Eigen  3.3.0
 
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FullPivHouseholderQR.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_FULLPIVOTINGHOUSEHOLDERQR_H
12#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
13
14namespace Eigen {
15
16namespace internal {
17
18template<typename _MatrixType> struct traits<FullPivHouseholderQR<_MatrixType> >
19 : traits<_MatrixType>
20{
21 enum { Flags = 0 };
22};
23
24template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType;
25
26template<typename MatrixType>
27struct traits<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
28{
29 typedef typename MatrixType::PlainObject ReturnType;
30};
31
32} // end namespace internal
33
57template<typename _MatrixType> class FullPivHouseholderQR
58{
59 public:
60
61 typedef _MatrixType MatrixType;
62 enum {
63 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
64 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
65 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
66 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
67 };
68 typedef typename MatrixType::Scalar Scalar;
69 typedef typename MatrixType::RealScalar RealScalar;
70 // FIXME should be int
71 typedef typename MatrixType::StorageIndex StorageIndex;
72 typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> MatrixQReturnType;
73 typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
74 typedef Matrix<StorageIndex, 1,
75 EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime,RowsAtCompileTime), RowMajor, 1,
76 EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime,MaxRowsAtCompileTime)> IntDiagSizeVectorType;
78 typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
79 typedef typename internal::plain_col_type<MatrixType>::type ColVectorType;
80 typedef typename MatrixType::PlainObject PlainObject;
81
88 : m_qr(),
89 m_hCoeffs(),
90 m_rows_transpositions(),
91 m_cols_transpositions(),
92 m_cols_permutation(),
93 m_temp(),
94 m_isInitialized(false),
95 m_usePrescribedThreshold(false) {}
96
104 : m_qr(rows, cols),
105 m_hCoeffs((std::min)(rows,cols)),
106 m_rows_transpositions((std::min)(rows,cols)),
107 m_cols_transpositions((std::min)(rows,cols)),
108 m_cols_permutation(cols),
109 m_temp(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_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
130 m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
131 m_cols_permutation(matrix.cols()),
132 m_temp(matrix.cols()),
133 m_isInitialized(false),
134 m_usePrescribedThreshold(false)
135 {
136 compute(matrix.derived());
137 }
138
145 template<typename InputType>
147 : m_qr(matrix.derived()),
148 m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
149 m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
150 m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
151 m_cols_permutation(matrix.cols()),
152 m_temp(matrix.cols()),
153 m_isInitialized(false),
154 m_usePrescribedThreshold(false)
155 {
156 computeInPlace();
157 }
158
174 template<typename Rhs>
176 solve(const MatrixBase<Rhs>& b) const
177 {
178 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
179 return Solve<FullPivHouseholderQR, Rhs>(*this, b.derived());
180 }
181
184 MatrixQReturnType matrixQ(void) const;
185
188 const MatrixType& matrixQR() const
189 {
190 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
191 return m_qr;
192 }
193
194 template<typename InputType>
195 FullPivHouseholderQR& compute(const EigenBase<InputType>& matrix);
196
199 {
200 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
201 return m_cols_permutation;
202 }
203
206 {
207 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
208 return m_rows_transpositions;
209 }
210
224 typename MatrixType::RealScalar absDeterminant() const;
225
238 typename MatrixType::RealScalar logAbsDeterminant() const;
239
246 inline Index rank() const
247 {
248 using std::abs;
249 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
250 RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
251 Index result = 0;
252 for(Index i = 0; i < m_nonzero_pivots; ++i)
253 result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
254 return result;
255 }
256
264 {
265 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
266 return cols() - rank();
267 }
268
276 inline bool isInjective() const
277 {
278 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
279 return rank() == cols();
280 }
281
289 inline bool isSurjective() const
290 {
291 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
292 return rank() == rows();
293 }
294
301 inline bool isInvertible() const
302 {
303 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
304 return isInjective() && isSurjective();
305 }
306
313 {
314 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
315 return Inverse<FullPivHouseholderQR>(*this);
316 }
317
318 inline Index rows() const { return m_qr.rows(); }
319 inline Index cols() const { return m_qr.cols(); }
320
325 const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
326
345 {
346 m_usePrescribedThreshold = true;
347 m_prescribedThreshold = threshold;
348 return *this;
349 }
350
360 {
361 m_usePrescribedThreshold = false;
362 return *this;
363 }
369 RealScalar threshold() const
370 {
371 eigen_assert(m_isInitialized || m_usePrescribedThreshold);
372 return m_usePrescribedThreshold ? m_prescribedThreshold
373 // this formula comes from experimenting (see "LU precision tuning" thread on the list)
374 // and turns out to be identical to Higham's formula used already in LDLt.
