1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
5// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
6// Copyright (C) 2010 Vincent Lejeune
7//
8// This Source Code Form is subject to the terms of the Mozilla
9// Public License v. 2.0. If a copy of the MPL was not distributed
10// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
11
12#ifndef EIGEN_QR_H
13#define EIGEN_QR_H
14
15namespace Eigen {
16
17/** \ingroup QR_Module
18  *
19  *
20  * \class HouseholderQR
21  *
22  * \brief Householder QR decomposition of a matrix
23  *
24  * \param MatrixType the type of the matrix of which we are computing the QR decomposition
25  *
26  * This class performs a QR decomposition of a matrix \b A into matrices \b Q and \b R
27  * such that
28  * \f[
29  *  \mathbf{A} = \mathbf{Q} \, \mathbf{R}
30  * \f]
31  * by using Householder transformations. Here, \b Q a unitary matrix and \b R an upper triangular matrix.
32  * The result is stored in a compact way compatible with LAPACK.
33  *
34  * Note that no pivoting is performed. This is \b not a rank-revealing decomposition.
35  * If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.
36  *
37  * This Householder QR decomposition is faster, but less numerically stable and less feature-full than
38  * FullPivHouseholderQR or ColPivHouseholderQR.
39  *
40  * \sa MatrixBase::householderQr()
41  */
42template<typename _MatrixType> class HouseholderQR
43{
44  public:
45
46    typedef _MatrixType MatrixType;
47    enum {
48      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
49      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
50      Options = MatrixType::Options,
51      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
52      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
53    };
54    typedef typename MatrixType::Scalar Scalar;
55    typedef typename MatrixType::RealScalar RealScalar;
56    typedef typename MatrixType::Index Index;
57    typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, (MatrixType::Flags&RowMajorBit) ? RowMajor : ColMajor, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
58    typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
59    typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
60    typedef typename HouseholderSequence<MatrixType,HCoeffsType>::ConjugateReturnType HouseholderSequenceType;
61
62    /**
63    * \brief Default Constructor.
64    *
65    * The default constructor is useful in cases in which the user intends to
66    * perform decompositions via HouseholderQR::compute(const MatrixType&).
67    */
68    HouseholderQR() : m_qr(), m_hCoeffs(), m_temp(), m_isInitialized(false) {}
69
70    /** \brief Default Constructor with memory preallocation
71      *
72      * Like the default constructor but with preallocation of the internal data
73      * according to the specified problem \a size.
74      * \sa HouseholderQR()
75      */
76    HouseholderQR(Index rows, Index cols)
77      : m_qr(rows, cols),
78        m_hCoeffs((std::min)(rows,cols)),
79        m_temp(cols),
80        m_isInitialized(false) {}
81
82    HouseholderQR(const MatrixType& matrix)
83      : m_qr(matrix.rows(), matrix.cols()),
84        m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
85        m_temp(matrix.cols()),
86        m_isInitialized(false)
87    {
88      compute(matrix);
89    }
90
91    /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
92      * *this is the QR decomposition, if any exists.
93      *
94      * \param b the right-hand-side of the equation to solve.
95      *
96      * \returns a solution.
97      *
98      * \note The case where b is a matrix is not yet implemented. Also, this
99      *       code is space inefficient.
100      *
101      * \note_about_checking_solutions
102      *
103      * \note_about_arbitrary_choice_of_solution
104      *
105      * Example: \include HouseholderQR_solve.cpp
106      * Output: \verbinclude HouseholderQR_solve.out
107      */
108    template<typename Rhs>
109    inline const internal::solve_retval<HouseholderQR, Rhs>
110    solve(const MatrixBase<Rhs>& b) const
111    {
112      eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
113      return internal::solve_retval<HouseholderQR, Rhs>(*this, b.derived());
114    }
115
116    HouseholderSequenceType householderQ() const
117    {
118      eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
119      return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
120    }
121
122    /** \returns a reference to the matrix where the Householder QR decomposition is stored
123      * in a LAPACK-compatible way.
