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
|
/* This file is public domain. Author: Fredrik Johansson. */
#include <stdlib.h>
#include <string.h>
#include <flint/profiler.h>
#include <flint/fmpq_mat.h>
#include <flint/calcium.h>
#include <flint/ca.h>
#include <flint/ca_vec.h>
#include <flint/ca_mat.h>
#include <flint/qqbar.h>
int main(int argc, char *argv[])
{
slong n, i;
int qqbar, vieta, novieta;
if (argc < 2)
{
flint_printf("usage: hilbert_matrix [-qqbar] [-vieta | -novieta] n\n");
return 1;
}
qqbar = 0;
vieta = 0;
novieta = 0;
n = 0;
for (i = 1; i < argc; i++)
{
if (!strcmp(argv[i], "-qqbar"))
{
qqbar = 1;
}
else if (!strcmp(argv[i], "-vieta"))
{
vieta = 1;
}
else if (!strcmp(argv[i], "-novieta"))
{
novieta = 1;
}
else
{
n = atol(argv[i]);
if (n < 0 || n > 100)
flint_abort();
}
}
TIMEIT_ONCE_START;
if (qqbar)
{
fmpq_mat_t mat;
qqbar_ptr eig;
qqbar_t trace, det;
fmpq_mat_init(mat, n, n);
qqbar_init(trace);
qqbar_init(det);
eig = _qqbar_vec_init(n);
fmpq_mat_hilbert_matrix(mat);
qqbar_eigenvalues_fmpq_mat(eig, mat, 0);
flint_printf("Trace:\n");
qqbar_zero(trace);
for (i = 0; i < n; i++)
{
qqbar_add(trace, trace, eig + i);
flint_printf("%wd/%wd: degree %wd\n", i, n, qqbar_degree(trace));
}
qqbar_print(trace);
flint_printf("\n");
flint_printf("Determinant:\n");
qqbar_one(det);
for (i = 0; i < n; i++)
{
qqbar_mul(det, det, eig + i);
flint_printf("%wd/%wd: degree %wd\n", i, n, qqbar_degree(det));
}
qqbar_print(det);
flint_printf("\n");
fmpq_mat_clear(mat);
qqbar_clear(trace);
qqbar_clear(det);
_qqbar_vec_clear(eig, n);
}
else
{
ca_ctx_t ctx;
ca_mat_t mat;
ca_vec_t eig;
ca_t trace, det, t;
ulong * mul;
ca_ctx_init(ctx);
/* Verification requires high-degree algebraics. */
ctx->options[CA_OPT_QQBAR_DEG_LIMIT] = 10000;
if (vieta)
ctx->options[CA_OPT_VIETA_LIMIT] = n;
if (novieta)
ctx->options[CA_OPT_VIETA_LIMIT] = 0;
ca_mat_init(mat, n, n, ctx);
ca_vec_init(eig, 0, ctx);
mul = flint_malloc(sizeof(ulong) * n);
ca_init(trace, ctx);
ca_init(det, ctx);
ca_init(t, ctx);
ca_mat_hilbert(mat, ctx);
ca_mat_eigenvalues(eig, mul, mat, ctx);
ca_mat_trace(trace, mat, ctx);
ca_mat_det(det, mat, ctx);
/* note: in general, we should use the multiplicities, but
we happen to know that the eigenvalues are simple here */
flint_printf("Trace:\n");
ca_zero(t, ctx);
for (i = 0; i < n; i++)
ca_add(t, t, ca_vec_entry(eig, i), ctx);
ca_print(trace, ctx); flint_printf("\n");
ca_print(t, ctx); flint_printf("\n");
flint_printf("Equal: "); truth_print(ca_check_equal(trace, t, ctx)); flint_printf("\n\n");
flint_printf("Det:\n");
ca_one(t, ctx);
for (i = 0; i < n; i++)
ca_mul(t, t, ca_vec_entry(eig, i), ctx);
ca_print(det, ctx); flint_printf("\n");
ca_print(t, ctx); flint_printf("\n");
flint_printf("Equal: "); truth_print(ca_check_equal(det, t, ctx)); flint_printf("\n\n");
ca_mat_clear(mat, ctx);
ca_vec_clear(eig, ctx);
flint_free(mul);
ca_clear(trace, ctx);
ca_clear(det, ctx);
ca_clear(t, ctx);
ca_ctx_clear(ctx);
}
flint_printf("\n");
TIMEIT_ONCE_STOP;
print_memory_usage();
flint_cleanup();
return 0;
}
|
