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

#[cfg(feature = "serde-serialize-no-std")]
use serde::{Deserialize, Serialize};

use crate::allocator::Allocator;
use crate::base::{DefaultAllocator, OMatrix, OVector};
use crate::dimension::{Const, DimDiff, DimSub, U1};
use simba::scalar::ComplexField;

use crate::linalg::householder;
use crate::Matrix;
use std::mem::MaybeUninit;

/// Tridiagonalization of a symmetric matrix.
#[cfg_attr(feature = "serde-serialize-no-std", derive(Serialize, Deserialize))]
#[cfg_attr(
    feature = "serde-serialize-no-std",
    serde(bound(serialize = "DefaultAllocator: Allocator<T, D, D> +
                           Allocator<T, DimDiff<D, U1>>,
         OMatrix<T, D, D>: Serialize,
         OVector<T, DimDiff<D, U1>>: Serialize"))
)]
#[cfg_attr(
    feature = "serde-serialize-no-std",
    serde(bound(deserialize = "DefaultAllocator: Allocator<T, D, D> +
                           Allocator<T, DimDiff<D, U1>>,
         OMatrix<T, D, D>: Deserialize<'de>,
         OVector<T, DimDiff<D, U1>>: Deserialize<'de>"))
)]
#[derive(Clone, Debug)]
pub struct SymmetricTridiagonal<T: ComplexField, D: DimSub<U1>>
where
    DefaultAllocator: Allocator<T, D, D> + Allocator<T, DimDiff<D, U1>>,
{
    tri: OMatrix<T, D, D>,
    off_diagonal: OVector<T, DimDiff<D, U1>>,
}

impl<T: ComplexField, D: DimSub<U1>> Copy for SymmetricTridiagonal<T, D>
where
    DefaultAllocator: Allocator<T, D, D> + Allocator<T, DimDiff<D, U1>>,
    OMatrix<T, D, D>: Copy,
    OVector<T, DimDiff<D, U1>>: Copy,
{
}

impl<T: ComplexField, D: DimSub<U1>> SymmetricTridiagonal<T, D>
where
    DefaultAllocator: Allocator<T, D, D> + Allocator<T, DimDiff<D, U1>>,
{
    /// Computes the tridiagonalization of the symmetric matrix `m`.
    ///
    /// Only the lower-triangular part (including the diagonal) of `m` is read.
    pub fn new(mut m: OMatrix<T, D, D>) -> Self {
        let dim = m.shape_generic().0;

        assert!(
            m.is_square(),
            "Unable to compute the symmetric tridiagonal decomposition of a non-square matrix."
        );
        assert!(
            dim.value() != 0,
            "Unable to compute the symmetric tridiagonal decomposition of an empty matrix."
        );

        let mut off_diagonal = Matrix::uninit(dim.sub(Const::<1>), Const::<1>);
        let mut p = Matrix::zeros_generic(dim.sub(Const::<1>), Const::<1>);

        for i in 0..dim.value() - 1 {
            let mut m = m.rows_range_mut(i + 1..);
            let (mut axis, mut m) = m.columns_range_pair_mut(i, i + 1..);

            let (norm, not_zero) = householder::reflection_axis_mut(&mut axis);
            off_diagonal[i] = MaybeUninit::new(norm);

            if not_zero {
                let mut p = p.rows_range_mut(i..);

                p.hegemv(crate::convert(2.0), &m, &axis, T::zero());

                let dot = axis.dotc(&p);
                m.hegerc(-T::one(), &p, &axis, T::one());
                m.hegerc(-T::one(), &axis, &p, T::one());
                m.hegerc(dot * crate::convert(2.0), &axis, &axis, T::one());
            }
        }

        // Safety: off_diagonal has been fully initialized.
        let off_diagonal = unsafe { off_diagonal.assume_init() };
        Self {
            tri: m,
            off_diagonal,
        }
    }

    #[doc(hidden)]
    // For debugging.
    pub fn internal_tri(&self) -> &OMatrix<T, D, D> {
        &self.tri
    }

    /// Retrieve the orthogonal transformation, diagonal, and off diagonal elements of this
    /// decomposition.
    pub fn unpack(
        self,
    ) -> (
        OMatrix<T, D, D>,
        OVector<T::RealField, D>,
        OVector<T::RealField, DimDiff<D, U1>>,
    )
    where
        DefaultAllocator: Allocator<T::RealField, D> + Allocator<T::RealField, DimDiff<D, U1>>,
    {
        let diag = self.diagonal();
        let q = self.q();

        (q, diag, self.off_diagonal.map(T::modulus))
    }

    /// Retrieve the diagonal, and off diagonal elements of this decomposition.
    pub fn unpack_tridiagonal(
        self,
    ) -> (
        OVector<T::RealField, D>,
        OVector<T::RealField, DimDiff<D, U1>>,
    )
    where
        DefaultAllocator: Allocator<T::RealField, D> + Allocator<T::RealField, DimDiff<D, U1>>,
    {
        (self.diagonal(), self.off_diagonal.map(T::modulus))
    }

    /// The diagonal components of this decomposition.
    #[must_use]
    pub fn diagonal(&self) -> OVector<T::RealField, D>
    where
        DefaultAllocator: Allocator<T::RealField, D>,
    {
        self.tri.map_diagonal(|e| e.real())
    }

    /// The off-diagonal components of this decomposition.
    #[must_use]
    pub fn off_diagonal(&self) -> OVector<T::RealField, DimDiff<D, U1>>
    where
        DefaultAllocator: Allocator<T::RealField, DimDiff<D, U1>>,
    {
        self.off_diagonal.map(T::modulus)
    }

    /// Computes the orthogonal matrix `Q` of this decomposition.
    #[must_use]
    pub fn q(&self) -> OMatrix<T, D, D> {
        householder::assemble_q(&self.tri, self.off_diagonal.as_slice())
    }

    /// Recomputes the original symmetric matrix.
    pub fn recompose(mut self) -> OMatrix<T, D, D> {
        let q = self.q();
        self.tri.fill_lower_triangle(T::zero(), 2);
        self.tri.fill_upper_triangle(T::zero(), 2);

        for i in 0..self.off_diagonal.len() {
            let val = T::from_real(self.off_diagonal[i].clone().modulus());
            self.tri[(i + 1, i)] = val.clone();
            self.tri[(i, i + 1)] = val;
        }

        &q * self.tri * q.adjoint()
    }
}