Eigen  3.2.10
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 
14 namespace Eigen {
15 
37 template<typename _MatrixType> class ColPivHouseholderQR
38 {
39  public:
40 
41  typedef _MatrixType MatrixType;
42  enum {
43  RowsAtCompileTime = MatrixType::RowsAtCompileTime,
44  ColsAtCompileTime = MatrixType::ColsAtCompileTime,
45  Options = MatrixType::Options,
46  MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
47  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
48  };
49  typedef typename MatrixType::Scalar Scalar;
50  typedef typename MatrixType::RealScalar RealScalar;
51  typedef typename MatrixType::Index Index;
52  typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
53  typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
54  typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
55  typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
56  typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
57  typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
58  typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
59 
60  private:
61 
62  typedef typename PermutationType::Index PermIndexType;
63 
64  public:
65 
73  : m_qr(),
74  m_hCoeffs(),
75  m_colsPermutation(),
76  m_colsTranspositions(),
77  m_temp(),
78  m_colSqNorms(),
79  m_isInitialized(false),
80  m_usePrescribedThreshold(false) {}
81 
88  ColPivHouseholderQR(Index rows, Index cols)
89  : m_qr(rows, cols),
90  m_hCoeffs((std::min)(rows,cols)),
91  m_colsPermutation(PermIndexType(cols)),
92  m_colsTranspositions(cols),
93  m_temp(cols),
94  m_colSqNorms(cols),
95  m_isInitialized(false),
96  m_usePrescribedThreshold(false) {}
97 
110  ColPivHouseholderQR(const MatrixType& matrix)
111  : m_qr(matrix.rows(), matrix.cols()),
112  m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
113  m_colsPermutation(PermIndexType(matrix.cols())),
114  m_colsTranspositions(matrix.cols()),
115  m_temp(matrix.cols()),
116  m_colSqNorms(matrix.cols()),
117  m_isInitialized(false),
118  m_usePrescribedThreshold(false)
119  {
120  compute(matrix);
121  }
122 
137  template<typename Rhs>
138  inline const internal::solve_retval<ColPivHouseholderQR, Rhs>
139  solve(const MatrixBase<Rhs>& b) const
140  {
141  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
142  return internal::solve_retval<ColPivHouseholderQR, Rhs>(*this, b.derived());
143  }
144 
145  HouseholderSequenceType householderQ(void) const;
146  HouseholderSequenceType matrixQ(void) const
147  {
148  return householderQ();
149  }
150 
153  const MatrixType& matrixQR() const
154  {
155  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
156  return m_qr;
157  }
158 
168  const MatrixType& matrixR() const
169  {
170  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
171  return m_qr;
172  }
173 
174  ColPivHouseholderQR& compute(const MatrixType& matrix);
175 
177  const PermutationType& colsPermutation() const
178  {
179  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
180  return m_colsPermutation;
181  }
182 
196  typename MatrixType::RealScalar absDeterminant() const;
197 
210  typename MatrixType::RealScalar logAbsDeterminant() const;
211 
218  inline Index rank() const
219  {
220  using std::abs;
221  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
222  RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
223  Index result = 0;
224  for(Index i = 0; i < m_nonzero_pivots; ++i)
225  result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
226  return result;
227  }
228 
235  inline Index dimensionOfKernel() const
236  {
237  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
238  return cols() - rank();
239  }
240 
248  inline bool isInjective() const
249  {
250  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
251  return rank() == cols();
252  }
253 
261  inline bool isSurjective() const
262  {
263  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
264  return rank() == rows();
265  }
266 
273  inline bool isInvertible() const
274  {
275  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
276  return isInjective() && isSurjective();
277  }
278 
284  inline const
285  internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType>
286  inverse() const
287  {
288  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
289  return internal::solve_retval<ColPivHouseholderQR,typename MatrixType::IdentityReturnType>
290  (*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
291  }
292 
293  inline Index rows() const { return m_qr.rows(); }
294  inline Index cols() const { return m_qr.cols(); }
295 
300  const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
301 
320  {
321  m_usePrescribedThreshold = true;
322  m_prescribedThreshold = threshold;
323  return *this;
324  }
325 
335  {
336  m_usePrescribedThreshold = false;
337  return *this;
338  }
339 
344  RealScalar threshold() const
345  {
346  eigen_assert(m_isInitialized || m_usePrescribedThreshold);
347  return m_usePrescribedThreshold ? m_prescribedThreshold
348  // this formula comes from experimenting (see "LU precision tuning" thread on the list)
349  // and turns out to be identical to Higham's formula used already in LDLt.
