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
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
#[cfg(feature = "serde-serialize")]
use serde::{Deserialize, Serialize};

use approx::AbsDiffEq;
use num_complex::Complex as NumComplex;
use simba::scalar::{ComplexField, RealField};
use std::cmp;

use crate::allocator::Allocator;
use crate::base::dimension::{Dim, DimDiff, DimSub, Dynamic, U1, U2, U3};
use crate::base::storage::Storage;
use crate::base::{DefaultAllocator, MatrixN, SquareMatrix, Unit, Vector2, Vector3, VectorN};

use crate::geometry::Reflection;
use crate::linalg::givens::GivensRotation;
use crate::linalg::householder;
use crate::linalg::Hessenberg;

/// Schur decomposition of a square matrix.
///
/// If this is a real matrix, this will be a RealField Schur decomposition.
#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
#[cfg_attr(
    feature = "serde-serialize",
    serde(bound(serialize = "DefaultAllocator: Allocator<N, D, D>,
         MatrixN<N, D>: Serialize"))
)]
#[cfg_attr(
    feature = "serde-serialize",
    serde(bound(deserialize = "DefaultAllocator: Allocator<N, D, D>,
         MatrixN<N, D>: Deserialize<'de>"))
)]
#[derive(Clone, Debug)]
pub struct Schur<N: ComplexField, D: Dim>
where
    DefaultAllocator: Allocator<N, D, D>,
{
    q: MatrixN<N, D>,
    t: MatrixN<N, D>,
}

impl<N: ComplexField, D: Dim> Copy for Schur<N, D>
where
    DefaultAllocator: Allocator<N, D, D>,
    MatrixN<N, D>: Copy,
{
}

impl<N: ComplexField, D: Dim> Schur<N, D>
where
    D: DimSub<U1>, // For Hessenberg.
    DefaultAllocator: Allocator<N, D, DimDiff<D, U1>>
        + Allocator<N, DimDiff<D, U1>>
        + Allocator<N, D, D>
        + Allocator<N, D>,
{
    /// Computes the Schur decomposition of a square matrix.
    pub fn new(m: MatrixN<N, D>) -> Self {
        Self::try_new(m, N::RealField::default_epsilon(), 0).unwrap()
    }

    /// Attempts to compute the Schur decomposition of a square matrix.
    ///
    /// If only eigenvalues are needed, it is more efficient to call the matrix method
    /// `.eigenvalues()` instead.
    ///
    /// # Arguments
    ///
    /// * `eps`       − tolerance used to determine when a value converged to 0.
    /// * `max_niter` − maximum total number of iterations performed by the algorithm. If this
    /// number of iteration is exceeded, `None` is returned. If `niter == 0`, then the algorithm
    /// continues indefinitely until convergence.
    pub fn try_new(m: MatrixN<N, D>, eps: N::RealField, max_niter: usize) -> Option<Self> {
        let mut work =
            unsafe { crate::unimplemented_or_uninitialized_generic!(m.data.shape().0, U1) };

        Self::do_decompose(m, &mut work, eps, max_niter, true)
            .map(|(q, t)| Schur { q: q.unwrap(), t })
    }

    fn do_decompose(
        mut m: MatrixN<N, D>,
        work: &mut VectorN<N, D>,
        eps: N::RealField,
        max_niter: usize,
        compute_q: bool,
    ) -> Option<(Option<MatrixN<N, D>>, MatrixN<N, D>)> {
        assert!(
            m.is_square(),
            "Unable to compute the eigenvectors and eigenvalues of a non-square matrix."
        );

        let dim = m.data.shape().0;

        // Specialization would make this easier.
        if dim.value() == 0 {
            let vecs = Some(MatrixN::from_element_generic(dim, dim, N::zero()));
            let vals = MatrixN::from_element_generic(dim, dim, N::zero());
            return Some((vecs, vals));
        } else if dim.value() == 1 {
            if compute_q {
                let q = MatrixN::from_element_generic(dim, dim, N::one());
                return Some((Some(q), m));
            } else {
                return Some((None, m));
            }
        } else if dim.value() == 2 {
            return decompose_2x2(m, compute_q);
        }

        let amax_m = m.camax();
        m.unscale_mut(amax_m);

        let hess = Hessenberg::new_with_workspace(m, work);
        let mut q;
        let mut t;

        if compute_q {
            // TODO: could we work without unpacking? Using only the internal representation of
            // hessenberg decomposition.
            let (vecs, vals) = hess.unpack();
            q = Some(vecs);
            t = vals;
        } else {
            q = None;
            t = hess.unpack_h()
        }

