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
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
use num::Zero;
#[cfg(feature = "serde-serialize-no-std")]
use serde::{Deserialize, Serialize};

use crate::allocator::{Allocator, Reallocator};
use crate::base::{DefaultAllocator, Matrix, OMatrix, OVector, Unit};
use crate::constraint::{SameNumberOfRows, ShapeConstraint};
use crate::dimension::{Const, Dim, DimMin, DimMinimum};
use crate::storage::{Storage, StorageMut};
use simba::scalar::ComplexField;

use crate::geometry::Reflection;
use crate::linalg::householder;

/// The QR decomposition of a general matrix.
#[cfg_attr(feature = "serde-serialize-no-std", derive(Serialize, Deserialize))]
#[cfg_attr(
    feature = "serde-serialize-no-std",
    serde(bound(serialize = "DefaultAllocator: Allocator<T, R, C> +
                           Allocator<T, DimMinimum<R, C>>,
         OMatrix<T, R, C>: Serialize,
         OVector<T, DimMinimum<R, C>>: Serialize"))
)]
#[cfg_attr(
    feature = "serde-serialize-no-std",
    serde(bound(deserialize = "DefaultAllocator: Allocator<T, R, C> +
                           Allocator<T, DimMinimum<R, C>>,
         OMatrix<T, R, C>: Deserialize<'de>,
         OVector<T, DimMinimum<R, C>>: Deserialize<'de>"))
)]
#[derive(Clone, Debug)]
pub struct QR<T: ComplexField, R: DimMin<C>, C: Dim>
where
    DefaultAllocator: Allocator<T, R, C> + Allocator<T, DimMinimum<R, C>>,
{
    qr: OMatrix<T, R, C>,
    diag: OVector<T, DimMinimum<R, C>>,
}

impl<T: ComplexField, R: DimMin<C>, C: Dim> Copy for QR<T, R, C>
where
    DefaultAllocator: Allocator<T, R, C> + Allocator<T, DimMinimum<R, C>>,
    OMatrix<T, R, C>: Copy,
    OVector<T, DimMinimum<R, C>>: Copy,
{
}

impl<T: ComplexField, R: DimMin<C>, C: Dim> QR<T, R, C>
where
    DefaultAllocator: Allocator<T, R, C> + Allocator<T, R> + Allocator<T, DimMinimum<R, C>>,
{
    /// Computes the QR decomposition using householder reflections.
    pub fn new(mut matrix: OMatrix<T, R, C>) -> Self {
        let (nrows, ncols) = matrix.data.shape();
        let min_nrows_ncols = nrows.min(ncols);

        let mut diag =
            unsafe { crate::unimplemented_or_uninitialized_generic!(min_nrows_ncols, Const::<1>) };

        if min_nrows_ncols.value() == 0 {
            return QR { qr: matrix, diag };
        }

        for i in 0..min_nrows_ncols.value() {
            householder::clear_column_unchecked(&mut matrix, &mut diag[i], i, 0, None);
        }

        QR { qr: matrix, diag }
    }

    /// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
    #[inline]
    pub fn r(&self) -> OMatrix<T, DimMinimum<R, C>, C>
    where
        DefaultAllocator: Allocator<T, DimMinimum<R, C>, C>,
    {
        let (nrows, ncols) = self.qr.data.shape();
        let mut res = self.qr.rows_generic(0, nrows.min(ncols)).upper_triangle();
        res.set_partial_diagonal(self.diag.iter().map(|e| T::from_real(e.modulus())));
        res
    }

    /// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
    ///
    /// This is usually faster than `r` but consumes `self`.
    #[inline]
    pub fn unpack_r(self) -> OMatrix<T, DimMinimum<R, C>, C>
    where
        DefaultAllocator: Reallocator<T, R, C, DimMinimum<R, C>, C>,
    {
        let (nrows, ncols) = self.qr.data.shape();
        let mut res = self.qr.resize_generic(nrows.min(ncols), ncols, T::zero());
        res.fill_lower_triangle(T::zero(), 1);
        res.set_partial_diagonal(self.diag.iter().map(|e| T::from_real(e.modulus())));
        res
    }

    /// Computes the orthogonal matrix `Q` of this decomposition.
    pub fn q(&self) -> OMatrix<T, R, DimMinimum<R, C>>
    where
        DefaultAllocator: Allocator<T, R, DimMinimum<R, C>>,
    {
        let (nrows, ncols) = self.qr.data.shape();

        // NOTE: we could build the identity matrix and call q_mul on it.
        // Instead we don't so that we take in account the matrix sparseness.
        let mut res = Matrix::identity_generic(nrows, nrows.min(ncols));
        let dim = self.diag.len();

        for i in (0..dim).rev() {
            let axis = self.qr.slice_range(i.., i);
            // TODO: sometimes, the axis might have a zero magnitude.
            let refl = Reflection::new(Unit::new_unchecked(axis), T::zero());

            let mut res_rows = res.slice_range_mut(i.., i..);
            refl.reflect_with_sign(&mut res_rows, self.diag[i].signum());
        }

        res
    }

    /// Unpacks this decomposition into its two matrix factors.
    pub fn unpack(
        self,
    ) -> (
        OMatrix<T, R, DimMinimum<R, C>>,
        OMatrix<T, DimMinimum<R, C>, C>,
    )
    where
        DimMinimum<R, C>: DimMin<C, Output = DimMinimum<R, C>>,
        DefaultAllocator:
            Allocator<T, R, DimMinimum<R, C>> + Reallocator<T, R, C, DimMinimum<R, C>, C>,
    {
        (self.q(), self.unpack_r())
    }

