t, lp);
arb_div(x, x, t, lp);
arb_div_si(x, x, prec, lp);
arb_log(x, x, lp);
arb_div(x, x, t, lp);
if (!arb_is_finite(x) || arf_cmpabs_2exp_si(arb_midref(x), FLINT_BITS - 4) > 0)
{
/* Should not happen in tests */
arb_clear(x);
arb_clear(t);
return 0;
}
rough[s] = -arf_get_si(arb_midref(x), ARF_RND_NEAR);
}
/* Experimental data from p-acb_theta_ql_exact: rough -> desired pattern */
/* Genus 1 theta constants */
/* 0, ..., 9 -> 0
10 -> 3
11 -> 4
12, 13 -> 5
14, 15 -> 6
16 -> 7
17 -> 8 */
/* Genus 1 general theta values */
/* 0, ..., 8 -> 0
9 -> 0 or 4
10 -> 5
11 -> 6, ...,
13, 14 -> 8
15, 16 -> 9
17 -> 10 */
/* Genus 2 */
/* 4 3, 5 3 -> 0 0
6 4 -> 3 3 or 4 4
6 5 -> 4 4 or 5 5
8 6 -> 6 6
11 10 -> 10 10
12 10 -> 10 10 or 11 11
16 14 -> 14 14 or 15 15
8 2 -> 3 2 or 4 2
9 3 -> 4 2
10 4 -> 6 5 or 7 6
11 5 -> 7 6
12 6 -> 8 7
15 9 -> 11 10
8 0 -> 2 0 or 0 0
9 1 -> 4 2
10 2 -> 6 2
12 4 -> 8 6
14 6 -> 8 7 */
/* Start adapting the pattern from s = g-1 downwards. This is because the
choice of whether to trigger dimension-lowering formulas in low
dimensions will depend on whether or not duplications/dimension-lowerings
have already been applied. */
/* See /path/to/flint/build/acb_theta/profile/p-acb_theta_ql_exact */
/* Some of these branches will not show up in tests. */
for (s = g - 1; s >= 0; s--)
{
/* Find out how many duplication steps have been performed so far
(could be negative if s < g - 1) */
if (s == g - 1)
{
dupl = 0;
}
else
{
dupl = pattern[s + 1];
}
pattern[s] = rough[s];
/* Force trigger dimension-lowering at that point ? We only do this if
s = 0 and dupl is negative as the ellipsoid really contains very few
points. */
if (s == 0 && dupl < 0)
{
pattern[s] = FLINT_MAX(1, pattern[s]);
}
/* Force more duplication steps ? We only do this for s = 1 if there
will be a dimension-lowering later on, or for s >= 2. */
if (s == 1 && pattern[s] < rough[0] && pattern[s] >= 1)
{
pattern[s] = FLINT_MIN(rough[0] - 1, pattern[s] + 2);
}
else if (s >= 2 && pattern[s] >= 1)
{
pattern[s] += 2;
}
/* Remove further duplication steps in genus 1 if it doesn't mess with
the dimension-lowering strategy. */
if (s == 0)
{
nb = pattern[s] - 5;
if (nb >= 10)
{
nb -= 2;
}
else if (nb >= 8)
{
nb -= 1;
}
if (g == 1 && cst)
{
nb -= 2;
}
pattern[s] = FLINT_MAX(FLINT_MAX(0, dupl) + 1, nb);
}
/* Remove duplication steps to avoid dimension-lowering altogether ? We
only do this if the number of steps is <= dupl + 2. In genus 1, we
additionally demand that rough[s] be less than dupl + 1, unless dupl
== 0 */
if (pattern[s] <= dupl + 2
&& (s > 0 || dupl == 0))
{
pattern[s] = dupl;
}
else if (s == 0 && rough[s] <= dupl + 1)
{
pattern[s] = dupl;
}
/* In any case, make pattern a nonincreasing vector */
if (s < g - 1)
{
pattern[s] = FLINT_MAX(dupl, pattern[s]);
}
}
/* Clean up: make pattern a nonnegative vector */
for (s = 0; s < g; s++)
{
pattern[s] = FLINT_MAX(0, pattern[s]);
}
arb_clear(x);
arb_clear(t);
arb_mat_clear(cho);
arb_mat_clear(yinv);
flint_free(rough);
return 1;
}