124      */
125    const MatrixType& matrixQR() const
126    {
127        eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
128        return m_qr;
129    }
130
131    HouseholderQR& compute(const MatrixType& matrix);
132
133    /** \returns the absolute value of the determinant of the matrix of which
134      * *this is the QR decomposition. It has only linear complexity
135      * (that is, O(n) where n is the dimension of the square matrix)
136      * as the QR decomposition has already been computed.
137      *
138      * \note This is only for square matrices.
139      *
140      * \warning a determinant can be very big or small, so for matrices
141      * of large enough dimension, there is a risk of overflow/underflow.
142      * One way to work around that is to use logAbsDeterminant() instead.
143      *
144      * \sa logAbsDeterminant(), MatrixBase::determinant()
145      */
146    typename MatrixType::RealScalar absDeterminant() const;
147
148    /** \returns the natural log of the absolute value of the determinant of the matrix of which
149      * *this is the QR decomposition. It has only linear complexity
150      * (that is, O(n) where n is the dimension of the square matrix)
151      * as the QR decomposition has already been computed.
152      *
153      * \note This is only for square matrices.
154      *
155      * \note This method is useful to work around the risk of overflow/underflow that's inherent
156      * to determinant computation.
157      *
158      * \sa absDeterminant(), MatrixBase::determinant()
159      */
160    typename MatrixType::RealScalar logAbsDeterminant() const;
161
162    inline Index rows() const { return m_qr.rows(); }
163    inline Index cols() const { return m_qr.cols(); }
164    const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
165
166  protected:
167    MatrixType m_qr;
168    HCoeffsType m_hCoeffs;
169    RowVectorType m_temp;
170    bool m_isInitialized;
171};
172
173template<typename MatrixType>
174typename MatrixType::RealScalar HouseholderQR<MatrixType>::absDeterminant() const
175{
176  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
177  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
178  return internal::abs(m_qr.diagonal().prod());
179}
180
181template<typename MatrixType>
182typename MatrixType::RealScalar HouseholderQR<MatrixType>::logAbsDeterminant() const
183{
184  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
185  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
186  return m_qr.diagonal().cwiseAbs().array().log().sum();
187}
188
189namespace internal {
190
191/** \internal */
192template<typename MatrixQR, typename HCoeffs>
193void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0)
194{
195  typedef typename MatrixQR::Index Index;
196  typedef typename MatrixQR::Scalar Scalar;
197  typedef typename MatrixQR::RealScalar RealScalar;
198  Index rows = mat.rows();
199  Index cols = mat.cols();
200  Index size = (std::min)(rows,cols);
201
202  eigen_assert(hCoeffs.size() == size);
203
204  typedef Matrix<Scalar,MatrixQR::ColsAtCompileTime,1> TempType;
205  TempType tempVector;
206  if(tempData==0)
207  {
208    tempVector.resize(cols);
209    tempData = tempVector.data();
210  }
211
212  for(Index k = 0; k < size; ++k)
213  {
214    Index remainingRows = rows - k;
215    Index remainingCols = cols - k - 1;
216
217    RealScalar beta;
218    mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
219    mat.coeffRef(k,k) = beta;
220
221    // apply H to remaining part of m_qr from the left
222    mat.bottomRightCorner(remainingRows, remainingCols)
223        .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows-1), hCoeffs.coeffRef(k), tempData+k+1);
224  }
225}
