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Eigen  3.4.0
 
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ColPivHouseholderQR.h
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 typedef MatrixXpr XprKind;
21 typedef SolverStorage StorageKind;
22 typedef int StorageIndex;
23 enum { Flags = 0 };
24};
25
26} // end namespace internal
27
51template<typename _MatrixType> class ColPivHouseholderQR
52 : public SolverBase<ColPivHouseholderQR<_MatrixType> >
53{
54 public:
55
56 typedef _MatrixType MatrixType;
58 friend class SolverBase<ColPivHouseholderQR>;
59
60 EIGEN_GENERIC_PUBLIC_INTERFACE(ColPivHouseholderQR)
61 enum {
62 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
63 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
64 };
65 typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
67 typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
68 typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
69 typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
71 typedef typename MatrixType::PlainObject PlainObject;
72
73 private:
74
75 typedef typename PermutationType::StorageIndex PermIndexType;
76
77 public:
78
86 : m_qr(),
87 m_hCoeffs(),
88 m_colsPermutation(),
89 m_colsTranspositions(),
90 m_temp(),
91 m_colNormsUpdated(),
92 m_colNormsDirect(),
93 m_isInitialized(false),
94 m_usePrescribedThreshold(false) {}
95
103 : m_qr(rows, cols),
104 m_hCoeffs((std::min)(rows,cols)),
105 m_colsPermutation(PermIndexType(cols)),
106 m_colsTranspositions(cols),
107 m_temp(cols),
108 m_colNormsUpdated(cols),
109 m_colNormsDirect(cols),
110 m_isInitialized(false),
111 m_usePrescribedThreshold(false) {}
112
125 template<typename InputType>
127 : m_qr(matrix.rows(), matrix.cols()),
128 m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
129 m_colsPermutation(PermIndexType(matrix.cols())),
130 m_colsTranspositions(matrix.cols()),
131 m_temp(matrix.cols()),
132 m_colNormsUpdated(matrix.cols()),
133 m_colNormsDirect(matrix.cols()),
134 m_isInitialized(false),
135 m_usePrescribedThreshold(false)
136 {
137 compute(matrix.derived());
138 }
139
146 template<typename InputType>
148 : m_qr(matrix.derived()),
149 m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
150 m_colsPermutation(PermIndexType(matrix.cols())),
151 m_colsTranspositions(matrix.cols()),
152 m_temp(matrix.cols()),
153 m_colNormsUpdated(matrix.cols()),
154 m_colNormsDirect(matrix.cols()),
155 m_isInitialized(false),
156 m_usePrescribedThreshold(false)
157 {
158 computeInPlace();
159 }
160
161 #ifdef EIGEN_PARSED_BY_DOXYGEN
176 template<typename Rhs>
178 solve(const MatrixBase<Rhs>& b) const;
179 #endif
180
182 HouseholderSequenceType matrixQ() const
183 {
184 return householderQ();
185 }
186
189 const MatrixType& matrixQR() const
190 {
191 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
192 return m_qr;
193 }
194
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
215 {
216 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
217 return m_colsPermutation;
218 }
219
233 typename MatrixType::RealScalar absDeterminant() const;
234
247 typename MatrixType::RealScalar logAbsDeterminant() const;
248
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
273 {
274 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
275 return cols() - rank();
276 }
277
285 inline bool isInjective() const
286 {
287 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
288 return rank() == cols();
289 }
290
298 inline bool isSurjective() const
299 {
300 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
301 return rank() == rows();
302 }
303
310 inline bool isInvertible() const
311 {
312 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
313 return isInjective() && isSurjective();
314 }
315
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
334 const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
335
354 {
355 m_usePrescribedThreshold = true;
356 m_prescribedThreshold = threshold;
357 return *this;
358 }
359
369 {
370 m_usePrescribedThreshold = false;
371 return *this;
372 }
373
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
394 inline Index nonzeroPivots() const
395 {
396 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
397 return m_nonzero_pivots;
398 }
399
403 RealScalar maxPivot() const { return m_maxpivot; }
404
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 void _solve_impl(const RhsType &rhs, DstType &dst) const;
420
421 template<bool Conjugate, typename RhsType, typename DstType>
