1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
5// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
6//
7// This Source Code Form is subject to the terms of the Mozilla
8// Public License v. 2.0. If a copy of the MPL was not distributed
9// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10
11#ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
12#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
13
14namespace Eigen {
15
16namespace internal {
17template<typename _MatrixType> struct traits<ColPivHouseholderQR<_MatrixType> >
18 : traits<_MatrixType>
19{
20 enum { Flags = 0 };
21};
22
23} // end namespace internal
24
25/** \ingroup QR_Module
26 *
27 * \class ColPivHouseholderQR
28 *
29 * \brief Householder rank-revealing QR decomposition of a matrix with column-pivoting
30 *
31 * \tparam _MatrixType the type of the matrix of which we are computing the QR decomposition
32 *
33 * This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b P, \b Q and \b R
34 * such that
35 * \f[
36 * \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
37 * \f]
38 * by using Householder transformations. Here, \b P is a permutation matrix, \b Q a unitary matrix and \b R an
39 * upper triangular matrix.
40 *
41 * This decomposition performs column pivoting in order to be rank-revealing and improve
42 * numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
43 *
44 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
45 *
46 * \sa MatrixBase::colPivHouseholderQr()
47 */
48template<typename _MatrixType> class ColPivHouseholderQR
49{
50 public:
51
52 typedef _MatrixType MatrixType;
53 enum {
54 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
55 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
56 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
57 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
58 };
59 typedef typename MatrixType::Scalar Scalar;
60 typedef typename MatrixType::RealScalar RealScalar;
61 // FIXME should be int
62 typedef typename MatrixType::StorageIndex StorageIndex;
63 typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
64 typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
65 typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
66 typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
67 typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
68 typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
69 typedef typename MatrixType::PlainObject PlainObject;
70
71 private:
72
73 typedef typename PermutationType::StorageIndex PermIndexType;
74
75 public:
76
77 /**
78 * \brief Default Constructor.
79 *
80 * The default constructor is useful in cases in which the user intends to
81 * perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).
82 */
83 ColPivHouseholderQR()
84 : m_qr(),
85 m_hCoeffs(),
86 m_colsPermutation(),
87 m_colsTranspositions(),
88 m_temp(),
89 m_colNormsUpdated(),
90 m_colNormsDirect(),
91 m_isInitialized(false),
92 m_usePrescribedThreshold(false) {}
93
94 /** \brief Default Constructor with memory preallocation
95 *
96 * Like the default constructor but with preallocation of the internal data
97 * according to the specified problem \a size.
98 * \sa ColPivHouseholderQR()
99 */
100 ColPivHouseholderQR(Index rows, Index cols)
101 : m_qr(rows, cols),
102 m_hCoeffs((std::min)(rows,cols)),
103 m_colsPermutation(PermIndexType(cols)),
104 m_colsTranspositions(cols),
105 m_temp(cols),
106 m_colNormsUpdated(cols),
107 m_colNormsDirect(cols),
108 m_isInitialized(false),
109 m_usePrescribedThreshold(false) {}
110
111 /** \brief Constructs a QR factorization from a given matrix
112 *
113 * This constructor computes the QR factorization of the matrix \a matrix by calling
114 * the method compute(). It is a short cut for:
115 *
116 * \code
117 * ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
118 * qr.compute(matrix);
119 * \endcode
120 *
121 * \sa compute()
122 */
123 template<typename InputType>
124 explicit ColPivHouseholderQR(const EigenBase<InputType>& matrix)
125 : m_qr(matrix.rows(), matrix.cols()),
126 m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
127 m_colsPermutation(PermIndexType(matrix.cols())),
128 m_colsTranspositions(matrix.cols()),
129 m_temp(matrix.cols()),
130 m_colNormsUpdated(matrix.cols()),
131 m_colNormsDirect(matrix.cols()),
132 m_isInitialized(false),
133 m_usePrescribedThreshold(false)
134 {
135 compute(matrix.derived());
136 }
137
138 /** \brief Constructs a QR factorization from a given matrix
139 *
140 * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref.
