ldl 0.1.0

LDL factorisation for quasi-definite linear systems
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
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
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242

use crate::factor_solve;
use num_traits::{Float, PrimInt, ToPrimitive};

const TESTS_TOL: f64 = 1e-4;

#[test]
fn basic() {
    let a_n = 10;
    let a_p = vec![0, 1, 2, 4, 5, 6, 8, 10, 12, 14, 17];
    let a_i = vec![0, 1, 1, 2, 3, 4, 1, 5, 0, 6, 3, 7, 6, 8, 1, 2, 9];
    let a_x = vec![
        1.0, 0.460641, -0.121189, 0.417928, 0.177828, 0.1, -0.0290058, -1.0, 0.350321, -0.441092,
        -0.0845395, -0.316228, 0.178663, -0.299077, 0.182452, -1.56506, -0.1,
    ];
    let mut b = vec![1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0];

    // RHS and solution to Ax = b
    let xsol = vec![
        10.2171, 3.9416, -5.69096, 9.28661, 50.0, -6.11433, -26.3104, -27.7809, -45.8099, -3.74178,
    ];

    // x replaces b during solve
    let status = factor_solve::<f64, usize>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_ok(), "Factorisation failed");
    assert!(
        vec_diff_norm(&b, &xsol, a_n) < TESTS_TOL,
        "Solve accuracy failed"
    );
}

#[test]
fn basic_f32_i64() {
    let a_n = 10;
    let a_p = vec![0, 1, 2, 4, 5, 6, 8, 10, 12, 14, 17];
    let a_i = vec![0, 1, 1, 2, 3, 4, 1, 5, 0, 6, 3, 7, 6, 8, 1, 2, 9];
    let a_x = vec![
        1.0, 0.460641, -0.121189, 0.417928, 0.177828, 0.1, -0.0290058, -1.0, 0.350321, -0.441092,
        -0.0845395, -0.316228, 0.178663, -0.299077, 0.182452, -1.56506, -0.1,
    ];
    let mut b = vec![1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0];

    // RHS and solution to Ax = b
    let xsol = vec![
        10.2171, 3.9416, -5.69096, 9.28661, 50.0, -6.11433, -26.3104, -27.7809, -45.8099, -3.74178,
    ];

    // x replaces b during solve
    let status = factor_solve::<f32, i64>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_ok(), "Factorisation failed");
    assert!(
        vec_diff_norm(&b, &xsol, a_n) < TESTS_TOL as f32,
        "Solve accuracy failed"
    );
}

#[test]
fn identity() {
    // A matrix data
    let a_p = vec![0, 1, 2, 3, 4];
    let a_i = vec![0, 1, 2, 3];
    let a_x = vec![1.0, 1.0, 1.0, 1.0];
    let a_n = 4;

    // RHS and solution to Ax = b
    let mut b = vec![2.0, 2.0, 2.0, 2.0];
    let xsol = vec![2.0, 2.0, 2.0, 2.0];

    // x replaces b during solve
    let status = factor_solve::<f64, usize>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_ok(), "Factorisation failed");
    assert!(
        vec_diff_norm(&b, &xsol, a_n) < TESTS_TOL,
        "Solve accuracy failed"
    );
}

#[test]
fn osqp_kkt() {
    // Unordered A
    let a_p = vec![0, 1, 2, 5, 6, 7, 8, 12];
    let a_i = vec![0, 1, 2, 1, 0, 3, 4, 5, 5, 6, 4, 3];
    let a_x = vec![
        -0.25, -0.25, 1.0, 0.513578, 0.529142, -0.25, -0.25, 1.10274, 0.15538, 1.25883, 0.13458,
        0.621134,
    ];
    let a_n = 7;

    // RHS and solution to Ax = b
    let mut b = vec![
        -0.595598, -0.0193715, -0.576156, -0.168746, 0.61543, 0.419073, 1.31087,
    ];
    let xsol = vec![
        1.13141, -1.1367, -0.591044, 1.68867, -2.24209, 0.32254, 0.407998,
    ];

