blu 0.2.1

LU factorization with dynamic Markowitz search and columnwise threshold pivoting
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

// Copyright (C) 2016-2018 ERGO-Code
// Copyright (C) 2022-2023 Richard Lincoln

use crate::LUInt;

// LINPACK condition number estimate //

// Given `m`-by-`m` matrix `U` such that `U[perm,perm]` is upper triangular,
// return estimate for 1-norm condition number of `U`.
// If `norm` is not None, it holds the 1-norm of the matrix on return.
// If `norminv` is not None, it holds the estimated 1-norm of the inverse on
// return.
//
// The other function arguments are the same as in [`normest()`].
pub(crate) fn condest(
    m: usize,
    u_begin: &[LUInt],
    u_i: &[LUInt],
    u_x: &[f64],
    pivot: Option<&[f64]>,
    perm: Option<&[LUInt]>,
    upper: i32,
    work: &mut [f64],
    norm: Option<&mut f64>,
    norminv: Option<&mut f64>,
) -> f64 {
    // compute 1-norm of U
    let mut u_norm = 0.0;
    for j in 0..m {
        let mut colsum: f64 = if let Some(pivot) = pivot {
            pivot[j].abs()
        } else {
            1.0
        };
        let mut p = u_begin[j] as usize;
        while u_i[p] >= 0 {
            colsum += u_x[p].abs();
            p += 1;
        }
        u_norm = f64::max(u_norm, colsum);
    }

    // estimate 1-norm of U^{-1}
    let u_invnorm = normest(m, u_begin, u_i, u_x, pivot, perm, upper, work);

    if let Some(norm) = norm {
        *norm = u_norm;
    }
    if let Some(norminv) = norminv {
        *norminv = u_invnorm;
    }

    u_norm * u_invnorm
}

// Given `m`-by-`m` matrix `U` such that `U[perm,perm]` is triangular,
// estimate 1-norm of `U^{-1}` by computing
//
//     U'x = b, Uy = x, normest = max{norm(y)_1/norm(x)_1, norm(x)_inf},
//
// where the entries of `b` are +/-1 chosen dynamically to make `x` large.
// The method is described in [1].
//
// - `u_begin`, `u_`, `u_x` matrix `U` in compressed column format without pivots,
//                          columns are terminated by a negative index
// - `pivot` pivot elements by column index of `U`; None if unit pivots
// - `perm` permutation to triangular form; None if identity
// - `upper` nonzero if permuted matrix is upper triangular; zero if lower
// - `work` size `m` workspace, uninitialized on entry/return
//
// Return: estimate for 1-norm of `U^{-1}`
//
// [1] I. Duff, A. Erisman, J. Reid, "Direct Methods for Sparse Matrices"
pub(crate) fn normest(
    m: usize,
    u_begin: &[LUInt],
    u_i: &[LUInt],
    u_x: &[f64],
    pivot: Option<&[f64]>,
    perm: Option<&[LUInt]>,
    upper: i32,
    work: &mut [f64],
) -> f64 {
    let mut x1norm = 0.0;
    let mut xinfnorm = 0.0;
    let (kbeg, kend, kinc): (LUInt, LUInt, LUInt) = if upper != 0 {
        let kbeg = 0;
        let kend = m as LUInt;
        let kinc = 1;
        (kbeg, kend, kinc)
    } else {
        let kbeg = (m as LUInt) - 1;
        let kend = -1;
        let kinc = -1;
        (kbeg, kend, kinc)
    };
    let mut k = kbeg;
    while k != kend {
        let j = if let Some(perm) = perm {
            perm[k as usize] as usize
        } else {
            k as usize
        };
        let mut temp = 0.0;
        // for (p = u_begin[j]; (i = u_i[p]) >= 0; p++) {
        let mut p = u_begin[j] as usize;
        while u_i[p] >= 0 {
            temp -= work[u_i[p] as usize] * u_x[p];
            p += 1;
        }
        temp += if temp >= 0.0 { 1.0 } else { -1.0 }; // choose b[i] = 1 or b[i] = -1
        if let Some(pivot) = pivot {
            temp /= pivot[j];
        }
        work[j] = temp;
        x1norm += temp.abs();
        xinfnorm = f64::max(xinfnorm, temp.abs());
        k += kinc;
    }

    let mut y1norm = 0.0;
    let (kbeg, kend, kinc): (LUInt, LUInt, LUInt) = if upper != 0 {
        let kbeg = (m as LUInt) - 1;
        let kend = -1;
        let kinc = -1;
        (kbeg, kend, kinc)
    } else {
        let kbeg = 0;
        let kend = m as LUInt;
        let kinc = 1;
        (kbeg, kend, kinc)
    };
    // for (k = kbeg; k != kend; k += kinc)
    let mut k = kbeg;
    while k != kend {
        let j = if let Some(perm) = perm {
            perm[k as usize] as usize
        } else {
            k as usize
        };
        if let Some(pivot) = pivot {
            work[j] /= pivot[j];
        }
        let temp = work[j];
        // for (p = u_begin[j]; (i = u_i[p]) >= 0; p++) {
        let mut p = u_begin[j] as usize;
        while u_i[p] >= 0 {
            work[u_i[p] as usize] -= temp * u_x[p];
            p += 1;
        }
        y1norm += temp.abs();

        k += kinc;
    }

    f64::max(y1norm / x1norm, xinfnorm)
}