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 HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> 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    /** \brief Constructs a QR factorization from a given matrix
83      *
84      * This constructor computes the QR factorization of the matrix \a matrix by calling
85      * the method compute(). It is a short cut for:
86      *
87      * \code
88      * HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
89      * qr.compute(matrix);
90      * \endcode
91      *
92      * \sa compute()
93      */
94    HouseholderQR(const MatrixType& matrix)
95      : m_qr(matrix.rows(), matrix.cols()),
96        m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
97        m_temp(matrix.cols()),
98        m_isInitialized(false)
99    {
100      compute(matrix);
101    }
102
103    /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
104      * *this is the QR decomposition, if any exists.
105      *
106      * \param b the right-hand-side of the equation to solve.
107      *
108      * \returns a solution.
109      *
110      * \note The case where b is a matrix is not yet implemented. Also, this
111      *       code is space inefficient.
112      *
113      * \note_about_checking_solutions
114      *
115      * \note_about_arbitrary_choice_of_solution
116      *
117      * Example: \include HouseholderQR_solve.cpp
118      * Output: \verbinclude HouseholderQR_solve.out
119      */
120    template<typename Rhs>
121    inline const internal::solve_retval<HouseholderQR, Rhs>
122    solve(const MatrixBase<Rhs>& b) const
123    {
124      eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
125      return internal::solve_retval<HouseholderQR, Rhs>(*this, b.derived());
126    }
127
128    /** This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.
129      *
130      * The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix object.
131      * Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:
132      *
133      * Example: \include HouseholderQR_householderQ.cpp
134      * Output: \verbinclude HouseholderQR_householderQ.out
135      */
136    HouseholderSequenceType householderQ() const
137    {
138      eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
139      return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
140    }
141
142    /** \returns a reference to the matrix where the Householder QR decomposition is stored
143      * in a LAPACK-compatible way.
144      */
145    const MatrixType& matrixQR() const
146    {
147        eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
148        return m_qr;
149    }
150
151    HouseholderQR& compute(const MatrixType& matrix);
152
153    /** \returns the absolute value of the determinant of the matrix of which
154      * *this is the QR decomposition. It has only linear complexity
155      * (that is, O(n) where n is the dimension of the square matrix)
156      * as the QR decomposition has already been computed.
157      *
158      * \note This is only for square matrices.
159      *
160      * \warning a determinant can be very big or small, so for matrices
161      * of large enough dimension, there is a risk of overflow/underflow.
162      * One way to work around that is to use logAbsDeterminant() instead.
163      *
164      * \sa logAbsDeterminant(), MatrixBase::determinant()
165      */
166    typename MatrixType::RealScalar absDeterminant() const;
167
168    /** \returns the natural log of the absolute value of the determinant of the matrix of which
169      * *this is the QR decomposition. It has only linear complexity
170      * (that is, O(n) where n is the dimension of the square matrix)
171      * as the QR decomposition has already been computed.
172      *
173      * \note This is only for square matrices.
174      *
175      * \note This method is useful to work around the risk of overflow/underflow that's inherent
176      * to determinant computation.
177      *
178      * \sa absDeterminant(), MatrixBase::determinant()
179      */
180    typename MatrixType::RealScalar logAbsDeterminant() const;
181
182    inline Index rows() const { return m_qr.rows(); }
183    inline Index cols() const { return m_qr.cols(); }
184
185    /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
186      *
187      * For advanced uses only.
188      */
189    const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
190
191  protected:
192    MatrixType m_qr;
193    HCoeffsType m_hCoeffs;
194    RowVectorType m_temp;
195    bool m_isInitialized;
196};
197
198template<typename MatrixType>
199typename MatrixType::RealScalar HouseholderQR<MatrixType>::absDeterminant() const
200{
201  using std::abs;
202  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
203  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
204  return abs(m_qr.diagonal().prod());
205}
206
207template<typename MatrixType>
208typename MatrixType::RealScalar HouseholderQR<MatrixType>::logAbsDeterminant() const
209{
210  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
211  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