375 : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
376 }
377
385 inline Index nonzeroPivots() const
386 {
387 eigen_assert(m_isInitialized && "LU is not initialized.");
388 return m_nonzero_pivots;
389 }
390
394 RealScalar maxPivot() const { return m_maxpivot; }
395
396 #ifndef EIGEN_PARSED_BY_DOXYGEN
397 template<typename RhsType, typename DstType>
398 EIGEN_DEVICE_FUNC
399 void _solve_impl(const RhsType &rhs, DstType &dst) const;
400 #endif
401
402 protected:
403
404 static void check_template_parameters()
405 {
406 EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
407 }
408
409 void computeInPlace();
410
411 MatrixType m_qr;
412 HCoeffsType m_hCoeffs;
413 IntDiagSizeVectorType m_rows_transpositions;
414 IntDiagSizeVectorType m_cols_transpositions;
415 PermutationType m_cols_permutation;
416 RowVectorType m_temp;
417 bool m_isInitialized, m_usePrescribedThreshold;
418 RealScalar m_prescribedThreshold, m_maxpivot;
419 Index m_nonzero_pivots;
420 RealScalar m_precision;
421 Index m_det_pq;
422};
423
424template<typename MatrixType>
425typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::absDeterminant() const
426{
427 using std::abs;
428 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
429 eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
430 return abs(m_qr.diagonal().prod());
431}
432
433template<typename MatrixType>
434typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::logAbsDeterminant() const
435{
436 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
437 eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
438 return m_qr.diagonal().cwiseAbs().array().log().sum();
439}
440
447template<typename MatrixType>
448template<typename InputType>
450{
451 m_qr = matrix.derived();
452 computeInPlace();
453 return *this;
454}
455
456template<typename MatrixType>
458{
459 check_template_parameters();
460
461 using std::abs;
462 Index rows = m_qr.rows();
463 Index cols = m_qr.cols();
464 Index size = (std::min)(rows,cols);
465
466
467 m_hCoeffs.resize(size);
468
469 m_temp.resize(cols);
470
471 m_precision = NumTraits<Scalar>::epsilon() * RealScalar(size);
472
473 m_rows_transpositions.resize(size);
474 m_cols_transpositions.resize(size);
475 Index number_of_transpositions = 0;
476
477 RealScalar biggest(0);
478
479 m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
480 m_maxpivot = RealScalar(0);
481
482 for (Index k = 0; k < size; ++k)
483 {
484 Index row_of_biggest_in_corner, col_of_biggest_in_corner;
485 typedef internal::scalar_score_coeff_op<Scalar> Scoring;
486 typedef typename Scoring::result_type Score;
487
488 Score score = m_qr.bottomRightCorner(rows-k, cols-k)
489 .unaryExpr(Scoring())
490 .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
491 row_of_biggest_in_corner += k;
492 col_of_biggest_in_corner += k;
493 RealScalar biggest_in_corner = internal::abs_knowing_score<Scalar>()(m_qr(row_of_biggest_in_corner, col_of_biggest_in_corner), score);
494 if(k==0) biggest = biggest_in_corner;
495
496 // if the corner is negligible, then we have less than full rank, and we can finish early
497 if(internal::isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
498 {
499 m_nonzero_pivots = k;
500 for(Index i = k; i < size; i++)
501 {
502 m_rows_transpositions.coeffRef(i) = i;
503 m_cols_transpositions.coeffRef(i) = i;
504 m_hCoeffs.coeffRef(i) = Scalar(0);
505 }
506 break;
507 }
508
509 m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
510 m_cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
511 if(k != row_of_biggest_in_corner) {
512 m_qr.row(k).tail(cols-k).swap(m_qr.row(row_of_biggest_in_corner).tail(cols-k));
513 ++number_of_transpositions;
514 }
515 if(k != col_of_biggest_in_corner) {
516 m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
517 ++number_of_transpositions;
518 }
519
520 RealScalar beta;
521 m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
522 m_qr.coeffRef(k,k) = beta;
523
524 // remember the maximum absolute value of diagonal coefficients
525 if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
526
527 m_qr.bottomRightCorner(rows-k, cols-k-1)
528 .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
529 }
530
531 m_cols_permutation.setIdentity(cols);
532 for(Index k = 0; k < size; ++k)
533 m_cols_permutation.applyTranspositionOnTheRight(k, m_cols_transpositions.coeff(k));
534
535 m_det_pq = (number_of_transpositions%2) ? -1 : 1;
536 m_isInitialized = true;
537}
538
539#ifndef EIGEN_PARSED_BY_DOXYGEN
540template<typename _MatrixType>
541template<typename RhsType, typename DstType>
542void FullPivHouseholderQR<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const
543{
544 eigen_assert(rhs.rows() == rows());
545 const Index l_rank = rank();
546
547 // FIXME introduce nonzeroPivots() and use it here. and more generally,
548 // make the same improvements in this dec as in FullPivLU.
