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

//! Tridiagonal decomposition of a symmetric matrix

use ndarray::{
    linalg::{general_mat_mul, general_mat_vec_mul},
    s, Array1, Array2, ArrayBase, Axis, DataMut, Ix2, NdFloat, RawDataClone,
};

use crate::{
    check_square, householder,
    triangular::{IntoTriangular, UPLO},
    LinalgError, Result,
};

/// Tridiagonal decomposition of a non-empty symmetric matrix
pub trait SymmetricTridiagonal {
    type Decomp;

    /// Calculate the tridiagonal decomposition of a symmetric matrix, consisting of symmetric
    /// tridiagonal matrix `T` and orthogonal matrix `Q`, such that `Q * T * Q.t` yields the
    /// original matrix.
    fn sym_tridiagonal(self) -> Result<Self::Decomp>;
}

impl<S, A> SymmetricTridiagonal for ArrayBase<S, Ix2>
where
    A: NdFloat,
    S: DataMut<Elem = A>,
{
    type Decomp = TridiagonalDecomp<A, S>;

    fn sym_tridiagonal(mut self) -> Result<Self::Decomp> {
        let n = check_square(&self)?;
        if n < 1 {
            return Err(LinalgError::EmptyMatrix);
        }

        let mut off_diagonal = Array1::zeros(n - 1); // TODO can be uninit
        let mut p = Array1::zeros(n - 1);

        for i in 0..n - 1 {
            let mut m = self.slice_mut(s![i + 1.., ..]);
            let (mut axis, mut m) = m.multi_slice_mut((s![.., i], s![.., i + 1..]));

            let norm = householder::reflection_axis_mut(&mut axis);
            *off_diagonal.get_mut(i).unwrap() = norm.unwrap_or_else(A::zero);

            if norm.is_some() {
                let mut p = p.slice_mut(s![i..]);
                general_mat_vec_mul(A::from(2.0f64).unwrap(), &m, &axis, A::zero(), &mut p);
                let dot = axis.dot(&p);

                let p_row = p.view().insert_axis(Axis(0));
                let p_col = p.view().insert_axis(Axis(1));
                let ax_row = axis.view().insert_axis(Axis(0));
                let ax_col = axis.view().insert_axis(Axis(1));
                general_mat_mul(-A::one(), &p_col, &ax_row, A::one(), &mut m);
                general_mat_mul(-A::one(), &ax_col, &p_row, A::one(), &mut m);
                general_mat_mul(dot + dot, &ax_col, &ax_row, A::one(), &mut m);
            }
        }

        Ok(TridiagonalDecomp {
            diag_matrix: self,
            off_diagonal,
        })
    }
}

/// Full tridiagonal decomposition, containing the symmetric tridiagonal matrix `T`
#[derive(Debug)]
pub struct TridiagonalDecomp<A, S: DataMut<Elem = A>> {
    // This matrix is only useful for its diagonal, which is the diagonal of the tridiagonal matrix
    // Guaranteed to be square matrix
    diag_matrix: ArrayBase<S, Ix2>,
    // The off-diagonal elements of the tridiagonal matrix
    off_diagonal: Array1<A>,
}

impl<A: Clone, S: DataMut<Elem = A> + RawDataClone> Clone for TridiagonalDecomp<A, S> {
    fn clone(&self) -> Self {
        Self {
            diag_matrix: self.diag_matrix.clone(),
            off_diagonal: self.off_diagonal.clone(),
        }
    }
}

impl<A: NdFloat, S: DataMut<Elem = A>> TridiagonalDecomp<A, S> {
    /// Construct the orthogonal matrix `Q`, where `Q * T * Q.t` results in the original matrix
    pub fn generate_q(&self) -> Array2<A> {
        householder::assemble_q(&self.diag_matrix, 1, |i| self.off_diagonal[i])
    }

    /// Return the diagonal elements and off-diagonal elements of the tridiagonal matrix as 1D
    /// arrays
    pub fn into_diagonals(self) -> (Array1<A>, Array1<A>) {
        (
            self.diag_matrix.diag().to_owned(),
            self.off_diagonal.mapv_into(A::abs),
        )
    }

    /// Return the full tridiagonal matrix `T`
    pub fn into_tridiag_matrix(mut self) -> ArrayBase<S, Ix2> {
        self.diag_matrix.triangular_inplace(UPLO::Upper).unwrap();
        self.diag_matrix.triangular_inplace(UPLO::Lower).unwrap();
        for (i, off) in self.off_diagonal.into_iter().enumerate() {
            let off = off.abs();
            self.diag_matrix[(i + 1, i)] = off;
            self.diag_matrix[(i, i + 1)] = off;
        }
        self.diag_matrix
    }
}

#[cfg(test)]
mod tests {
    use approx::assert_abs_diff_eq;
    use ndarray::array;

    use super::*;

    #[test]
    fn sym_tridiagonal() {
        let arr = array![
            [4.0f64, 1., -2., 2.],
            [1., 2., 0., 1.],
            [-2., 0., 3., -2.],
            [2., 1., -2., -1.]
        ];

        let decomp = arr.clone().sym_tridiagonal().unwrap();
        let (diag, offdiag) = decomp.into_diagonals();
        assert_abs_diff_eq!(
            diag,
            array![4., 10. / 3., -33. / 25., 149. / 75.],
            epsilon = 1e-5
        );
        assert_abs_diff_eq!(offdiag, array![3., 5. / 3., 68. / 75.], epsilon = 1e-5);

        let decomp = arr.clone().sym_tridiagonal().unwrap();
        let q = decomp.generate_q();
        let tri = decomp.into_tridiag_matrix();
        assert_abs_diff_eq!(q.dot(&tri).dot(&q.t()), arr, epsilon = 1e-9);
        // Q must be orthogonal
        assert_abs_diff_eq!(q.dot(&q.t()), Array2::eye(4), epsilon = 1e-9);

        let one = array![[1.1f64]].sym_tridiagonal().unwrap();
        let (one_diag, one_offdiag) = one.into_diagonals();
        assert_abs_diff_eq!(one_diag, array![1.1f64]);
        assert!(one_offdiag.is_empty());
    }

    #[test]
    fn sym_tridiag_error() {
        assert!(matches!(
            array![[1., 2., 3.], [5., 4., 3.0f64]].sym_tridiagonal(),
            Err(LinalgError::NotSquare { rows: 2, cols: 3 })
        ));
        assert!(matches!(
            Array2::<f64>::zeros((0, 0)).sym_tridiagonal(),
            Err(LinalgError::EmptyMatrix)
        ));
    }
}