        // Implicit double-shift QR method.
        let mut niter = 0;
        let (mut start, mut end) = Self::delimit_subproblem(&mut t, eps, dim.value() - 1);

        while end != start {
            let subdim = end - start + 1;

            if subdim > 2 {
                let m = end - 1;
                let n = end;

                let h11 = t[(start, start)];
                let h12 = t[(start, start + 1)];
                let h21 = t[(start + 1, start)];
                let h22 = t[(start + 1, start + 1)];
                let h32 = t[(start + 2, start + 1)];

                let hnn = t[(n, n)];
                let hmm = t[(m, m)];
                let hnm = t[(n, m)];
                let hmn = t[(m, n)];

                let tra = hnn + hmm;
                let det = hnn * hmm - hnm * hmn;

                let mut axis = Vector3::new(
                    h11 * h11 + h12 * h21 - tra * h11 + det,
                    h21 * (h11 + h22 - tra),
                    h21 * h32,
                );

                for k in start..n - 1 {
                    let (norm, not_zero) = householder::reflection_axis_mut(&mut axis);

                    if not_zero {
                        if k > start {
                            t[(k, k - 1)] = norm;
                            t[(k + 1, k - 1)] = N::zero();
                            t[(k + 2, k - 1)] = N::zero();
                        }

                        let refl = Reflection::new(Unit::new_unchecked(axis), N::zero());

                        {
                            let krows = cmp::min(k + 4, end + 1);
                            let mut work = work.rows_mut(0, krows);
                            refl.reflect(
                                &mut t
                                    .generic_slice_mut((k, k), (U3, Dynamic::new(dim.value() - k))),
                            );
                            refl.reflect_rows(
                                &mut t.generic_slice_mut((0, k), (Dynamic::new(krows), U3)),
                                &mut work,
                            );
                        }

                        if let Some(ref mut q) = q {
                            refl.reflect_rows(&mut q.generic_slice_mut((0, k), (dim, U3)), work);
                        }
                    }

                    axis.x = t[(k + 1, k)];
                    axis.y = t[(k + 2, k)];

                    if k < n - 2 {
                        axis.z = t[(k + 3, k)];
                    }
                }

                let mut axis = Vector2::new(axis.x, axis.y);
                let (norm, not_zero) = householder::reflection_axis_mut(&mut axis);

                if not_zero {
                    let refl = Reflection::new(Unit::new_unchecked(axis), N::zero());

                    t[(m, m - 1)] = norm;
                    t[(n, m - 1)] = N::zero();

                    {
                        let mut work = work.rows_mut(0, end + 1);
                        refl.reflect(
                            &mut t.generic_slice_mut((m, m), (U2, Dynamic::new(dim.value() - m))),
                        );
                        refl.reflect_rows(
                            &mut t.generic_slice_mut((0, m), (Dynamic::new(end + 1), U2)),
                            &mut work,
                        );
                    }

                    if let Some(ref mut q) = q {
                        refl.reflect_rows(&mut q.generic_slice_mut((0, m), (dim, U2)), work);
                    }
                }
            } else {
                // Decouple the 2x2 block if it has real eigenvalues.
                if let Some(rot) = compute_2x2_basis(&t.fixed_slice::<U2, U2>(start, start)) {
                    let inv_rot = rot.inverse();
                    inv_rot.rotate(&mut t.generic_slice_mut(
                        (start, start),
                        (U2, Dynamic::new(dim.value() - start)),
                    ));
                    rot.rotate_rows(
                        &mut t.generic_slice_mut((0, start), (Dynamic::new(end + 1), U2)),
                    );
                    t[(end, start)] = N::zero();

                    if let Some(ref mut q) = q {
                        rot.rotate_rows(&mut q.generic_slice_mut((0, start), (dim, U2)));
                    }
                }

                // Check if we reached the beginning of the matrix.
                if end > 2 {
                    end -= 2;
                } else {
                    break;
                }
            }

            let sub = Self::delimit_subproblem(&mut t, eps, end);

            start = sub.0;
            end = sub.1;

            niter += 1;
            if niter == max_niter {
                return None;
            }
        }

        t.scale_mut(amax_m);