141 *
142 * \sa ColPivHouseholderQR(const EigenBase&)
143 */
144 template<typename InputType>
145 explicit ColPivHouseholderQR(EigenBase<InputType>& matrix)
146 : m_qr(matrix.derived()),
147 m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
148 m_colsPermutation(PermIndexType(matrix.cols())),
149 m_colsTranspositions(matrix.cols()),
150 m_temp(matrix.cols()),
151 m_colNormsUpdated(matrix.cols()),
152 m_colNormsDirect(matrix.cols()),
153 m_isInitialized(false),
154 m_usePrescribedThreshold(false)
155 {
156 computeInPlace();
157 }
158
159 /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
160 * *this is the QR decomposition, if any exists.
161 *
162 * \param b the right-hand-side of the equation to solve.
163 *
164 * \returns a solution.
165 *
166 * \note_about_checking_solutions
167 *
168 * \note_about_arbitrary_choice_of_solution
169 *
170 * Example: \include ColPivHouseholderQR_solve.cpp
171 * Output: \verbinclude ColPivHouseholderQR_solve.out
172 */
173 template<typename Rhs>
174 inline const Solve<ColPivHouseholderQR, Rhs>
175 solve(const MatrixBase<Rhs>& b) const
176 {
177 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
178 return Solve<ColPivHouseholderQR, Rhs>(*this, b.derived());
179 }
180
181 HouseholderSequenceType householderQ() const;
182 HouseholderSequenceType matrixQ() const
183 {
184 return householderQ();
185 }
186
187 /** \returns a reference to the matrix where the Householder QR decomposition is stored
188 */
189 const MatrixType& matrixQR() const
190 {
191 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
192 return m_qr;
193 }
194
195 /** \returns a reference to the matrix where the result Householder QR is stored
196 * \warning The strict lower part of this matrix contains internal values.
197 * Only the upper triangular part should be referenced. To get it, use
198 * \code matrixR().template triangularView<Upper>() \endcode
199 * For rank-deficient matrices, use
200 * \code
201 * matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
202 * \endcode
203 */
204 const MatrixType& matrixR() const
205 {
206 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
207 return m_qr;
208 }
209
210 template<typename InputType>
211 ColPivHouseholderQR& compute(const EigenBase<InputType>& matrix);
212
213 /** \returns a const reference to the column permutation matrix */
214 const PermutationType& colsPermutation() const
215 {
216 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
217 return m_colsPermutation;
218 }
219
220 /** \returns the absolute value of the determinant of the matrix of which
221 * *this is the QR decomposition. It has only linear complexity
222 * (that is, O(n) where n is the dimension of the square matrix)
223 * as the QR decomposition has already been computed.
224 *
225 * \note This is only for square matrices.
226 *
227 * \warning a determinant can be very big or small, so for matrices
228 * of large enough dimension, there is a risk of overflow/underflow.
229 * One way to work around that is to use logAbsDeterminant() instead.
230 *
231 * \sa logAbsDeterminant(), MatrixBase::determinant()
232 */
233 typename MatrixType::RealScalar absDeterminant() const;
234
235 /** \returns the natural log of the absolute value of the determinant of the matrix of which
236 * *this is the QR decomposition. It has only linear complexity
237 * (that is, O(n) where n is the dimension of the square matrix)
238 * as the QR decomposition has already been computed.
239 *
240 * \note This is only for square matrices.
241 *
242 * \note This method is useful to work around the risk of overflow/underflow that's inherent
243 * to determinant computation.
244 *
245 * \sa absDeterminant(), MatrixBase::determinant()
246 */
247 typename MatrixType::RealScalar logAbsDeterminant() const;
248
249 /** \returns the rank of the matrix of which *this is the QR decomposition.
250 *
251 * \note This method has to determine which pivots should be considered nonzero.
252 * For that, it uses the threshold value that you can control by calling
253 * setThreshold(const RealScalar&).
254 */
255 inline Index rank() const
256 {
257 using std::abs;
258 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
259 RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
260 Index result = 0;
261 for(Index i = 0; i < m_nonzero_pivots; ++i)
262 result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
263 return result;
264 }
265
266 /** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
267 *
268 * \note This method has to determine which pivots should be considered nonzero.
269 * For that, it uses the threshold value that you can control by calling
270 * setThreshold(const RealScalar&).
271 */
272 inline Index dimensionOfKernel() const
273 {
274 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
275 return cols() - rank();
276 }
277
278 /** \returns true if the matrix of which *this is the QR decomposition represents an injective
279 * linear map, i.e. has trivial kernel; false otherwise.
280 *
281 * \note This method has to determine which pivots should be considered nonzero.