212  return m_qr.diagonal().cwiseAbs().array().log().sum();
213}
214
215namespace internal {
216
217/** \internal */
218template<typename MatrixQR, typename HCoeffs>
219void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0)
220{
221  typedef typename MatrixQR::Index Index;
222  typedef typename MatrixQR::Scalar Scalar;
223  typedef typename MatrixQR::RealScalar RealScalar;
224  Index rows = mat.rows();
225  Index cols = mat.cols();
226  Index size = (std::min)(rows,cols);
227
228  eigen_assert(hCoeffs.size() == size);
229
230  typedef Matrix<Scalar,MatrixQR::ColsAtCompileTime,1> TempType;
231  TempType tempVector;
232  if(tempData==0)
233  {
234    tempVector.resize(cols);
235    tempData = tempVector.data();
236  }
237
238  for(Index k = 0; k < size; ++k)
239  {
240    Index remainingRows = rows - k;
241    Index remainingCols = cols - k - 1;
242
243    RealScalar beta;
244    mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
245    mat.coeffRef(k,k) = beta;
246
247    // apply H to remaining part of m_qr from the left
248    mat.bottomRightCorner(remainingRows, remainingCols)
249        .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows-1), hCoeffs.coeffRef(k), tempData+k+1);
250  }
251}
252
253/** \internal */
254template<typename MatrixQR, typename HCoeffs>
255void householder_qr_inplace_blocked(MatrixQR& mat, HCoeffs& hCoeffs,
256                                       typename MatrixQR::Index maxBlockSize=32,
257                                       typename MatrixQR::Scalar* tempData = 0)
258{
259  typedef typename MatrixQR::Index Index;
260  typedef typename MatrixQR::Scalar Scalar;
261  typedef Block<MatrixQR,Dynamic,Dynamic> BlockType;
262
263  Index rows = mat.rows();
264  Index cols = mat.cols();
265  Index size = (std::min)(rows, cols);
266
267  typedef Matrix<Scalar,Dynamic,1,ColMajor,MatrixQR::MaxColsAtCompileTime,1> TempType;
268  TempType tempVector;
269  if(tempData==0)
270  {
271    tempVector.resize(cols);
272    tempData = tempVector.data();
273  }
274
275  Index blockSize = (std::min)(maxBlockSize,size);
276
277  Index k = 0;
278  for (k = 0; k < size; k += blockSize)
279  {
280    Index bs = (std::min)(size-k,blockSize);  // actual size of the block
281    Index tcols = cols - k - bs;            // trailing columns
282    Index brows = rows-k;                   // rows of the block
283
284    // partition the matrix:
285    //        A00 | A01 | A02
286    // mat  = A10 | A11 | A12
287    //        A20 | A21 | A22
288    // and performs the qr dec of [A11^T A12^T]^T
289    // and update [A21^T A22^T]^T using level 3 operations.
290    // Finally, the algorithm continue on A22
291
292    BlockType A11_21 = mat.block(k,k,brows,bs);
293    Block<HCoeffs,Dynamic,1> hCoeffsSegment = hCoeffs.segment(k,bs);
294
295    householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);
296
297    if(tcols)
298    {
299      BlockType A21_22 = mat.block(k,k+bs,brows,tcols);
300      apply_block_householder_on_the_left(A21_22,A11_21,hCoeffsSegment.adjoint());
301    }
302  }
303}
304
305template<typename _MatrixType, typename Rhs>
306struct solve_retval<HouseholderQR<_MatrixType>, Rhs>
307  : solve_retval_base<HouseholderQR<_MatrixType>, Rhs>
308{
309  EIGEN_MAKE_SOLVE_HELPERS(HouseholderQR<_MatrixType>,Rhs)
310
311  template<typename Dest> void evalTo(Dest& dst) const
312  {
313    const Index rows = dec().rows(), cols = dec().cols();
314    const Index rank = (std::min)(rows, cols);
315    eigen_assert(rhs().rows() == rows);
316
317    typename Rhs::PlainObject c(rhs());
318
319    // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
320    c.applyOnTheLeft(householderSequence(
321      dec().matrixQR().leftCols(rank),
322      dec().hCoeffs().head(rank)).transpose()
323    );
324
325    dec().matrixQR()
326       .topLeftCorner(rank, rank)
327       .template triangularView<Upper>()
328       .solveInPlace(c.topRows(rank));
329
330    dst.topRows(rank) = c.topRows(rank);
331    dst.bottomRows(cols-rank).setZero();
332  }
333};
334
335} // end namespace internal
336
337/** Performs the QR factorization of the given matrix \a matrix. The result of
338  * the factorization is stored into \c *this, and a reference to \c *this
339  * is returned.
340  *
341  * \sa class HouseholderQR, HouseholderQR(const MatrixType&)
342  */
343template<typename MatrixType>
344HouseholderQR<MatrixType>& HouseholderQR<MatrixType>::compute(const MatrixType& matrix)
345{
346  Index rows = matrix.rows();
347  Index cols = matrix.cols();
348  Index size = (std::min)(rows,cols);
349
350  m_qr = matrix;
351  m_hCoeffs.resize(size);
352
353  m_temp.resize(cols);
354
355  internal::householder_qr_inplace_blocked(m_qr, m_hCoeffs, 48, m_temp.data());
356
357  m_isInitialized = true;
358  return *this;
359}
360
361/** \return the Householder QR decomposition of \c *this.
362  *
363  * \sa class HouseholderQR
364  */
365template<typename Derived>
366const HouseholderQR<typename MatrixBase<Derived>::PlainObject>
367MatrixBase<Derived>::householderQr() const
368{
369  return HouseholderQR<PlainObject>(eval());
370}
371
372} // end namespace Eigen
373
374#endif // EIGEN_QR_H
375