350  : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
351  }
352 
360  inline Index nonzeroPivots() const
361  {
362  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
363  return m_nonzero_pivots;
364  }
365 
369  RealScalar maxPivot() const { return m_maxpivot; }
370 
378  {
379  eigen_assert(m_isInitialized && "Decomposition is not initialized.");
380  return Success;
381  }
382 
383  protected:
384 
385  static void check_template_parameters()
386  {
387  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
388  }
389 
390  MatrixType m_qr;
391  HCoeffsType m_hCoeffs;
392  PermutationType m_colsPermutation;
393  IntRowVectorType m_colsTranspositions;
394  RowVectorType m_temp;
395  RealRowVectorType m_colSqNorms;
396  bool m_isInitialized, m_usePrescribedThreshold;
397  RealScalar m_prescribedThreshold, m_maxpivot;
398  Index m_nonzero_pivots;
399  Index m_det_pq;
400 };
401 
402 template<typename MatrixType>
403 typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() const
404 {
405  using std::abs;
406  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
407  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
408  return abs(m_qr.diagonal().prod());
409 }
410 
411 template<typename MatrixType>
412 typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
413 {
414  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
415  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
416  return m_qr.diagonal().cwiseAbs().array().log().sum();
417 }
418 
425 template<typename MatrixType>
427 {
428  check_template_parameters();
429 
430  using std::abs;
431  Index rows = matrix.rows();
432  Index cols = matrix.cols();
433  Index size = matrix.diagonalSize();
434 
435  // the column permutation is stored as int indices, so just to be sure:
436  eigen_assert(cols<=NumTraits<int>::highest());
437 
438  m_qr = matrix;
439  m_hCoeffs.resize(size);
440 
441  m_temp.resize(cols);
442 
443  m_colsTranspositions.resize(matrix.cols());
444  Index number_of_transpositions = 0;
445 
446  m_colSqNorms.resize(cols);
447  for(Index k = 0; k < cols; ++k)
448  m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();
449 
450  RealScalar threshold_helper = m_colSqNorms.maxCoeff() * numext::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);
451 
452  m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
453  m_maxpivot = RealScalar(0);
454 
455  for(Index k = 0; k < size; ++k)
456  {
457  // first, we look up in our table m_colSqNorms which column has the biggest squared norm
458  Index biggest_col_index;
459  RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
460  biggest_col_index += k;
461 
462  // since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
463  // the actual squared norm of the selected column.
464  // Note that not doing so does result in solve() sometimes returning inf/nan values
465  // when running the unit test with 1000 repetitions.
466  biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();
467 
468  // we store that back into our table: it can't hurt to correct our table.
469  m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;
470 
471  // Track the number of meaningful pivots but do not stop the decomposition to make
472  // sure that the initial matrix is properly reproduced. See bug 941.
473  if(m_nonzero_pivots==size && biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
474  m_nonzero_pivots = k;
475 
476  // apply the transposition to the columns
477  m_colsTranspositions.coeffRef(k) = biggest_col_index;
478  if(k != biggest_col_index) {
479  m_qr.col(k).swap(m_qr.col(biggest_col_index));
480  std::swap(m_colSqNorms.coeffRef(k), m_colSqNorms.coeffRef(biggest_col_index));
481  ++number_of_transpositions;
482  }
483 
484  // generate the householder vector, store it below the diagonal
485  RealScalar beta;
486  m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
487 
488  // apply the householder transformation to the diagonal coefficient
489  m_qr.coeffRef(k,k) = beta;
490 
491  // remember the maximum absolute value of diagonal coefficients
492  if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
493 
494  // apply the householder transformation
495  m_qr.bottomRightCorner(rows-k, cols-k-1)
496  .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
497 
498  // update our table of squared norms of the columns
499  m_colSqNorms.tail(cols-k-1) -= m_qr.row(k).tail(cols-k-1).cwiseAbs2();
500  }
501 
502  m_colsPermutation.setIdentity(PermIndexType(cols));
503  for(PermIndexType k = 0; k < size/*m_nonzero_pivots*/; ++k)
504  m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));
505 
506  m_det_pq = (number_of_transpositions%2) ? -1 : 1;
507  m_isInitialized = true;
508 
509  return *this;