    #[doc(hidden)]
    pub fn qr_internal(&self) -> &OMatrix<T, R, C> {
        &self.qr
    }

    /// Multiplies the provided matrix by the transpose of the `Q` matrix of this decomposition.
    pub fn q_tr_mul<R2: Dim, C2: Dim, S2>(&self, rhs: &mut Matrix<T, R2, C2, S2>)
    // TODO: do we need a static constraint on the number of rows of rhs?
    where
        S2: StorageMut<T, R2, C2>,
    {
        let dim = self.diag.len();

        for i in 0..dim {
            let axis = self.qr.slice_range(i.., i);
            let refl = Reflection::new(Unit::new_unchecked(axis), T::zero());

            let mut rhs_rows = rhs.rows_range_mut(i..);
            refl.reflect_with_sign(&mut rhs_rows, self.diag[i].signum().conjugate());
        }
    }
}

impl<T: ComplexField, D: DimMin<D, Output = D>> QR<T, D, D>
where
    DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
{
    /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
    ///
    /// Returns `None` if `self` is not invertible.
    pub fn solve<R2: Dim, C2: Dim, S2>(
        &self,
        b: &Matrix<T, R2, C2, S2>,
    ) -> Option<OMatrix<T, R2, C2>>
    where
        S2: Storage<T, R2, C2>,
        ShapeConstraint: SameNumberOfRows<R2, D>,
        DefaultAllocator: Allocator<T, R2, C2>,
    {
        let mut res = b.clone_owned();

        if self.solve_mut(&mut res) {
            Some(res)
        } else {
            None
        }
    }

    /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
    ///
    /// If the decomposed matrix is not invertible, this returns `false` and its input `b` is
    /// overwritten with garbage.
    pub fn solve_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<T, R2, C2, S2>) -> bool
    where
        S2: StorageMut<T, R2, C2>,
        ShapeConstraint: SameNumberOfRows<R2, D>,
    {
        assert_eq!(
            self.qr.nrows(),
            b.nrows(),
            "QR solve matrix dimension mismatch."
        );
        assert!(
            self.qr.is_square(),
            "QR solve: unable to solve a non-square system."
        );

        self.q_tr_mul(b);
        self.solve_upper_triangular_mut(b)
    }

    // TODO: duplicate code from the `solve` module.
    fn solve_upper_triangular_mut<R2: Dim, C2: Dim, S2>(
        &self,
        b: &mut Matrix<T, R2, C2, S2>,
    ) -> bool
    where
        S2: StorageMut<T, R2, C2>,
        ShapeConstraint: SameNumberOfRows<R2, D>,
    {
        let dim = self.qr.nrows();

        for k in 0..b.ncols() {
            let mut b = b.column_mut(k);
            for i in (0..dim).rev() {
                let coeff;

                unsafe {
                    let diag = self.diag.vget_unchecked(i).modulus();

                    if diag.is_zero() {
                        return false;
                    }

                    coeff = b.vget_unchecked(i).unscale(diag);
                    *b.vget_unchecked_mut(i) = coeff;
                }

                b.rows_range_mut(..i)
                    .axpy(-coeff, &self.qr.slice_range(..i, i), T::one());
            }
        }

        true
    }

    /// Computes the inverse of the decomposed matrix.
    ///
    /// Returns `None` if the decomposed matrix is not invertible.
    pub fn try_inverse(&self) -> Option<OMatrix<T, D, D>> {
        assert!(
            self.qr.is_square(),
            "QR inverse: unable to compute the inverse of a non-square matrix."
        );

        // TODO: is there a less naive method ?
        let (nrows, ncols) = self.qr.data.shape();
        let mut res = OMatrix::identity_generic(nrows, ncols);

        if self.solve_mut(&mut res) {
            Some(res)
        } else {
            None
        }
    }

    /// Indicates if the decomposed matrix is invertible.
    pub fn is_invertible(&self) -> bool {
        assert!(
            self.qr.is_square(),
            "QR: unable to test the invertibility of a non-square matrix."
        );

        for i in 0..self.diag.len() {
            if self.diag[i].is_zero() {
                return false;
            }
        }

        true
    }

    // /// Computes the determinant of the decomposed matrix.
    // pub fn determinant(&self) -> T {
    //     let dim = self.qr.nrows();
    //     assert!(self.qr.is_square(), "QR determinant: unable to compute the determinant of a non-square matrix.");

    //     let mut res = T::one();
    //     for i in 0 .. dim {
    //         res *= unsafe { *self.diag.vget_unchecked(i) };
    //     }

    //     res self.q_determinant()
    // }
}