549 if(l_rank==0)
550 {
551 dst.setZero();
552 return;
553 }
554
555 typename RhsType::PlainObject c(rhs);
556
557 Matrix<Scalar,1,RhsType::ColsAtCompileTime> temp(rhs.cols());
558 for (Index k = 0; k < l_rank; ++k)
559 {
560 Index remainingSize = rows()-k;
561 c.row(k).swap(c.row(m_rows_transpositions.coeff(k)));
562 c.bottomRightCorner(remainingSize, rhs.cols())
563 .applyHouseholderOnTheLeft(m_qr.col(k).tail(remainingSize-1),
564 m_hCoeffs.coeff(k), &temp.coeffRef(0));
565 }
566
567 m_qr.topLeftCorner(l_rank, l_rank)
568 .template triangularView<Upper>()
569 .solveInPlace(c.topRows(l_rank));
570
571 for(Index i = 0; i < l_rank; ++i) dst.row(m_cols_permutation.indices().coeff(i)) = c.row(i);
572 for(Index i = l_rank; i < cols(); ++i) dst.row(m_cols_permutation.indices().coeff(i)).setZero();
573}
574#endif
575
576namespace internal {
577
578template<typename DstXprType, typename MatrixType>
579struct Assignment<DstXprType, Inverse<FullPivHouseholderQR<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename FullPivHouseholderQR<MatrixType>::Scalar>, Dense2Dense>
580{
581 typedef FullPivHouseholderQR<MatrixType> QrType;
582 typedef Inverse<QrType> SrcXprType;
583 static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename QrType::Scalar> &)
584 {
585 dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
586 }
587};
588
595template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType
596 : public ReturnByValue<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
597{
598public:
599 typedef typename FullPivHouseholderQR<MatrixType>::IntDiagSizeVectorType IntDiagSizeVectorType;
600 typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
601 typedef Matrix<typename MatrixType::Scalar, 1, MatrixType::RowsAtCompileTime, RowMajor, 1,
602 MatrixType::MaxRowsAtCompileTime> WorkVectorType;
603
604 FullPivHouseholderQRMatrixQReturnType(const MatrixType& qr,
605 const HCoeffsType& hCoeffs,
606 const IntDiagSizeVectorType& rowsTranspositions)
607 : m_qr(qr),
608 m_hCoeffs(hCoeffs),
609 m_rowsTranspositions(rowsTranspositions)
610 {}
611
612 template <typename ResultType>
613 void evalTo(ResultType& result) const
614 {
615 const Index rows = m_qr.rows();
616 WorkVectorType workspace(rows);
617 evalTo(result, workspace);
618 }
619
620 template <typename ResultType>
621 void evalTo(ResultType& result, WorkVectorType& workspace) const
622 {
623 using numext::conj;
624 // compute the product H'_0 H'_1 ... H'_n-1,
625 // where H_k is the k-th Householder transformation I - h_k v_k v_k'
626 // and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
627 const Index rows = m_qr.rows();
628 const Index cols = m_qr.cols();
629 const Index size = (std::min)(rows, cols);
630 workspace.resize(rows);
631 result.setIdentity(rows, rows);
632 for (Index k = size-1; k >= 0; k--)
633 {
634 result.block(k, k, rows-k, rows-k)
635 .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), conj(m_hCoeffs.coeff(k)), &workspace.coeffRef(k));
636 result.row(k).swap(result.row(m_rowsTranspositions.coeff(k)));
637 }
638 }
639
640 Index rows() const { return m_qr.rows(); }
641 Index cols() const { return m_qr.rows(); }
642
643protected:
644 typename MatrixType::Nested m_qr;
645 typename HCoeffsType::Nested m_hCoeffs;
646 typename IntDiagSizeVectorType::Nested m_rowsTranspositions;
647};
648
649// template<typename MatrixType>
650// struct evaluator<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
651// : public evaluator<ReturnByValue<FullPivHouseholderQRMatrixQReturnType<MatrixType> > >
652// {};
653
654} // end namespace internal
655
656template<typename MatrixType>
657inline typename FullPivHouseholderQR<MatrixType>::MatrixQReturnType FullPivHouseholderQR<MatrixType>::matrixQ() const
658{
659 eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
660 return MatrixQReturnType(m_qr, m_hCoeffs, m_rows_transpositions);
661}
662
667template<typename Derived>
670{
672}
673
674} // end namespace Eigen
675
676#endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
Derived & derived()
Definition: EigenBase.h:44
Householder rank-revealing QR decomposition of a matrix with full pivoting.
