flint3-sys 3.2.2-1

Rust bindings to the FLINT C library
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

/*
    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 "acb_poly.h"
#include "acb_mat.h"
#include "acb_theta.h"

static void
acb_theta_sqrt_branch(acb_t res, const acb_t x, acb_srcptr rts_neg, slong nb_neg,
    acb_srcptr rts_pos, slong nb_pos, const acb_t sqrt_lead, slong prec)
{
    acb_t s, t;
    slong k;

    acb_init(s);
    acb_init(t);

    acb_set(s, sqrt_lead);
    for (k = 0; k < nb_neg; k++)
    {
        acb_sub(t, x, &rts_neg[k], prec);
        acb_sqrt_analytic(t, t, 1, prec);
        acb_mul(s, s, t, prec);
    }
    for (k = 0; k < nb_pos; k++)
    {
        acb_sub(t, &rts_pos[k], x, prec);
        acb_sqrt_analytic(t, t, 1, prec);
        acb_mul(s, s, t, prec);
    }

    acb_set(res, s);

    acb_clear(s);
    acb_clear(t);
}

void
acb_theta_transform_sqrtdet(acb_t res, const acb_mat_t tau, slong prec)
{
    slong g = acb_mat_nrows(tau);
    flint_rand_t state;
    acb_mat_t A, B, C;
    acb_poly_t pol, h;
    acb_ptr rts, rts_neg, rts_pos;
    acb_t z, rt, mu;
    arb_t x;
    slong k, j, nb_neg, nb_pos;
    int success = 0;

    flint_rand_init(state);
    acb_mat_init(A, g, g);
    acb_mat_init(B, g, g);
    acb_mat_init(C, g, g);
    acb_poly_init(pol);
    acb_poly_init(h);
    rts = _acb_vec_init(g);
    rts_neg = _acb_vec_init(g);
    rts_pos = _acb_vec_init(g);
    acb_init(z);
    acb_init(rt);
    acb_init(mu);
    arb_init(x);

    /* Choose a purely imaginary matrix A and compute pol s.t. pol(-1) = det(A)
       and pol(1) = det(tau): pol(t) is

       det(A + (t+1)/2 (tau - A)) = det(A) det(I - (t+1)/2 (I - A^{-1}tau))

       We want to get the g roots of this polynomial to compute the branch
       cuts. This can fail e.g. when det(tau - A) = 0, so pick A at random
       until the roots can be found */
    for (k = 0; (k < 100) && !success; k++)
    {
        acb_mat_onei(A);
        for (j = 0; j < g; j++)
        {
            arb_urandom(x, state, prec);
            arb_add(acb_imagref(acb_mat_entry(A, j, j)),
                acb_imagref(acb_mat_entry(A, j, j)), x, prec);
        }
        acb_mat_inv(B, A, prec);
        acb_mat_mul(B, B, tau, prec);
        acb_mat_one(C);
        acb_mat_sub(C, C, B, prec);

        /* Get reverse of charpoly */
        acb_mat_charpoly(h, C, prec);
        acb_poly_zero(pol);
        for (j = 0; j <= g; j++)
        {
            acb_poly_get_coeff_acb(z, h, j);
            acb_poly_set_coeff_acb(pol, g - j, z);
        }
        acb_poly_one(h);
        acb_poly_set_coeff_si(h, 1, 1);
        acb_poly_scalar_mul_2exp_si(h, h, -1);
        acb_poly_compose(pol, pol, h, prec);

        success = (acb_poly_find_roots(rts, pol, NULL, 0, prec) == g);

        /* Check that no root intersects the [-1,1] segment */
        for (j = 0; (j < g) && success; j++)
        {
            if (arb_contains_zero(acb_imagref(&rts[j])))
            {
                arb_abs(x, acb_realref(&rts[j]));
                arb_sub_si(x, x, 1, prec);
                success = arb_is_positive(x);
            }
        }
    }

    if (success)
    {
        /* Partition the roots between positive & negative real parts to
           compute branch for sqrt(pol) */
        nb_neg = 0;
        nb_pos = 0;
        for (k = 0; k < g; k++)
        {
            if (arb_is_negative(acb_realref(&rts[k])))
            {
                acb_set(&rts_neg[nb_neg], &rts[k]);
                nb_neg++;
            }
            else
            {
                acb_set(&rts_pos[nb_pos], &rts[k]);
                nb_pos++;
            }
        }
        acb_mat_det(rt, A, prec);
        acb_mul(rt, rt, acb_poly_get_coeff_ptr(pol, g), prec);
        acb_sqrts(rt, z, rt, prec);

        /* Set mu to +-1 such that mu*sqrt_branch gives the correct value at A,
           i.e. i^(g/2) * something positive */
        acb_mat_det(mu, A, prec);
        acb_mul_i_pow_si(mu, mu, -g);
        acb_sqrt(mu, mu, prec);
        acb_set_si(z, g);
        acb_mul_2exp_si(z, z, -2);
        acb_exp_pi_i(z, z, prec);
        acb_mul(mu, mu, z, prec);
        acb_set_si(z, -1);
        acb_theta_sqrt_branch(z, z, rts_neg, nb_neg, rts_pos, nb_pos, rt, prec);
        acb_div(mu, mu, z, prec);

        /* Compute square root branch at z=1 to get sqrtdet */
        acb_set_si(z, 1);
        acb_theta_sqrt_branch(rt, z, rts_neg, nb_neg, rts_pos, nb_pos, rt, prec);
        acb_mul(rt, rt, mu, prec);
        acb_mat_det(res, tau, prec);
        acb_theta_agm_sqrt(res, res, rt, 1, prec);
    }
    else
    {
        acb_mat_det(res, tau, prec);
        acb_sqrts(res, z, res, prec);
        acb_union(res, res, z, prec);
    }

    flint_rand_clear(state);
    acb_mat_clear(A);
    acb_mat_clear(B);
    acb_mat_clear(C);
    acb_poly_clear(pol);
    acb_poly_clear(h);
    _acb_vec_clear(rts, g);
    _acb_vec_clear(rts_pos, g);
    _acb_vec_clear(rts_neg, g);
    acb_clear(z);
    acb_clear(rt);
    acb_clear(mu);
    arb_clear(x);
}