        Some((q, t))
    }

    /// Computes the eigenvalues of the decomposed matrix.
    fn do_eigenvalues(t: &MatrixN<N, D>, out: &mut VectorN<N, D>) -> bool {
        let dim = t.nrows();
        let mut m = 0;

        while m < dim - 1 {
            let n = m + 1;

            if t[(n, m)].is_zero() {
                out[m] = t[(m, m)];
                m += 1;
            } else {
                // Complex eigenvalue.
                return false;
            }
        }

        if m == dim - 1 {
            out[m] = t[(m, m)];
        }

        true
    }

    /// Computes the complex eigenvalues of the decomposed matrix.
    fn do_complex_eigenvalues(t: &MatrixN<N, D>, out: &mut VectorN<NumComplex<N>, D>)
    where
        N: RealField,
        DefaultAllocator: Allocator<NumComplex<N>, D>,
    {
        let dim = t.nrows();
        let mut m = 0;

        while m < dim - 1 {
            let n = m + 1;

            if t[(n, m)].is_zero() {
                out[m] = NumComplex::new(t[(m, m)], N::zero());
                m += 1;
            } else {
                // Solve the 2x2 eigenvalue subproblem.
                let hmm = t[(m, m)];
                let hnm = t[(n, m)];
                let hmn = t[(m, n)];
                let hnn = t[(n, n)];

                // NOTE: use the same algorithm as in compute_2x2_eigvals.
                let val = (hmm - hnn) * crate::convert(0.5);
                let discr = hnm * hmn + val * val;

                // All 2x2 blocks have negative discriminant because we already decoupled those
                // with positive eigenvalues.
                let sqrt_discr = NumComplex::new(N::zero(), (-discr).sqrt());

                let half_tra = (hnn + hmm) * crate::convert(0.5);
                out[m] = NumComplex::new(half_tra, N::zero()) + sqrt_discr;
                out[m + 1] = NumComplex::new(half_tra, N::zero()) - sqrt_discr;

                m += 2;
            }
        }

        if m == dim - 1 {
            out[m] = NumComplex::new(t[(m, m)], N::zero());
        }
    }

    fn delimit_subproblem(t: &mut MatrixN<N, D>, eps: N::RealField, end: usize) -> (usize, usize)
    where
        D: DimSub<U1>,
        DefaultAllocator: Allocator<N, DimDiff<D, U1>>,
    {
        let mut n = end;

        while n > 0 {
            let m = n - 1;

            if t[(n, m)].norm1() <= eps * (t[(n, n)].norm1() + t[(m, m)].norm1()) {
                t[(n, m)] = N::zero();
            } else {
                break;
            }

            n -= 1;
        }

        if n == 0 {
            return (0, 0);
        }

        let mut new_start = n - 1;
        while new_start > 0 {
            let m = new_start - 1;

            let off_diag = t[(new_start, m)];
            if off_diag.is_zero()
                || off_diag.norm1() <= eps * (t[(new_start, new_start)].norm1() + t[(m, m)].norm1())
            {
                t[(new_start, m)] = N::zero();
                break;
            }

            new_start -= 1;
        }

        (new_start, n)
    }

    /// Retrieves the unitary matrix `Q` and the upper-quasitriangular matrix `T` such that the
    /// decomposed matrix equals `Q * T * Q.transpose()`.
    pub fn unpack(self) -> (MatrixN<N, D>, MatrixN<N, D>) {
        (self.q, self.t)
    }

    /// Computes the real eigenvalues of the decomposed matrix.
    ///
    /// Return `None` if some eigenvalues are complex.
    pub fn eigenvalues(&self) -> Option<VectorN<N, D>> {
        let mut out =
            unsafe { crate::unimplemented_or_uninitialized_generic!(self.t.data.shape().0, U1) };
        if Self::do_eigenvalues(&self.t, &mut out) {
            Some(out)
        } else {
            None
        }
    }

    /// Computes the complex eigenvalues of the decomposed matrix.
    pub fn complex_eigenvalues(&self) -> VectorN<NumComplex<N>, D>
    where
        N: RealField,
        DefaultAllocator: Allocator<NumComplex<N>, D>,
    {
        let mut out =
            unsafe { crate::unimplemented_or_uninitialized_generic!(self.t.data.shape().0, U1) };
        Self::do_complex_eigenvalues(&self.t, &mut out);
        out
    }
}

fn decompose_2x2<N: ComplexField, D: Dim>(
    mut m: MatrixN<N, D>,
    compute_q: bool,
) -> Option<(Option<MatrixN<N, D>>, MatrixN<N, D>)>
where
    DefaultAllocator: Allocator<N, D, D>,
{
    let dim = m.data.shape().0;
    let mut q = None;
    match compute_2x2_basis(&m.fixed_slice::<U2, U2>(0, 0)) {
        Some(rot) => {
            let mut m = m.fixed_slice_mut::<U2, U2>(0, 0);
            let inv_rot = rot.inverse();
            inv_rot.rotate(&mut m);
            rot.rotate_rows(&mut m);
            m[(1, 0)] = N::zero();

            if compute_q {
                // XXX: we have to build the matrix manually because
                // rot.to_rotation_matrix().unwrap() causes an ICE.
                let c = N::from_real(rot.c());
                q = Some(MatrixN::from_column_slice_generic(
                    dim,
                    dim,
                    &[c, rot.s(), -rot.s().conjugate(), c],
                ));
            }
        }
        None => {
            if compute_q {
                q = Some(MatrixN::identity_generic(dim, dim));
            }
        }
    };