282 * For that, it uses the threshold value that you can control by calling
283 * setThreshold(const RealScalar&).
284 */
285 inline bool isInjective() const
286 {
287 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
288 return rank() == cols();
289 }
290
291 /** \returns true if the matrix of which *this is the QR decomposition represents a surjective
292 * linear map; false otherwise.
293 *
294 * \note This method has to determine which pivots should be considered nonzero.
295 * For that, it uses the threshold value that you can control by calling
296 * setThreshold(const RealScalar&).
297 */
298 inline bool isSurjective() const
299 {
300 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
301 return rank() == rows();
302 }
303
304 /** \returns true if the matrix of which *this is the QR decomposition is invertible.
305 *
306 * \note This method has to determine which pivots should be considered nonzero.
307 * For that, it uses the threshold value that you can control by calling
308 * setThreshold(const RealScalar&).
309 */
310 inline bool isInvertible() const
311 {
312 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
313 return isInjective() && isSurjective();
314 }
315
316 /** \returns the inverse of the matrix of which *this is the QR decomposition.
317 *
318 * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
319 * Use isInvertible() to first determine whether this matrix is invertible.
320 */
321 inline const Inverse<ColPivHouseholderQR> inverse() const
322 {
323 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
324 return Inverse<ColPivHouseholderQR>(*this);
325 }
326
327 inline Index rows() const { return m_qr.rows(); }
328 inline Index cols() const { return m_qr.cols(); }
329
330 /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
331 *
332 * For advanced uses only.
333 */
334 const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
335
336 /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
337 * who need to determine when pivots are to be considered nonzero. This is not used for the
338 * QR decomposition itself.
339 *
340 * When it needs to get the threshold value, Eigen calls threshold(). By default, this
341 * uses a formula to automatically determine a reasonable threshold.
342 * Once you have called the present method setThreshold(const RealScalar&),
343 * your value is used instead.
344 *
345 * \param threshold The new value to use as the threshold.
346 *
347 * A pivot will be considered nonzero if its absolute value is strictly greater than
348 * \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
349 * where maxpivot is the biggest pivot.
350 *
351 * If you want to come back to the default behavior, call setThreshold(Default_t)
352 */
353 ColPivHouseholderQR& setThreshold(const RealScalar& threshold)
354 {
355 m_usePrescribedThreshold = true;
356 m_prescribedThreshold = threshold;
357 return *this;
358 }
359
360 /** Allows to come back to the default behavior, letting Eigen use its default formula for
361 * determining the threshold.
362 *
363 * You should pass the special object Eigen::Default as parameter here.
364 * \code qr.setThreshold(Eigen::Default); \endcode
365 *
366 * See the documentation of setThreshold(const RealScalar&).
367 */
368 ColPivHouseholderQR& setThreshold(Default_t)
369 {
370 m_usePrescribedThreshold = false;
371 return *this;
372 }
373
374 /** Returns the threshold that will be used by certain methods such as rank().
375 *
376 * See the documentation of setThreshold(const RealScalar&).
377 */
378 RealScalar threshold() const
379 {
380 eigen_assert(m_isInitialized || m_usePrescribedThreshold);
381 return m_usePrescribedThreshold ? m_prescribedThreshold
382 // this formula comes from experimenting (see "LU precision tuning" thread on the list)
383 // and turns out to be identical to Higham's formula used already in LDLt.
384 : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
385 }
386
387 /** \returns the number of nonzero pivots in the QR decomposition.
388 * Here nonzero is meant in the exact sense, not in a fuzzy sense.
389 * So that notion isn't really intrinsically interesting, but it is
390 * still useful when implementing algorithms.
391 *
392 * \sa rank()
393 */
394 inline Index nonzeroPivots() const
395 {
396 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
397 return m_nonzero_pivots;
398 }
399
400 /** \returns the absolute value of the biggest pivot, i.e. the biggest
401 * diagonal coefficient of R.
402 */
403 RealScalar maxPivot() const { return m_maxpivot; }
404
405 /** \brief Reports whether the QR factorization was succesful.
406 *
407 * \note This function always returns \c Success. It is provided for compatibility
408 * with other factorization routines.
