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
|
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Gael Guennebaud <g.gael@free.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.
#ifndef EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
/** \ingroup QR_Module
* \nonstableyet
*
* \class FullPivHouseholderQR
*
* \brief Householder rank-revealing QR decomposition of a matrix with full pivoting
*
* \param MatrixType the type of the matrix of which we are computing the QR decomposition
*
* This class performs a rank-revealing QR decomposition using Householder transformations.
*
* This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal
* numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.
*
* \sa MatrixBase::fullPivHouseholderQr()
*/
template<typename _MatrixType> class FullPivHouseholderQR
{
public:
typedef _MatrixType MatrixType;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
Options = MatrixType::Options,
DiagSizeAtCompileTime = EIGEN_ENUM_MIN(ColsAtCompileTime,RowsAtCompileTime)
};
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime> MatrixQType;
typedef Matrix<Scalar, DiagSizeAtCompileTime, 1> HCoeffsType;
typedef Matrix<int, 1, ColsAtCompileTime> IntRowVectorType;
typedef PermutationMatrix<ColsAtCompileTime> PermutationType;
typedef Matrix<int, RowsAtCompileTime, 1> IntColVectorType;
typedef Matrix<Scalar, 1, ColsAtCompileTime> RowVectorType;
typedef Matrix<Scalar, RowsAtCompileTime, 1> ColVectorType;
/** \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via FullPivHouseholderQR::compute(const MatrixType&).
*/
FullPivHouseholderQR() : m_isInitialized(false) {}
FullPivHouseholderQR(const MatrixType& matrix)
: m_isInitialized(false)
{
compute(matrix);
}
/** This method finds a solution x to the equation Ax=b, where A is the matrix of which
* *this is the QR decomposition, if any exists.
*
* \param b the right-hand-side of the equation to solve.
*
* \returns a solution.
*
* \note The case where b is a matrix is not yet implemented. Also, this
* code is space inefficient.
*
* \note_about_checking_solutions
*
* \note_about_arbitrary_choice_of_solution
*
* Example: \include FullPivHouseholderQR_solve.cpp
* Output: \verbinclude FullPivHouseholderQR_solve.out
*/
template<typename Rhs>
inline const ei_solve_retval<FullPivHouseholderQR, Rhs>
solve(const MatrixBase<Rhs>& b) const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return ei_solve_retval<FullPivHouseholderQR, Rhs>(*this, b.derived());
}
MatrixQType matrixQ(void) const;
/** \returns a reference to the matrix where the Householder QR decomposition is stored
*/
const MatrixType& matrixQR() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return m_qr;
}
FullPivHouseholderQR& compute(const MatrixType& matrix);
const PermutationType& colsPermutation() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return m_cols_permutation;
}
const IntColVectorType& rowsTranspositions() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return m_rows_transpositions;
}
/** \returns the absolute value of the determinant of the matrix of which
* *this is the QR decomposition. It has only linear complexity
* (that is, O(n) where n is the dimension of the square matrix)
* as the QR decomposition has already been computed.
*
* \note This is only for square matrices.
*
* \warning a determinant can be very big or small, so for matrices
* of large enough dimension, there is a risk of overflow/underflow.
* One way to work around that is to use logAbsDeterminant() instead.
*
* \sa logAbsDeterminant(), MatrixBase::determinant()
*/
typename MatrixType::RealScalar absDeterminant() const;
/** \returns the natural log of the absolute value of the determinant of the matrix of which
* *this is the QR decomposition. It has only linear complexity
* (that is, O(n) where n is the dimension of the square matrix)
* as the QR decomposition has already been computed.
*
* \note This is only for square matrices.
*
* \note This method is useful to work around the risk of overflow/underflow that's inherent
* to determinant computation.
*
* \sa absDeterminant(), MatrixBase::determinant()
*/
typename MatrixType::RealScalar logAbsDeterminant() const;
/** \returns the rank of the matrix of which *this is the QR decomposition.
*
* \note This is computed at the time of the construction of the QR decomposition. This
* method does not perform any further computation.