422 void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const;
423 #endif
424
425 protected:
426
427 friend class CompleteOrthogonalDecomposition<MatrixType>;
428
429 static void check_template_parameters()
430 {
431 EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
432 }
433
434 void computeInPlace();
435
436 MatrixType m_qr;
437 HCoeffsType m_hCoeffs;
438 PermutationType m_colsPermutation;
439 IntRowVectorType m_colsTranspositions;
440 RowVectorType m_temp;
441 RealRowVectorType m_colNormsUpdated;
442 RealRowVectorType m_colNormsDirect;
443 bool m_isInitialized, m_usePrescribedThreshold;
444 RealScalar m_prescribedThreshold, m_maxpivot;
445 Index m_nonzero_pivots;
446 Index m_det_pq;
447};
448
449template<typename MatrixType>
450typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() const
451{
452 using std::abs;
453 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
454 eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
455 return abs(m_qr.diagonal().prod());
456}
457
458template<typename MatrixType>
459typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
460{
461 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
462 eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
463 return m_qr.diagonal().cwiseAbs().array().log().sum();
464}
465
472template<typename MatrixType>
473template<typename InputType>
475{
476 m_qr = matrix.derived();
477 computeInPlace();
478 return *this;
479}
480
481template<typename MatrixType>
483{
484 check_template_parameters();
485
486 // the column permutation is stored as int indices, so just to be sure:
487 eigen_assert(m_qr.cols()<=NumTraits<int>::highest());
488
489 using std::abs;
490
491 Index rows = m_qr.rows();
492 Index cols = m_qr.cols();
493 Index size = m_qr.diagonalSize();
494
495 m_hCoeffs.resize(size);
496
497 m_temp.resize(cols);
498
499 m_colsTranspositions.resize(m_qr.cols());
500 Index number_of_transpositions = 0;
501
502 m_colNormsUpdated.resize(cols);
503 m_colNormsDirect.resize(cols);
504 for (Index k = 0; k < cols; ++k) {
505 // colNormsDirect(k) caches the most recent directly computed norm of
506 // column k.
507 m_colNormsDirect.coeffRef(k) = m_qr.col(k).norm();
508 m_colNormsUpdated.coeffRef(k) = m_colNormsDirect.coeffRef(k);
509 }
510
511 RealScalar threshold_helper = numext::abs2<RealScalar>(m_colNormsUpdated.maxCoeff() * NumTraits<RealScalar>::epsilon()) / RealScalar(rows);
512 RealScalar norm_downdate_threshold = numext::sqrt(NumTraits<RealScalar>::epsilon());
513
514 m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
515 m_maxpivot = RealScalar(0);
516
517 for(Index k = 0; k < size; ++k)
518 {
519 // first, we look up in our table m_colNormsUpdated which column has the biggest norm
520 Index biggest_col_index;
521 RealScalar biggest_col_sq_norm = numext::abs2(m_colNormsUpdated.tail(cols-k).maxCoeff(&biggest_col_index));
522 biggest_col_index += k;
523
524 // Track the number of meaningful pivots but do not stop the decomposition to make
525 // sure that the initial matrix is properly reproduced. See bug 941.
526 if(m_nonzero_pivots==size && biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
527 m_nonzero_pivots = k;
528
529 // apply the transposition to the columns
530 m_colsTranspositions.coeffRef(k) = biggest_col_index;
531 if(k != biggest_col_index) {
532 m_qr.col(k).swap(m_qr.col(biggest_col_index));
533 std::swap(m_colNormsUpdated.coeffRef(k), m_colNormsUpdated.coeffRef(biggest_col_index));
534 std::swap(m_colNormsDirect.coeffRef(k), m_colNormsDirect.coeffRef(biggest_col_index));
535 ++number_of_transpositions;
536 }
537
538 // generate the householder vector, store it below the diagonal
539 RealScalar beta;
540 m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
541
542 // apply the householder transformation to the diagonal coefficient
543 m_qr.coeffRef(k,k) = beta;
544
545 // remember the maximum absolute value of diagonal coefficients
546 if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
547
548 // apply the householder transformation
549 m_qr.bottomRightCorner(rows-k, cols-k-1)
550 .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
551
552 // update our table of norms of the columns
553 for (Index j = k + 1; j < cols; ++j) {
554 // The following implements the stable norm downgrade step discussed in
555 // http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
556 // and used in LAPACK routines xGEQPF and xGEQP3.