409 * \returns \c Success
410 */
411 ComputationInfo info() const
412 {
413 eigen_assert(m_isInitialized && "Decomposition is not initialized.");
414 return Success;
415 }
416
417 #ifndef EIGEN_PARSED_BY_DOXYGEN
418 template<typename RhsType, typename DstType>
419 EIGEN_DEVICE_FUNC
420 void _solve_impl(const RhsType &rhs, DstType &dst) const;
421 #endif
422
423 protected:
424
425 friend class CompleteOrthogonalDecomposition<MatrixType>;
426
427 static void check_template_parameters()
428 {
429 EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
430 }
431
432 void computeInPlace();
433
434 MatrixType m_qr;
435 HCoeffsType m_hCoeffs;
436 PermutationType m_colsPermutation;
437 IntRowVectorType m_colsTranspositions;
438 RowVectorType m_temp;
439 RealRowVectorType m_colNormsUpdated;
440 RealRowVectorType m_colNormsDirect;
441 bool m_isInitialized, m_usePrescribedThreshold;
442 RealScalar m_prescribedThreshold, m_maxpivot;
443 Index m_nonzero_pivots;
444 Index m_det_pq;
445};
446
447template<typename MatrixType>
448typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() const
449{
450 using std::abs;
451 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
452 eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
453 return abs(m_qr.diagonal().prod());
454}
455
456template<typename MatrixType>
457typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
458{
459 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
460 eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
461 return m_qr.diagonal().cwiseAbs().array().log().sum();
462}
463
464/** Performs the QR factorization of the given matrix \a matrix. The result of
465 * the factorization is stored into \c *this, and a reference to \c *this
466 * is returned.
467 *
468 * \sa class ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)
469 */
470template<typename MatrixType>
471template<typename InputType>
472ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const EigenBase<InputType>& matrix)
473{
474 m_qr = matrix.derived();
475 computeInPlace();
476 return *this;
477}
478
479template<typename MatrixType>
480void ColPivHouseholderQR<MatrixType>::computeInPlace()
481{
482 check_template_parameters();
483
484 // the column permutation is stored as int indices, so just to be sure:
485 eigen_assert(m_qr.cols()<=NumTraits<int>::highest());
486
487 using std::abs;
488
489 Index rows = m_qr.rows();
490 Index cols = m_qr.cols();
491 Index size = m_qr.diagonalSize();
492
493 m_hCoeffs.resize(size);
494
495 m_temp.resize(cols);
496
497 m_colsTranspositions.resize(m_qr.cols());
498 Index number_of_transpositions = 0;
499
500 m_colNormsUpdated.resize(cols);
501 m_colNormsDirect.resize(cols);
502 for (Index k = 0; k < cols; ++k) {
503 // colNormsDirect(k) caches the most recent directly computed norm of
504 // column k.
505 m_colNormsDirect.coeffRef(k) = m_qr.col(k).norm();
506 m_colNormsUpdated.coeffRef(k) = m_colNormsDirect.coeffRef(k);
507 }
508
509 RealScalar threshold_helper = numext::abs2<RealScalar>(m_colNormsUpdated.maxCoeff() * NumTraits<RealScalar>::epsilon()) / RealScalar(rows);
510 RealScalar norm_downdate_threshold = numext::sqrt(NumTraits<RealScalar>::epsilon());
511
512 m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
513 m_maxpivot = RealScalar(0);
514
515 for(Index k = 0; k < size; ++k)
516 {
517 // first, we look up in our table m_colNormsUpdated which column has the biggest norm
518 Index biggest_col_index;
519 RealScalar biggest_col_sq_norm = numext::abs2(m_colNormsUpdated.tail(cols-k).maxCoeff(&biggest_col_index));
520 biggest_col_index += k;
521
522 // Track the number of meaningful pivots but do not stop the decomposition to make
523 // sure that the initial matrix is properly reproduced. See bug 941.
524 if(m_nonzero_pivots==size && biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
525 m_nonzero_pivots = k;
526
527 // apply the transposition to the columns
528 m_colsTranspositions.coeffRef(k) = biggest_col_index;
529 if(k != biggest_col_index) {
530 m_qr.col(k).swap(m_qr.col(biggest_col_index));
531 std::swap(m_colNormsUpdated.coeffRef(k), m_colNormsUpdated.coeffRef(biggest_col_index));
532 std::swap(m_colNormsDirect.coeffRef(k), m_colNormsDirect.coeffRef(biggest_col_index));
533 ++number_of_transpositions;
534 }
535
536 // generate the householder vector, store it below the diagonal
537 RealScalar beta;
538 m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
539
540 // apply the householder transformation to the diagonal coefficient
541 m_qr.coeffRef(k,k) = beta;
542
543 // remember the maximum absolute value of diagonal coefficients
544 if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
545
546 // apply the householder transformation
547 m_qr.bottomRightCorner(rows-k, cols-k-1)
548 .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
549
550 // update our table of norms of the columns
551 for (Index j = k + 1; j < cols; ++j) {
552 // The following implements the stable norm downgrade step discussed in
553 // http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
554 // and used in LAPACK routines xGEQPF and xGEQP3.
