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
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
#[cfg(feature = "serde-serialize-no-std")]
use serde::{Deserialize, Serialize};

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

use crate::geometry::Reflection;
use crate::linalg::householder;
use crate::num::Zero;
use std::mem::MaybeUninit;

/// The bidiagonalization of a general matrix.
#[cfg_attr(feature = "serde-serialize-no-std", derive(Serialize, Deserialize))]
#[cfg_attr(
    feature = "serde-serialize-no-std",
    serde(bound(serialize = "DimMinimum<R, C>: DimSub<U1>,
         DefaultAllocator: Allocator<R, C>             +
                           Allocator<DimMinimum<R, C>> +
                           Allocator<DimDiff<DimMinimum<R, C>, U1>>,
         OMatrix<T, R, C>: Serialize,
         OVector<T, DimMinimum<R, C>>: Serialize,
         OVector<T, DimDiff<DimMinimum<R, C>, U1>>: Serialize"))
)]
#[cfg_attr(
    feature = "serde-serialize-no-std",
    serde(bound(deserialize = "DimMinimum<R, C>: DimSub<U1>,
         DefaultAllocator: Allocator<R, C>             +
                           Allocator<DimMinimum<R, C>> +
                           Allocator<DimDiff<DimMinimum<R, C>, U1>>,
         OMatrix<T, R, C>: Deserialize<'de>,
         OVector<T, DimMinimum<R, C>>: Deserialize<'de>,
         OVector<T, DimDiff<DimMinimum<R, C>, U1>>: Deserialize<'de>"))
)]
#[derive(Clone, Debug)]
pub struct Bidiagonal<T: ComplexField, R: DimMin<C>, C: Dim>
where
    DimMinimum<R, C>: DimSub<U1>,
    DefaultAllocator:
        Allocator<R, C> + Allocator<DimMinimum<R, C>> + Allocator<DimDiff<DimMinimum<R, C>, U1>>,
{
    // TODO: perhaps we should pack the axes into different vectors so that axes for `v_t` are
    // contiguous. This prevents some useless copies.
    uv: OMatrix<T, R, C>,
    /// The diagonal elements of the decomposed matrix.
    diagonal: OVector<T, DimMinimum<R, C>>,
    /// The off-diagonal elements of the decomposed matrix.
    off_diagonal: OVector<T, DimDiff<DimMinimum<R, C>, U1>>,
    upper_diagonal: bool,
}

impl<T: ComplexField, R: DimMin<C>, C: Dim> Copy for Bidiagonal<T, R, C>
where
    DimMinimum<R, C>: DimSub<U1>,
    DefaultAllocator:
        Allocator<R, C> + Allocator<DimMinimum<R, C>> + Allocator<DimDiff<DimMinimum<R, C>, U1>>,
    OMatrix<T, R, C>: Copy,
    OVector<T, DimMinimum<R, C>>: Copy,
    OVector<T, DimDiff<DimMinimum<R, C>, U1>>: Copy,
{
}

impl<T: ComplexField, R: DimMin<C>, C: Dim> Bidiagonal<T, R, C>
where
    DimMinimum<R, C>: DimSub<U1>,
    DefaultAllocator: Allocator<R, C>
        + Allocator<C>
        + Allocator<R>
        + Allocator<DimMinimum<R, C>>
        + Allocator<DimDiff<DimMinimum<R, C>, U1>>,
{
    /// Computes the Bidiagonal decomposition using householder reflections.
    pub fn new(mut matrix: OMatrix<T, R, C>) -> Self {
        let (nrows, ncols) = matrix.shape_generic();
        let min_nrows_ncols = nrows.min(ncols);
        let dim = min_nrows_ncols.value();
        assert!(
            dim != 0,
            "Cannot compute the bidiagonalization of an empty matrix."
        );

        let mut diagonal = Matrix::uninit(min_nrows_ncols, Const::<1>);
        let mut off_diagonal = Matrix::uninit(min_nrows_ncols.sub(Const::<1>), Const::<1>);
        let mut axis_packed = Matrix::zeros_generic(ncols, Const::<1>);
        let mut work = Matrix::zeros_generic(nrows, Const::<1>);

        let upper_diagonal = nrows.value() >= ncols.value();
        if upper_diagonal {
            for ite in 0..dim - 1 {
                diagonal[ite] = MaybeUninit::new(householder::clear_column_unchecked(
                    &mut matrix,
                    ite,
                    0,
                    None,
                ));
                off_diagonal[ite] = MaybeUninit::new(householder::clear_row_unchecked(
                    &mut matrix,
                    &mut axis_packed,
                    &mut work,
                    ite,
                    1,
                ));
            }

