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
|
/*
Copyright (C) 2018 Fredrik Johansson
This file is part of Arb.
Arb 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 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb_mat.h"
static void
acb_approx_mul(acb_t res, const acb_t x, const acb_t y, slong prec)
{
arf_complex_mul(arb_midref(acb_realref(res)), arb_midref(acb_imagref(res)),
arb_midref(acb_realref(x)), arb_midref(acb_imagref(x)),
arb_midref(acb_realref(y)), arb_midref(acb_imagref(y)), prec, ARB_RND);
}
/* note: the tmp variable t should have zero radius */
static void
acb_approx_div(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
{
arf_set(arb_midref(acb_realref(t)), arb_midref(acb_realref(y)));
arf_set(arb_midref(acb_imagref(t)), arb_midref(acb_imagref(y)));
acb_inv(t, t, prec);
mag_zero(arb_radref(acb_realref(t)));
mag_zero(arb_radref(acb_imagref(t)));
acb_approx_mul(z, x, t, prec);
}
void
acb_mat_approx_solve_triu_classical(acb_mat_t X, const acb_mat_t U,
const acb_mat_t B, int unit, slong prec)
{
slong i, j, n, m;
acb_ptr tmp;
acb_t s, t;
n = U->r;
m = B->c;
acb_init(s);
acb_init(t);
tmp = flint_malloc(sizeof(acb_struct) * n);
for (i = 0; i < m; i++)
{
for (j = 0; j < n; j++)
tmp[j] = *acb_mat_entry(X, j, i);
for (j = n - 1; j >= 0; j--)
{
acb_approx_dot(s, acb_mat_entry(B, j, i), 1, U->rows[j] + j + 1, 1, tmp + j + 1, 1, n - j - 1, prec);
if (!unit)
acb_approx_div(tmp + j, s, arb_mat_entry(U, j, j), t, prec);
else
acb_swap(tmp + j, s);
}
for (j = 0; j < n; j++)
*acb_mat_entry(X, j, i) = tmp[j];
}
flint_free(tmp);
acb_clear(s);
acb_clear(t);
}
void
acb_mat_approx_solve_triu_recursive(acb_mat_t X,
const acb_mat_t U, const acb_mat_t B, int unit, slong prec)
{
acb_mat_t UA, UB, UD, XX, XY, BX, BY, T;
slong r, n, m;
n = U->r;
m = B->c;
r = n / 2;
if (n == 0 || m == 0)
return;
/*
Denoting inv(M) by M^, we have:
[A B]^ [X] == [A^ (X - B D^ Y)]
[0 D] [Y] == [ D^ Y ]
*/
acb_mat_window_init(UA, U, 0, 0, r, r);
acb_mat_window_init(UB, U, 0, r, r, n);
acb_mat_window_init(UD, U, r, r, n, n);
acb_mat_window_init(BX, B, 0, 0, r, m);
acb_mat_window_init(BY, B, r, 0, n, m);
acb_mat_window_init(XX, X, 0, 0, r, m);
acb_mat_window_init(XY, X, r, 0, n, m);
acb_mat_approx_solve_triu(XY, UD, BY, unit, prec);
acb_mat_init(T, UB->r, XY->c);
acb_mat_approx_mul(T, UB, XY, prec);
acb_mat_sub(XX, BX, T, prec);
acb_mat_get_mid(XX, XX);
acb_mat_clear(T);
acb_mat_approx_solve_triu(XX, UA, XX, unit, prec);
acb_mat_window_clear(UA);
acb_mat_window_clear(UB);
acb_mat_window_clear(UD);
acb_mat_window_clear(BX);
acb_mat_window_clear(BY);
acb_mat_window_clear(XX);
acb_mat_window_clear(XY);
}
void
acb_mat_approx_solve_triu(acb_mat_t X, const acb_mat_t U,
const acb_mat_t B, int unit, slong prec)
{
if (B->r < 40 || B->c < 40)
acb_mat_approx_solve_triu_classical(X, U, B, unit, prec);
else
acb_mat_approx_solve_triu_recursive(X, U, B, unit, prec);
}
|