Definition: FullPivHouseholderQR.h:58
FullPivHouseholderQR & setThreshold(const RealScalar &threshold)
Definition: FullPivHouseholderQR.h:344
MatrixType::RealScalar absDeterminant() const
Definition: FullPivHouseholderQR.h:425
const MatrixType & matrixQR() const
Definition: FullPivHouseholderQR.h:188
FullPivHouseholderQR & setThreshold(Default_t)
Definition: FullPivHouseholderQR.h:359
const PermutationType & colsPermutation() const
Definition: FullPivHouseholderQR.h:198
Index dimensionOfKernel() const
Definition: FullPivHouseholderQR.h:263
const IntDiagSizeVectorType & rowsTranspositions() const
Definition: FullPivHouseholderQR.h:205
bool isInjective() const
Definition: FullPivHouseholderQR.h:276
const HCoeffsType & hCoeffs() const
Definition: FullPivHouseholderQR.h:325
RealScalar maxPivot() const
Definition: FullPivHouseholderQR.h:394
bool isSurjective() const
Definition: FullPivHouseholderQR.h:289
MatrixType::RealScalar logAbsDeterminant() const
Definition: FullPivHouseholderQR.h:434
FullPivHouseholderQR(Index rows, Index cols)
Default Constructor with memory preallocation.
Definition: FullPivHouseholderQR.h:103
FullPivHouseholderQR(EigenBase< InputType > &matrix)
Constructs a QR factorization from a given matrix.
Definition: FullPivHouseholderQR.h:146
const Solve< FullPivHouseholderQR, Rhs > solve(const MatrixBase< Rhs > &b) const
Definition: FullPivHouseholderQR.h:176
MatrixQReturnType matrixQ(void) const
Definition: FullPivHouseholderQR.h:657
const Inverse< FullPivHouseholderQR > inverse() const
Definition: FullPivHouseholderQR.h:312
Index rank() const
Definition: FullPivHouseholderQR.h:246
FullPivHouseholderQR()
Default Constructor.
Definition: FullPivHouseholderQR.h:87
bool isInvertible() const
Definition: FullPivHouseholderQR.h:301
FullPivHouseholderQR(const EigenBase< InputType > &matrix)
Constructs a QR factorization from a given matrix.
Definition: FullPivHouseholderQR.h:126
Index nonzeroPivots() const
Definition: FullPivHouseholderQR.h:385
RealScalar threshold() const
Definition: FullPivHouseholderQR.h:369
Expression of the inverse of another expression.
Definition: Inverse.h:44
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:50
The matrix class, also used for vectors and row-vectors.
Definition: Matrix.h:180
Permutation matrix.
Definition: PermutationMatrix.h:309
Pseudo expression representing a solving operation.
Definition: Solve.h:63
@ RowMajor
Definition: Constants.h:322
Namespace containing all symbols from the Eigen library.
Definition: Core:287
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_conjugate_op< typename Derived::Scalar >, const Derived > conj(const Eigen::ArrayBase< Derived > &x)
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:33
Definition: EigenBase.h:29
Derived & derived()
Definition: EigenBase.h:44
Holds information about the various numeric (i.e. scalar) types allowed by Eigen.
Definition: NumTraits.h:151