    // x replaces b during solve
    let status = factor_solve::<f64, usize>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_ok(), "Factorisation failed");
    assert!(
        vec_diff_norm(&b, &xsol, a_n) < TESTS_TOL,
        "Solve accuracy failed"
    );
}

#[test]
fn rank_deficient() {
    // A matrix data
    let a_p = vec![0, 1, 3];
    let a_i = vec![0, 0, 1];
    let a_x = vec![1.0, 1.0, 1.0];
    let a_n = 2;

    // RHS for Ax = b (should fail to solve)
    let mut b = vec![1.0, 1.0];

    // x replaces b during solve
    let status = factor_solve::<f64, usize>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_err(), "Rank deficiency not detected");
}

#[test]
fn singleton() {
    // A matrix data
    let a_p = vec![0, 1];
    let a_i = vec![0];
    let a_x = vec![0.2];
    let a_n = 1;

    // RHS and solution to Ax = b
    let mut b = vec![2.0];
    let xsol = vec![10.0];

    // x replaces b during solve
    let status = factor_solve::<f64, usize>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_ok(), "Factorisation failed");
    assert!(
        vec_diff_norm(&b, &xsol, a_n) < TESTS_TOL,
        "Solve accuracy failed"
    );
}

#[test]
fn sym_structure() {
    // A matrix data
    let a_p = vec![0, 2, 4];
    let a_i = vec![0, 1, 0, 1];
    let a_x = vec![5.0, 1.0, 1.0, 5.0];
    let a_n = 2;

    // RHS for Ax = b
    let mut b = vec![1.0, 1.0];

    // x replaces b during solve
    let status = factor_solve::<f64, usize>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_err(), "Fully symmetric input not detected");
}

#[test]
fn tril_structure() {
    // A matrix data
    let a_p = vec![0, 2, 3];
    let a_i = vec![0, 1, 1];
    let a_x = vec![5.0, 1.0, 5.0];
    let a_n = 2;

    // RHS for Ax = b
    let mut b = vec![1.0, 1.0];

    // x replaces b during solve
    let status = factor_solve::<f64, usize>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_err(), "Tril input not detected");
}

#[test]
fn two_by_two() {
    // A matrix data
    let a_p = vec![0, 1, 3];
    let a_i = vec![0, 0, 1];
    let a_x = vec![1.0, 1.0, -1.0];
    let a_n = 2;

    // RHS and solution to Ax = b
    let mut b = vec![2.0, 4.0];
    let xsol = vec![3.0, -1.0];

    // x replaces b during solve
    let status = factor_solve::<f64, usize>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_ok(), "Factorisation failed");
    assert!(
        vec_diff_norm(&b, &xsol, a_n) < TESTS_TOL,
        "Solve accuracy failed"
    );
}

#[test]
fn zero_on_diag() {
    // A matrix data
    let a_p = vec![0, 1, 2, 5];
    let a_i = vec![0, 0, 0, 1, 2];
    let a_x = vec![4.0, 1.0, 2.0, 1.0, -3.0];
    let a_n = 3;

    // RHS and solution to Ax = b
    let mut b = vec![6.0, 9.0, 12.0];
    let xsol = vec![17.0, -46.0, -8.0];

    // x replaces b during solve (should fill due to zero in middle)
    // NB : this system is solvable, but not by LDL
    let status = factor_solve::<f64, usize>(a_n, &a_p, &a_i, &a_x, &mut b);

    assert!(status.is_ok(), "Factorisation failed");
    assert!(
        vec_diff_norm(&b, &xsol, a_n) < TESTS_TOL,
        "Solve accuracy failed"
    );
}

fn vec_diff_norm<S: Float, I: PrimInt + ToPrimitive>(x: &[S], y: &[S], len: I) -> S {
    let mut max_diff = S::zero();
    for i in 0..len.to_usize().unwrap() {
        let el_diff = x[i] - y[i];
        max_diff = if el_diff > max_diff {
            el_diff
        } else {
            if -el_diff > max_diff {
                -el_diff
            } else {
                max_diff
            }
        };
    }
    max_diff
}