557 // See lines 278-297 in http://www.netlib.org/lapack/explore-html/dc/df4/sgeqpf_8f_source.html
558 if (m_colNormsUpdated.coeffRef(j) != RealScalar(0)) {
559 RealScalar temp = abs(m_qr.coeffRef(k, j)) / m_colNormsUpdated.coeffRef(j);
560 temp = (RealScalar(1) + temp) * (RealScalar(1) - temp);
561 temp = temp < RealScalar(0) ? RealScalar(0) : temp;
562 RealScalar temp2 = temp * numext::abs2<RealScalar>(m_colNormsUpdated.coeffRef(j) /
563 m_colNormsDirect.coeffRef(j));
564 if (temp2 <= norm_downdate_threshold) {
565 // The updated norm has become too inaccurate so re-compute the column
566 // norm directly.
567 m_colNormsDirect.coeffRef(j) = m_qr.col(j).tail(rows - k - 1).norm();
568 m_colNormsUpdated.coeffRef(j) = m_colNormsDirect.coeffRef(j);
569 } else {
570 m_colNormsUpdated.coeffRef(j) *= numext::sqrt(temp);
571 }
572 }
573 }
574 }
575
576 m_colsPermutation.setIdentity(PermIndexType(cols));
577 for(PermIndexType k = 0; k < size/*m_nonzero_pivots*/; ++k)
578 m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));
579
580 m_det_pq = (number_of_transpositions%2) ? -1 : 1;
581 m_isInitialized = true;
582}
583
584#ifndef EIGEN_PARSED_BY_DOXYGEN
585template<typename _MatrixType>
586template<typename RhsType, typename DstType>
587void ColPivHouseholderQR<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const
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 c.applyOnTheLeft(householderQ().setLength(nonzero_pivots).adjoint() );
600
601 m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
602 .template triangularView<Upper>()
603 .solveInPlace(c.topRows(nonzero_pivots));
604
605 for(Index i = 0; i < nonzero_pivots; ++i) dst.row(m_colsPermutation.indices().coeff(i)) = c.row(i);
606 for(Index i = nonzero_pivots; i < cols(); ++i) dst.row(m_colsPermutation.indices().coeff(i)).setZero();
607}
608
609template<typename _MatrixType>
610template<bool Conjugate, typename RhsType, typename DstType>
611void ColPivHouseholderQR<_MatrixType>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const
612{
613 const Index nonzero_pivots = nonzeroPivots();
614
615 if(nonzero_pivots == 0)
616 {
617 dst.setZero();
618 return;
619 }
620
621 typename RhsType::PlainObject c(m_colsPermutation.transpose()*rhs);
622
623 m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
624 .template triangularView<Upper>()
625 .transpose().template conjugateIf<Conjugate>()
626 .solveInPlace(c.topRows(nonzero_pivots));
627
628 dst.topRows(nonzero_pivots) = c.topRows(nonzero_pivots);
629 dst.bottomRows(rows()-nonzero_pivots).setZero();
630
631 dst.applyOnTheLeft(householderQ().setLength(nonzero_pivots).template conjugateIf<!Conjugate>() );
632}
633#endif
634
635namespace internal {
636
637template<typename DstXprType, typename MatrixType>
638struct Assignment<DstXprType, Inverse<ColPivHouseholderQR<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename ColPivHouseholderQR<MatrixType>::Scalar>, Dense2Dense>
639{
640 typedef ColPivHouseholderQR<MatrixType> QrType;
641 typedef Inverse<QrType> SrcXprType;
642 static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename QrType::Scalar> &)
643 {
644 dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
645 }
646};
647
648} // end namespace internal
649
653template<typename MatrixType>
654typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>
655 ::householderQ() const
656{
657 eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
658 return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
659}
660
665template<typename Derived>
668{
670}
671
672} // end namespace Eigen
673
674#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
Householder rank-revealing QR decomposition of a matrix with column-pivoting.
