mathru 0.16.2

Fundamental algorithms for scientific computing in Rust
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

use crate::algebra::abstr::AbsDiffEq;
use crate::algebra::linear::matrix::{SchurDecomposition, UpperHessenberg};
use crate::algebra::linear::vector::Vector;
use crate::matrix;
use crate::{
    algebra::{
        abstr::{Field, Scalar},
        linear::matrix::{General, SchurDec, Transpose, UpperTriangular},
    },
    elementary::Power,
};

impl<T> SchurDecomposition<T> for UpperHessenberg<T>
where
    T: Field + Scalar + Power + AbsDiffEq<Epsilon = T>,
{
    /// Schur Decomposition
    /// ```math
    /// H = QUQ^{-1} \
    /// ```
    ///
    /// # Panics
    ///
    /// if H is not a square matrix
    ///
    /// # Example
    ///
    fn dec_schur(&self) -> Result<SchurDec<T>, ()> {
        let (q, u): (General<T>, UpperTriangular<T>) = if self.matrix.m > 2 {
            let h = self.clone();
            h.francis()
        } else if self.matrix.m == 2 {
            let a_11 = self[[0, 0]];
            let a_12 = self[[0, 1]];
            let a_21 = self[[1, 0]];
            let a_22 = self[[1, 1]];
            let b = -(a_11 + a_22);
            let c = a_11 * a_22 - a_21 * a_12;
            let t_1 = (b * b - T::from_f32(4.0) * c).sqrt();
            let x_1 = (-b - t_1) / T::from_f32(2.0);
            let x_2 = (-b + t_1) / T::from_f32(2.0);
            (
                General::zero(self.matrix.m, self.matrix.m),
                UpperTriangular::from(matrix![x_1, T::zero(); T::zero(), x_2]),
            )
        } else {
            (matrix![T::one()], self.matrix.clone().into())
        };

        Result::Ok(SchurDec::new(q, u))
    }
}

impl<T> UpperHessenberg<T>
where
    T: Field + Scalar + Power + AbsDiffEq<Epsilon = T>,
{
    //https://people.inf.ethz.ch/arbenz/ewp/Lnotes/chapter4.pdf
    fn francis(mut self) -> (General<T>, UpperTriangular<T>) {
        let epsilon: T = T::default_epsilon();

        let (m, n): (usize, usize) = self.matrix.dim();

        let mut u: General<T> = General::one(m);

        let mut p: usize = n;

        while p > 2 {
            let q = p - 1;

            // Bulge generating
            let h_qq = self[[q - 1, q - 1]];
            let h_pp = self[[p - 1, p - 1]];
            let s: T = h_qq + h_pp;

            let h_qp = self[[q - 1, p - 1]];
            let h_pq = self[[p - 1, q - 1]];
            let t: T = h_qq * h_pp - h_qp * h_pq;

            // compute first 3 elements of first column of M
            let h_11 = self[[0, 0]];
            let h_12 = self[[0, 1]];
            let h_21 = self[[1, 0]];
            let mut x: T = h_11 * h_11 + h_12 * h_21 - s * h_11 + t;

            let h_22 = self[[1, 1]];
            let mut y: T = h_21 * (h_11 + h_22 - s);

            let h_32 = self[[2, 1]];
            let mut z: T = h_21 * h_32;

            for k in 0..=(p - 3) {
                let b: Vector<T> = Vector::new_column(vec![x, y, z]);
                let hr: General<T> = General::householder(&b, 0);

                //Determine the Householder reflector P with P^T [x; y; z]^T = αe1 ;
                {
                    let r: usize = k.max(1);

                    let temp = &hr.clone().transpose() * &self.get_slice(k, k + 2, r - 1, n - 1);
                    self = self.set_slice(&temp, k, r - 1);
                }

                {
                    let r: usize = p.min(k + 4);
                    let temp: General<T> = &self.get_slice(0, r - 1, k, k + 2) * &hr;
                    self = self.set_slice(&temp, 0, k);

                    let temp1: General<T> = &u.get_slice(0, n - 1, k, k + 2) * &hr;

                    u = u.set_slice(&temp1, 0, k);
                }

                let h_k2_k1 = self[[k + 1, k]];
                x = h_k2_k1;
                let h_k3_k1 = self[[k + 2, k]];
                y = h_k3_k1;
                if k < (p - 3) {
                    let h_k4_k1 = self[[k + 3, k]];
                    z = h_k4_k1;
                }
            }

            // Determine the Givens rotation P with P [x; y]T = αe1 ;
            let (c, s): (T, T) = General::givens_cosine_sine_pair(x, y);
            let g: General<T> = General::givens(2, 0, 1, c, s);

            {
                let temp: General<T> = &g * &self.get_slice(q - 1, p - 1, p - 3, n - 1);
                self = self.set_slice(&temp, q - 1, p - 3);
            }

            {
                let g_trans: General<T> = g.transpose();
                let temp: General<T> = &self.get_slice(0, p - 1, p - 2, p - 1) * &g_trans;
                self = self.set_slice(&temp, 0, p - 2);

                let u_slice = &self.get_slice(0, p - 1, p - 2, p - 1) * &g_trans;
                u = u.set_slice(&u_slice, 0, p - 2);
            }

            // check for convergence
            let m: T = self[[q - 1, q - 1]].abs();
            let n: T = self[[p - 1, p - 1]].abs();
            if self[[p - 1, q - 1]].abs() < epsilon * (m + n) {
                self[[p - 1, q - 1]] = T::zero();
                p -= 1;
            } else {
                let k: T = self[[q - 2, q - 2]].abs();
                let l: T = self[[q - 1, q - 1]].abs();
                if self[[p - 2, q - 2]].abs() < epsilon * (k + l) {
                    self[[p - 2, q - 2]] = T::zero();
                    p -= 2;
                }
            }
        }

        (u, self.matrix.into())
    }
}