555 // See lines 278-297 in http://www.netlib.org/lapack/explore-html/dc/df4/sgeqpf_8f_source.html
556 if (m_colNormsUpdated.coeffRef(j) != RealScalar(0)) {
557 RealScalar temp = abs(m_qr.coeffRef(k, j)) / m_colNormsUpdated.coeffRef(j);
558 temp = (RealScalar(1) + temp) * (RealScalar(1) - temp);
559 temp = temp < RealScalar(0) ? RealScalar(0) : temp;
560 RealScalar temp2 = temp * numext::abs2<RealScalar>(m_colNormsUpdated.coeffRef(j) /
561 m_colNormsDirect.coeffRef(j));
562 if (temp2 <= norm_downdate_threshold) {
563 // The updated norm has become too inaccurate so re-compute the column
564 // norm directly.
565 m_colNormsDirect.coeffRef(j) = m_qr.col(j).tail(rows - k - 1).norm();
566 m_colNormsUpdated.coeffRef(j) = m_colNormsDirect.coeffRef(j);
567 } else {
568 m_colNormsUpdated.coeffRef(j) *= numext::sqrt(temp);
569 }
570 }
571 }
572 }
573
574 m_colsPermutation.setIdentity(PermIndexType(cols));
575 for(PermIndexType k = 0; k < size/*m_nonzero_pivots*/; ++k)
576 m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));
577
578 m_det_pq = (number_of_transpositions%2) ? -1 : 1;
579 m_isInitialized = true;
580}
581
582#ifndef EIGEN_PARSED_BY_DOXYGEN
583template<typename _MatrixType>
584template<typename RhsType, typename DstType>
585void ColPivHouseholderQR<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const
586{
587 eigen_assert(rhs.rows() == rows());
588
589 const Index nonzero_pivots = nonzeroPivots();
590
591 if(nonzero_pivots == 0)
592 {
593 dst.setZero();
594 return;
595 }
596
597 typename RhsType::PlainObject c(rhs);
598
599 // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
600 c.applyOnTheLeft(householderSequence(m_qr, m_hCoeffs)
601 .setLength(nonzero_pivots)
602 .transpose()
603 );
604
605 m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
606 .template triangularView<Upper>()
607 .solveInPlace(c.topRows(nonzero_pivots));
608
609 for(Index i = 0; i < nonzero_pivots; ++i) dst.row(m_colsPermutation.indices().coeff(i)) = c.row(i);
610 for(Index i = nonzero_pivots; i < cols(); ++i) dst.row(m_colsPermutation.indices().coeff(i)).setZero();
611}
612#endif
613
614namespace internal {
615
616template<typename DstXprType, typename MatrixType>
617struct Assignment<DstXprType, Inverse<ColPivHouseholderQR<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename ColPivHouseholderQR<MatrixType>::Scalar>, Dense2Dense>
618{
619 typedef ColPivHouseholderQR<MatrixType> QrType;
620 typedef Inverse<QrType> SrcXprType;
621 static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename QrType::Scalar> &)
622 {
623 dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
624 }
625};
626
627} // end namespace internal
628
629/** \returns the matrix Q as a sequence of householder transformations.
630 * You can extract the meaningful part only by using:
631 * \code qr.householderQ().setLength(qr.nonzeroPivots()) \endcode*/
632template<typename MatrixType>
633typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>
634 ::householderQ() const
635{
636 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
637 return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
638}
639
640/** \return the column-pivoting Householder QR decomposition of \c *this.
641 *
642 * \sa class ColPivHouseholderQR
643 */
644template<typename Derived>
645const ColPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
646MatrixBase<Derived>::colPivHouseholderQr() const
647{
648 return ColPivHouseholderQR<PlainObject>(eval());
649}
650
651} // end namespace Eigen
652
653#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
654