Definition: ColPivHouseholderQR.h:53
bool isInjective() const
Definition: ColPivHouseholderQR.h:285
const Solve< ColPivHouseholderQR, Rhs > solve(const MatrixBase< Rhs > &b) const
ColPivHouseholderQR(const EigenBase< InputType > &matrix)
Constructs a QR factorization from a given matrix.
Definition: ColPivHouseholderQR.h:126
const HCoeffsType & hCoeffs() const
Definition: ColPivHouseholderQR.h:334
HouseholderSequenceType householderQ() const
Definition: ColPivHouseholderQR.h:655
Index rank() const
Definition: ColPivHouseholderQR.h:255
ColPivHouseholderQR(Index rows, Index cols)
Default Constructor with memory preallocation.
Definition: ColPivHouseholderQR.h:102
ComputationInfo info() const
Reports whether the QR factorization was successful.
Definition: ColPivHouseholderQR.h:411
ColPivHouseholderQR(EigenBase< InputType > &matrix)
Constructs a QR factorization from a given matrix.
Definition: ColPivHouseholderQR.h:147
const Inverse< ColPivHouseholderQR > inverse() const
Definition: ColPivHouseholderQR.h:321
RealScalar threshold() const
Definition: ColPivHouseholderQR.h:378
Index nonzeroPivots() const
Definition: ColPivHouseholderQR.h:394
const PermutationType & colsPermutation() const
Definition: ColPivHouseholderQR.h:214
Index dimensionOfKernel() const
Definition: ColPivHouseholderQR.h:272
bool isSurjective() const
Definition: ColPivHouseholderQR.h:298
bool isInvertible() const
Definition: ColPivHouseholderQR.h:310
ColPivHouseholderQR()
Default Constructor.
Definition: ColPivHouseholderQR.h:85
RealScalar maxPivot() const
Definition: ColPivHouseholderQR.h:403
ColPivHouseholderQR & setThreshold(const RealScalar &threshold)
Definition: ColPivHouseholderQR.h:353
const MatrixType & matrixR() const
Definition: ColPivHouseholderQR.h:204
ColPivHouseholderQR & setThreshold(Default_t)
Definition: ColPivHouseholderQR.h:368
MatrixType::RealScalar absDeterminant() const
Definition: ColPivHouseholderQR.h:450
const MatrixType & matrixQR() const
Definition: ColPivHouseholderQR.h:189
MatrixType::RealScalar logAbsDeterminant() const
Definition: ColPivHouseholderQR.h:459
Complete orthogonal decomposition (COD) of a matrix.
Definition: CompleteOrthogonalDecomposition.h:52
Sequence of Householder reflections acting on subspaces with decreasing size.
Definition: HouseholderSequence.h:121
Expression of the inverse of another expression.
Definition: Inverse.h:44
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:50
Permutation matrix.
Definition: PermutationMatrix.h:298
Pseudo expression representing a solving operation.
Definition: Solve.h:63
A base class for matrix decomposition and solvers.
Definition: SolverBase.h:69
ColPivHouseholderQR< _MatrixType > & derived()
Definition: EigenBase.h:46
const Solve< Derived, Rhs > solve(const MatrixBase< Rhs > &b) const
Definition: SolverBase.h:106
ComputationInfo
Definition: Constants.h:440
@ Success
Definition: Constants.h:442
Namespace containing all symbols from the Eigen library.
Definition: Core:141
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:74
Definition: EigenBase.h:30
Eigen::Index Index
The interface type of indices.
Definition: EigenBase.h:39
Derived & derived()
Definition: EigenBase.h:46
Holds information about the various numeric (i.e. scalar) types allowed by Eigen.
Definition: NumTraits.h:233