*/
inline int rank() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return m_rank;
}
/** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
*
* \note Since the rank is computed at the time of the construction of the QR decomposition, this
* method almost does not perform any further computation.
*/
inline int dimensionOfKernel() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return m_qr.cols() - m_rank;
}
/** \returns true if the matrix of which *this is the QR decomposition represents an injective
* linear map, i.e. has trivial kernel; false otherwise.
*
* \note Since the rank is computed at the time of the construction of the QR decomposition, this
* method almost does not perform any further computation.
*/
inline bool isInjective() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return m_rank == m_qr.cols();
}
/** \returns true if the matrix of which *this is the QR decomposition represents a surjective
* linear map; false otherwise.
*
* \note Since the rank is computed at the time of the construction of the QR decomposition, this
* method almost does not perform any further computation.
*/
inline bool isSurjective() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return m_rank == m_qr.rows();
}
/** \returns true if the matrix of which *this is the QR decomposition is invertible.
*
* \note Since the rank is computed at the time of the construction of the QR decomposition, this
* method almost does not perform any further computation.
*/
inline bool isInvertible() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return isInjective() && isSurjective();
}
/** \returns the inverse of the matrix of which *this is the QR decomposition.
*
* \note If this matrix is not invertible, the returned matrix has undefined coefficients.
* Use isInvertible() to first determine whether this matrix is invertible.
*/ inline const
ei_solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType>
inverse() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return ei_solve_retval<FullPivHouseholderQR,typename MatrixType::IdentityReturnType>
(*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
}
inline int rows() const { return m_qr.rows(); }
inline int cols() const { return m_qr.cols(); }
const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
protected:
MatrixType m_qr;
HCoeffsType m_hCoeffs;
IntColVectorType m_rows_transpositions;
PermutationType m_cols_permutation;
bool m_isInitialized;
RealScalar m_precision;
int m_rank;
int m_det_pq;
};
#ifndef EIGEN_HIDE_HEAVY_CODE
template<typename MatrixType>
typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::absDeterminant() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
ei_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
return ei_abs(m_qr.diagonal().prod());
}
template<typename MatrixType>
typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::logAbsDeterminant() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
ei_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
return m_qr.diagonal().cwiseAbs().array().log().sum();
}
template<typename MatrixType>
FullPivHouseholderQR<MatrixType>& FullPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
{
int rows = matrix.rows();
int cols = matrix.cols();
int size = std::min(rows,cols);
m_rank = size;
m_qr = matrix;
m_hCoeffs.resize(size);
RowVectorType temp(cols);
m_precision = epsilon<Scalar>() * size;
m_rows_transpositions.resize(matrix.rows());
IntRowVectorType cols_transpositions(matrix.cols());
int number_of_transpositions = 0;
RealScalar biggest(0);
for (int k = 0; k < size; ++k)
{
int row_of_biggest_in_corner, col_of_biggest_in_corner;
RealScalar biggest_in_corner;
biggest_in_corner = m_qr.corner(Eigen::BottomRight, rows-k, cols-k)
.cwiseAbs()
.maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
row_of_biggest_in_corner += k;
col_of_biggest_in_corner += k;
if(k==0) biggest = biggest_in_corner;
// if the corner is negligible, then we have less than full rank, and we can finish early
if(ei_isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
{
m_rank = k;
for(int i = k; i < size; i++)
{
m_rows_transpositions.coeffRef(i) = i;
cols_transpositions.coeffRef(i) = i;
m_hCoeffs.coeffRef(i) = Scalar(0);
}
break;
}
m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
if(k != row_of_biggest_in_corner) {
m_qr.row(k).end(cols-k).swap(m_qr.row(row_of_biggest_in_corner).end(cols-k));
++number_of_transpositions;
}
if(k != col_of_biggest_in_corner) {
m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
++number_of_transpositions;
}
RealScalar beta;
m_qr.col(k).end(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
m_qr.coeffRef(k,k) = beta;
m_qr.corner(BottomRight, rows-k, cols-k-1)
.applyHouseholderOnTheLeft(m_qr.col(k).end(rows-k-1), m_hCoeffs.coeffRef(k), &temp.coeffRef(k+1));
}
m_cols_permutation.setIdentity(cols);
for(int k = 0; k < size; ++k)
m_cols_permutation.applyTranspositionOnTheRight(k, cols_transpositions.coeff(k));
m_det_pq = (number_of_transpositions%2) ? -1 : 1;
m_isInitialized = true;
return *this;
}
template<typename _MatrixType, typename Rhs>
struct ei_solve_retval<FullPivHouseholderQR<_MatrixType>, Rhs>
: ei_solve_retval_base<FullPivHouseholderQR<_MatrixType>, Rhs>
{
EIGEN_MAKE_SOLVE_HELPERS(FullPivHouseholderQR<_MatrixType>,Rhs)
template<typename Dest> void evalTo(Dest& dst) const
{
const int rows = dec().rows(), cols = dec().cols();
dst.resize(cols, rhs().cols());
ei_assert(rhs().rows() == rows);
// FIXME introduce nonzeroPivots() and use it here. and more generally,
// make the same improvements in this dec as in FullPivLU.