510 }
511 
512 namespace internal {
513 
514 template<typename _MatrixType, typename Rhs>
515 struct solve_retval<ColPivHouseholderQR<_MatrixType>, Rhs>
516  : solve_retval_base<ColPivHouseholderQR<_MatrixType>, Rhs>
517 {
518  EIGEN_MAKE_SOLVE_HELPERS(ColPivHouseholderQR<_MatrixType>,Rhs)
519 
520  template<typename Dest> void evalTo(Dest& dst) const
521  {
522  eigen_assert(rhs().rows() == dec().rows());
523 
524  const Index cols = dec().cols(),
525  nonzero_pivots = dec().nonzeroPivots();
526 
527  if(nonzero_pivots == 0)
528  {
529  dst.setZero();
530  return;
531  }
532 
533  typename Rhs::PlainObject c(rhs());
534 
535  // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
536  c.applyOnTheLeft(householderSequence(dec().matrixQR(), dec().hCoeffs())
537  .setLength(dec().nonzeroPivots())
538  .transpose()
539  );
540 
541  dec().matrixR()
542  .topLeftCorner(nonzero_pivots, nonzero_pivots)
543  .template triangularView<Upper>()
544  .solveInPlace(c.topRows(nonzero_pivots));
545 
546  for(Index i = 0; i < nonzero_pivots; ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
547  for(Index i = nonzero_pivots; i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
548  }
549 };
550 
551 } // end namespace internal
552 
556 template<typename MatrixType>
559 {
560  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
561  return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
562 }
563 
568 template<typename Derived>
571 {
572  return ColPivHouseholderQR<PlainObject>(eval());
573 }
574 
575 } // end namespace Eigen
576 
577 #endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
void setIdentity()
Definition: PermutationMatrix.h:148
MatrixType::RealScalar absDeterminant() const
Definition: ColPivHouseholderQR.h:403
MatrixType::RealScalar logAbsDeterminant() const
Definition: ColPivHouseholderQR.h:412
Derived & applyTranspositionOnTheRight(Index i, Index j)
Definition: PermutationMatrix.h:190
RealScalar maxPivot() const
Definition: ColPivHouseholderQR.h:369
RealScalar threshold() const
Definition: ColPivHouseholderQR.h:344
Definition: LDLT.h:16
Definition: StdDeque.h:50
Holds information about the various numeric (i.e. scalar) types allowed by Eigen. ...
Definition: NumTraits.h:88
const HCoeffsType & hCoeffs() const
Definition: ColPivHouseholderQR.h:300
const MatrixType & matrixQR() const
Definition: ColPivHouseholderQR.h:153
HouseholderSequence< VectorsType, CoeffsType > householderSequence(const VectorsType &v, const CoeffsType &h)
Convenience function for constructing a Householder sequence.
Definition: HouseholderSequence.h:423
ColPivHouseholderQR()
Default Constructor.
Definition: ColPivHouseholderQR.h:72
ComputationInfo info() const
Reports whether the QR factorization was succesful.
Definition: ColPivHouseholderQR.h:377
HouseholderSequenceType householderQ(void) const
Definition: ColPivHouseholderQR.h:558
ColPivHouseholderQR(const MatrixType &matrix)
Constructs a QR factorization from a given matrix.
Definition: ColPivHouseholderQR.h:110
Sequence of Householder reflections acting on subspaces with decreasing size.
Definition: ForwardDeclarations.h:227
ColPivHouseholderQR & compute(const MatrixType &matrix)
Definition: ColPivHouseholderQR.h:426
const PermutationType & colsPermutation() const
Definition: ColPivHouseholderQR.h:177
bool isSurjective() const
Definition: ColPivHouseholderQR.h:261
Householder rank-revealing QR decomposition of a matrix with column-pivoting.
Definition: ForwardDeclarations.h:222
bool isInjective() const
Definition: ColPivHouseholderQR.h:248
bool isInvertible() const
Definition: ColPivHouseholderQR.h:273
ColPivHouseholderQR & setThreshold(Default_t)
Definition: ColPivHouseholderQR.h:334
ColPivHouseholderQR & setThreshold(const RealScalar &threshold)
Definition: ColPivHouseholderQR.h:319
Definition: Eigen_Colamd.h:50
const internal::solve_retval< ColPivHouseholderQR, Rhs > solve(const MatrixBase< Rhs > &b) const
Definition: ColPivHouseholderQR.h:139
const MatrixType & matrixR() const
Definition: ColPivHouseholderQR.h:168
Index nonzeroPivots() const
Definition: ColPivHouseholderQR.h:360
Definition: Constants.h:376
const internal::solve_retval< ColPivHouseholderQR, typename MatrixType::IdentityReturnType > inverse() const
Definition: ColPivHouseholderQR.h:286
Index dimensionOfKernel() const
Definition: ColPivHouseholderQR.h:235
ColPivHouseholderQR(Index rows, Index cols)
Default Constructor with memory preallocation.
Definition: ColPivHouseholderQR.h:88
ComputationInfo
Definition: Constants.h:374
Index rank() const
Definition: ColPivHouseholderQR.h:218
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:48
const ColPivHouseholderQR< PlainObject > colPivHouseholderQr() const
Definition: ColPivHouseholderQR.h:570