            diagonal[dim - 1] = MaybeUninit::new(householder::clear_column_unchecked(
                &mut matrix,
                dim - 1,
                0,
                None,
            ));
        } else {
            for ite in 0..dim - 1 {
                diagonal[ite] = MaybeUninit::new(householder::clear_row_unchecked(
                    &mut matrix,
                    &mut axis_packed,
                    &mut work,
                    ite,
                    0,
                ));
                off_diagonal[ite] = MaybeUninit::new(householder::clear_column_unchecked(
                    &mut matrix,
                    ite,
                    1,
                    None,
                ));
            }

            diagonal[dim - 1] = MaybeUninit::new(householder::clear_row_unchecked(
                &mut matrix,
                &mut axis_packed,
                &mut work,
                dim - 1,
                0,
            ));
        }

        // Safety: diagonal and off_diagonal have been fully initialized.
        let (diagonal, off_diagonal) =
            unsafe { (diagonal.assume_init(), off_diagonal.assume_init()) };

        Bidiagonal {
            uv: matrix,
            diagonal,
            off_diagonal,
            upper_diagonal,
        }
    }

    /// Indicates whether this decomposition contains an upper-diagonal matrix.
    #[inline]
    #[must_use]
    pub fn is_upper_diagonal(&self) -> bool {
        self.upper_diagonal
    }

    #[inline]
    fn axis_shift(&self) -> (usize, usize) {
        if self.upper_diagonal {
            (0, 1)
        } else {
            (1, 0)
        }
    }

    /// Unpacks this decomposition into its three matrix factors `(U, D, V^t)`.
    ///
    /// The decomposed matrix `M` is equal to `U * D * V^t`.
    #[inline]
    pub fn unpack(
        self,
    ) -> (
        OMatrix<T, R, DimMinimum<R, C>>,
        OMatrix<T, DimMinimum<R, C>, DimMinimum<R, C>>,
        OMatrix<T, DimMinimum<R, C>, C>,
    )
    where
        DefaultAllocator: Allocator<DimMinimum<R, C>, DimMinimum<R, C>>
            + Allocator<R, DimMinimum<R, C>>
            + Allocator<DimMinimum<R, C>, C>,
    {
        // TODO: optimize by calling a reallocator.
        (self.u(), self.d(), self.v_t())
    }

    /// Retrieves the upper trapezoidal submatrix `R` of this decomposition.
    #[inline]
    #[must_use]
    pub fn d(&self) -> OMatrix<T, DimMinimum<R, C>, DimMinimum<R, C>>
    where
        DefaultAllocator: Allocator<DimMinimum<R, C>, DimMinimum<R, C>>,
    {
        let (nrows, ncols) = self.uv.shape_generic();

        let d = nrows.min(ncols);
        let mut res = OMatrix::identity_generic(d, d);
        res.set_partial_diagonal(
            self.diagonal
                .iter()
                .map(|e| T::from_real(e.clone().modulus())),
        );

        let start = self.axis_shift();
        res.view_mut(start, (d.value() - 1, d.value() - 1))
            .set_partial_diagonal(
                self.off_diagonal
                    .iter()
                    .map(|e| T::from_real(e.clone().modulus())),
            );
        res
    }

    /// Computes the orthogonal matrix `U` of this `U * D * V` decomposition.
    // TODO: code duplication with householder::assemble_q.
    // Except that we are returning a rectangular matrix here.
    #[must_use]
    pub fn u(&self) -> OMatrix<T, R, DimMinimum<R, C>>
    where
        DefaultAllocator: Allocator<R, DimMinimum<R, C>>,
    {
        let (nrows, ncols) = self.uv.shape_generic();

        let mut res = Matrix::identity_generic(nrows, nrows.min(ncols));
        let dim = self.diagonal.len();
        let shift = self.axis_shift().0;

        for i in (0..dim - shift).rev() {
            let axis = self.uv.view_range(i + shift.., i);

