flint-sys 0.9.0

Bindings to the FLINT C library
Documentation 
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

/*
    Copyright (C) 2012 Fredrik Johansson

    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"

static slong
arb_mat_gauss_partial(arb_mat_t A, slong prec)
{
    arb_t e;
    slong j, m, n, r, rank, row, col, sign;

    m = A->r;
    n = A->c;
    rank = row = col = 0;
    sign = 1;

    arb_init(e);

    while (row < m && col < n)
    {
        r = arb_mat_find_pivot_partial(A, row, m, col);

        if (r == -1)
        {
            break;
        }
        else if (r != row)
        {
            arb_mat_swap_rows(A, NULL, row, r);
            sign *= -1;
        }

        rank++;

        for (j = row + 1; j < m; j++)
        {
            arb_div(e, arb_mat_entry(A, j, col), arb_mat_entry(A, row, col), prec);
            arb_neg(e, e);
            _arb_vec_scalar_addmul(arb_mat_entry(A, j, col + 1), arb_mat_entry(A, row, col + 1), n - col - 1, e, prec);
        }

        row++;
        col++;
    }

    arb_clear(e);

    return rank * sign;
}

static void
arb_vec_get_arf_2norm_squared_bound(arf_t s, arb_srcptr vec, slong len, slong prec)
{
    slong i;
    arf_t t;

    arf_init(t);
    arf_zero(s);

    for (i = 0; i < len; i++)
    {
        arb_get_abs_ubound_arf(t, vec + i, prec);
        arf_addmul(s, t, t, prec, ARF_RND_UP);
    }

    arf_clear(t);
}

static void
arb_mat_det_lu_inplace(arb_t det, arb_mat_t A, slong prec)
{
    slong i, n, sign, rank;

    n = arb_mat_nrows(A);
    rank = arb_mat_gauss_partial(A, prec);
    sign = (rank < 0) ? -1 : 1;
    rank = FLINT_ABS(rank);

    arb_set_si(det, sign);
    for (i = 0; i < rank; i++)
        arb_mul(det, det, arb_mat_entry(A, i, i), prec);

    /* bound unreduced part using Hadamard's inequality */
    if (rank < n)
    {
        arf_t t;
        arf_t d;
        arb_t b;

        arf_init(t);
        arf_init(d);
        arb_init(b);

        arf_one(d);

        for (i = rank; i < n; i++)
        {
            arb_vec_get_arf_2norm_squared_bound(t, arb_mat_entry(A, i, rank),  n - rank, MAG_BITS);
            arf_mul(d, d, t, MAG_BITS, ARF_RND_UP);
        }

        arf_sqrt(d, d, MAG_BITS, ARF_RND_UP);
        arb_add_error_arf(b, d);

        arb_mul(det, det, b, prec);

        arf_clear(d);
        arf_clear(t);
        arb_clear(b);
    }
}

void
arb_mat_det_lu(arb_t det, const arb_mat_t A, slong prec)
{
    slong n;

    n = arb_mat_nrows(A);

    if (n == 0)
    {
        arb_one(det);
    }
    else
    {
        arb_mat_t T;
        arb_mat_init(T, arb_mat_nrows(A), arb_mat_ncols(A));
        arb_mat_set(T, A);
        arb_mat_det_lu_inplace(det, T, prec);
        arb_mat_clear(T);
    }
}