if(dec().rank()==0)
{
dst.setZero();
return;
}
typename Rhs::PlainMatrixType c(rhs());
Matrix<Scalar,1,Rhs::ColsAtCompileTime> temp(rhs().cols());
for (int k = 0; k < dec().rank(); ++k)
{
int remainingSize = rows-k;
c.row(k).swap(c.row(dec().rowsTranspositions().coeff(k)));
c.corner(BottomRight, remainingSize, rhs().cols())
.applyHouseholderOnTheLeft(dec().matrixQR().col(k).end(remainingSize-1),
dec().hCoeffs().coeff(k), &temp.coeffRef(0));
}
if(!dec().isSurjective())
{
// is c is in the image of R ?
RealScalar biggest_in_upper_part_of_c = c.corner(TopLeft, dec().rank(), c.cols()).cwiseAbs().maxCoeff();
RealScalar biggest_in_lower_part_of_c = c.corner(BottomLeft, rows-dec().rank(), c.cols()).cwiseAbs().maxCoeff();
// FIXME brain dead
const RealScalar m_precision = epsilon<Scalar>() * std::min(rows,cols);
if(!ei_isMuchSmallerThan(biggest_in_lower_part_of_c, biggest_in_upper_part_of_c, m_precision))
return;
}
dec().matrixQR()
.corner(TopLeft, dec().rank(), dec().rank())
.template triangularView<UpperTriangular>()
.solveInPlace(c.corner(TopLeft, dec().rank(), c.cols()));
for(int i = 0; i < dec().rank(); ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
for(int i = dec().rank(); i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
}
};
/** \returns the matrix Q */
template<typename MatrixType>
typename FullPivHouseholderQR<MatrixType>::MatrixQType FullPivHouseholderQR<MatrixType>::matrixQ() const
{
ei_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
// compute the product H'_0 H'_1 ... H'_n-1,
// where H_k is the k-th Householder transformation I - h_k v_k v_k'
// and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
int rows = m_qr.rows();
int cols = m_qr.cols();
int size = std::min(rows,cols);
MatrixQType res = MatrixQType::Identity(rows, rows);
Matrix<Scalar,1,MatrixType::RowsAtCompileTime> temp(rows);
for (int k = size-1; k >= 0; k--)
{
res.block(k, k, rows-k, rows-k)
.applyHouseholderOnTheLeft(m_qr.col(k).end(rows-k-1), ei_conj(m_hCoeffs.coeff(k)), &temp.coeffRef(k));
res.row(k).swap(res.row(m_rows_transpositions.coeff(k)));
}
return res;
}
#endif // EIGEN_HIDE_HEAVY_CODE
/** \return the full-pivoting Householder QR decomposition of \c *this.
*
* \sa class FullPivHouseholderQR
*/
template<typename Derived>
const FullPivHouseholderQR<typename MatrixBase<Derived>::PlainMatrixType>
MatrixBase<Derived>::fullPivHouseholderQr() const
{
return FullPivHouseholderQR<PlainMatrixType>(eval());
}
#endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
|