            // Sometimes, the axis might have a zero magnitude.
            if axis.norm_squared().is_zero() {
                continue;
            }
            let refl = Reflection::new(Unit::new_unchecked(axis), T::zero());

            let mut res_rows = res.view_range_mut(i + shift.., i..);

            let sign = if self.upper_diagonal {
                self.diagonal[i].clone().signum()
            } else {
                self.off_diagonal[i].clone().signum()
            };

            refl.reflect_with_sign(&mut res_rows, sign);
        }

        res
    }

    /// Computes the orthogonal matrix `V_t` of this `U * D * V_t` decomposition.
    #[must_use]
    pub fn v_t(&self) -> OMatrix<T, DimMinimum<R, C>, C>
    where
        DefaultAllocator: Allocator<DimMinimum<R, C>, C>,
    {
        let (nrows, ncols) = self.uv.shape_generic();
        let min_nrows_ncols = nrows.min(ncols);

        let mut res = Matrix::identity_generic(min_nrows_ncols, ncols);
        let mut work = Matrix::zeros_generic(min_nrows_ncols, Const::<1>);
        let mut axis_packed = Matrix::zeros_generic(ncols, Const::<1>);

        let shift = self.axis_shift().1;

        for i in (0..min_nrows_ncols.value() - shift).rev() {
            let axis = self.uv.view_range(i, i + shift..);
            let mut axis_packed = axis_packed.rows_range_mut(i + shift..);
            axis_packed.tr_copy_from(&axis);

            // Sometimes, the axis might have a zero magnitude.
            if axis_packed.norm_squared().is_zero() {
                continue;
            }
            let refl = Reflection::new(Unit::new_unchecked(axis_packed), T::zero());

            let mut res_rows = res.view_range_mut(i.., i + shift..);

            let sign = if self.upper_diagonal {
                self.off_diagonal[i].clone().signum()
            } else {
                self.diagonal[i].clone().signum()
            };

            refl.reflect_rows_with_sign(&mut res_rows, &mut work.rows_range_mut(i..), sign);
        }

        res
    }

    /// The diagonal part of this decomposed matrix.
    #[must_use]
    pub fn diagonal(&self) -> OVector<T::RealField, DimMinimum<R, C>>
    where
        DefaultAllocator: Allocator<DimMinimum<R, C>>,
    {
        self.diagonal.map(|e| e.modulus())
    }

    /// The off-diagonal part of this decomposed matrix.
    #[must_use]
    pub fn off_diagonal(&self) -> OVector<T::RealField, DimDiff<DimMinimum<R, C>, U1>>
    where
        DefaultAllocator: Allocator<DimDiff<DimMinimum<R, C>, U1>>,
    {
        self.off_diagonal.map(|e| e.modulus())
    }

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

// impl<T: ComplexField, D: DimMin<D, Output = D> + DimSub<Dyn>> Bidiagonal<T, D, D>
//     where DefaultAllocator: Allocator<D, D> +
//                             Allocator<D> {
//     /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
//     pub fn solve<R2: Dim, C2: Dim, S2>(&self, b: &Matrix<T, R2, C2, S2>) -> OMatrix<T, R2, C2>
//         where S2: StorageMut<T, R2, C2>,
//               ShapeConstraint: SameNumberOfRows<R2, D> {
//         let mut res = b.clone_owned();
//         self.solve_mut(&mut res);
//         res
//     }
//
//     /// Solves the linear system `self * x = b`, where `x` is the unknown to be determined.
//     pub fn solve_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<T, R2, C2, S2>)
//         where S2: StorageMut<T, R2, C2>,
//               ShapeConstraint: SameNumberOfRows<R2, D> {
//
//         assert_eq!(self.uv.nrows(), b.nrows(), "Bidiagonal solve matrix dimension mismatch.");
//         assert!(self.uv.is_square(), "Bidiagonal 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>)
//         where S2: StorageMut<T, R2, C2>,
//               ShapeConstraint: SameNumberOfRows<R2, D> {
//
//         let dim  = self.uv.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);
//                     coeff = *b.vget_unchecked(i) / diag;
//                     *b.vget_unchecked_mut(i) = coeff;
//                 }
//
//                 b.rows_range_mut(.. i).axpy(-coeff, &self.uv.view_range(.. i, i), T::one());
//             }
//         }
//     }
//
//     /// Computes the inverse of the decomposed matrix.
//     pub fn inverse(&self) -> OMatrix<T, D, D> {
//         assert!(self.uv.is_square(), "Bidiagonal inverse: unable to compute the inverse of a non-square matrix.");
//
//         // TODO: is there a less naive method ?
//         let (nrows, ncols) = self.uv.shape_generic();
//         let mut res = OMatrix::identity_generic(nrows, ncols);
//         self.solve_mut(&mut res);
//         res
//     }
//
//     // /// Computes the determinant of the decomposed matrix.
//     // pub fn determinant(&self) -> T {
//     //     let dim = self.uv.nrows();
//     //     assert!(self.uv.is_square(), "Bidiagonal 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()
//     // }
// }