226
227/** \internal */
228template<typename MatrixQR, typename HCoeffs>
229void householder_qr_inplace_blocked(MatrixQR& mat, HCoeffs& hCoeffs,
230                                       typename MatrixQR::Index maxBlockSize=32,
231                                       typename MatrixQR::Scalar* tempData = 0)
232{
233  typedef typename MatrixQR::Index Index;
234  typedef typename MatrixQR::Scalar Scalar;
235  typedef typename MatrixQR::RealScalar RealScalar;
236  typedef Block<MatrixQR,Dynamic,Dynamic> BlockType;
237
238  Index rows = mat.rows();
239  Index cols = mat.cols();
240  Index size = (std::min)(rows, cols);
241
242  typedef Matrix<Scalar,Dynamic,1,ColMajor,MatrixQR::MaxColsAtCompileTime,1> TempType;
243  TempType tempVector;
244  if(tempData==0)
245  {
246    tempVector.resize(cols);
247    tempData = tempVector.data();
248  }
249
250  Index blockSize = (std::min)(maxBlockSize,size);
251
252  Index k = 0;
253  for (k = 0; k < size; k += blockSize)
254  {
255    Index bs = (std::min)(size-k,blockSize);  // actual size of the block
256    Index tcols = cols - k - bs;            // trailing columns
257    Index brows = rows-k;                   // rows of the block
258
259    // partition the matrix:
260    //        A00 | A01 | A02
261    // mat  = A10 | A11 | A12
262    //        A20 | A21 | A22
263    // and performs the qr dec of [A11^T A12^T]^T
264    // and update [A21^T A22^T]^T using level 3 operations.
265    // Finally, the algorithm continue on A22
266
267    BlockType A11_21 = mat.block(k,k,brows,bs);
268    Block<HCoeffs,Dynamic,1> hCoeffsSegment = hCoeffs.segment(k,bs);
269
270    householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);
271
272    if(tcols)
273    {
274      BlockType A21_22 = mat.block(k,k+bs,brows,tcols);
275      apply_block_householder_on_the_left(A21_22,A11_21,hCoeffsSegment.adjoint());
276    }
277  }
278}
279
280template<typename _MatrixType, typename Rhs>
281struct solve_retval<HouseholderQR<_MatrixType>, Rhs>
282  : solve_retval_base<HouseholderQR<_MatrixType>, Rhs>
283{
284  EIGEN_MAKE_SOLVE_HELPERS(HouseholderQR<_MatrixType>,Rhs)
285
286  template<typename Dest> void evalTo(Dest& dst) const
287  {
288    const Index rows = dec().rows(), cols = dec().cols();
289    const Index rank = (std::min)(rows, cols);
290    eigen_assert(rhs().rows() == rows);
291
292    typename Rhs::PlainObject c(rhs());
293
294    // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
295    c.applyOnTheLeft(householderSequence(
296      dec().matrixQR().leftCols(rank),
297      dec().hCoeffs().head(rank)).transpose()
298    );
299
300    dec().matrixQR()
301       .topLeftCorner(rank, rank)
302       .template triangularView<Upper>()
303       .solveInPlace(c.topRows(rank));
304
305    dst.topRows(rank) = c.topRows(rank);
306    dst.bottomRows(cols-rank).setZero();
307  }
308};
309
310} // end namespace internal
311
312template<typename MatrixType>
313HouseholderQR<MatrixType>& HouseholderQR<MatrixType>::compute(const MatrixType& matrix)
314{
315  Index rows = matrix.rows();
316  Index cols = matrix.cols();
317  Index size = (std::min)(rows,cols);
318
319  m_qr = matrix;
320  m_hCoeffs.resize(size);
321
322  m_temp.resize(cols);
323
324  internal::householder_qr_inplace_blocked(m_qr, m_hCoeffs, 48, m_temp.data());
325
326  m_isInitialized = true;
327  return *this;
328}
329
330/** \return the Householder QR decomposition of \c *this.
331  *
332  * \sa class HouseholderQR
333  */
334template<typename Derived>
335const HouseholderQR<typename MatrixBase<Derived>::PlainObject>
336MatrixBase<Derived>::householderQr() const
337{
338  return HouseholderQR<PlainObject>(eval());
339}
340
341} // end namespace Eigen
342
343#endif // EIGEN_QR_H
344