    Some((q, m))
}

fn compute_2x2_eigvals<N: ComplexField, S: Storage<N, U2, U2>>(
    m: &SquareMatrix<N, U2, S>,
) -> Option<(N, N)> {
    // Solve the 2x2 eigenvalue subproblem.
    let h00 = m[(0, 0)];
    let h10 = m[(1, 0)];
    let h01 = m[(0, 1)];
    let h11 = m[(1, 1)];

    // NOTE: this discriminant computation is more stable than the
    // one based on the trace and determinant: 0.25 * tra * tra - det
    // because it ensures positiveness for symmetric matrices.
    let val = (h00 - h11) * crate::convert(0.5);
    let discr = h10 * h01 + val * val;

    discr.try_sqrt().map(|sqrt_discr| {
        let half_tra = (h00 + h11) * crate::convert(0.5);
        (half_tra + sqrt_discr, half_tra - sqrt_discr)
    })
}

// Computes the 2x2 transformation that upper-triangulates a 2x2 matrix with real eigenvalues.
/// Computes the singular vectors for a 2x2 matrix.
///
/// Returns `None` if the matrix has complex eigenvalues, or is upper-triangular. In both case,
/// the basis is the identity.
fn compute_2x2_basis<N: ComplexField, S: Storage<N, U2, U2>>(
    m: &SquareMatrix<N, U2, S>,
) -> Option<GivensRotation<N>> {
    let h10 = m[(1, 0)];

    if h10.is_zero() {
        return None;
    }

    if let Some((eigval1, eigval2)) = compute_2x2_eigvals(m) {
        let x1 = eigval1 - m[(1, 1)];
        let x2 = eigval2 - m[(1, 1)];

        // NOTE: Choose the one that yields a larger x component.
        // This is necessary for numerical stability of the normalization of the complex
        // number.
        if x1.norm1() > x2.norm1() {
            Some(GivensRotation::new(x1, h10).0)
        } else {
            Some(GivensRotation::new(x2, h10).0)
        }
    } else {
        None
    }
}

impl<N: ComplexField, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S>
where
    D: DimSub<U1>, // For Hessenberg.
    DefaultAllocator: Allocator<N, D, DimDiff<D, U1>>
        + Allocator<N, DimDiff<D, U1>>
        + Allocator<N, D, D>
        + Allocator<N, D>,
{
    /// Computes the eigenvalues of this matrix.
    pub fn eigenvalues(&self) -> Option<VectorN<N, D>> {
        assert!(
            self.is_square(),
            "Unable to compute eigenvalues of a non-square matrix."
        );

        let mut work =
            unsafe { crate::unimplemented_or_uninitialized_generic!(self.data.shape().0, U1) };

        // Special case for 2x2 matrices.
        if self.nrows() == 2 {
            // TODO: can we avoid this slicing
            // (which is needed here just to transform D to U2)?
            let me = self.fixed_slice::<U2, U2>(0, 0);
            return match compute_2x2_eigvals(&me) {
                Some((a, b)) => {
                    work[0] = a;
                    work[1] = b;
                    Some(work)
                }
                None => None,
            };
        }

        // TODO: add balancing?
        let schur = Schur::do_decompose(
            self.clone_owned(),
            &mut work,
            N::RealField::default_epsilon(),
            0,
            false,
        )
        .unwrap();
        if Schur::do_eigenvalues(&schur.1, &mut work) {
            Some(work)
        } else {
            None
        }
    }

    /// Computes the eigenvalues of this matrix.
    pub fn complex_eigenvalues(&self) -> VectorN<NumComplex<N>, D>
    // TODO: add balancing?
    where
        N: RealField,
        DefaultAllocator: Allocator<NumComplex<N>, D>,
    {
        let dim = self.data.shape().0;
        let mut work = unsafe { crate::unimplemented_or_uninitialized_generic!(dim, U1) };

        let schur = Schur::do_decompose(
            self.clone_owned(),
            &mut work,
            N::default_epsilon(),
            0,
            false,
        )
        .unwrap();
        let mut eig = unsafe { crate::unimplemented_or_uninitialized_generic!(dim, U1) };
        Schur::do_complex_eigenvalues(&schur.1, &mut eig);
        eig
    }
}