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  * \tparam _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  * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
41  *
42  * \sa MatrixBase::householderQr()
43  */
44template<typename _MatrixType> class HouseholderQR
45{
46  public:
47
48    typedef _MatrixType MatrixType;
49    enum {
50      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
51      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
52      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
53      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
54    };
55    typedef typename MatrixType::Scalar Scalar;
56    typedef typename MatrixType::RealScalar RealScalar;
57    // FIXME should be int
58    typedef typename MatrixType::StorageIndex StorageIndex;
59    typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, (MatrixType::Flags&RowMajorBit) ? RowMajor : ColMajor, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
60    typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
61    typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
62    typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
63
64    /**
65      * \brief Default Constructor.
66      *
67      * The default constructor is useful in cases in which the user intends to
68      * perform decompositions via HouseholderQR::compute(const MatrixType&).
69      */
70    HouseholderQR() : m_qr(), m_hCoeffs(), m_temp(), m_isInitialized(false) {}
71
72    /** \brief Default Constructor with memory preallocation
73      *
74      * Like the default constructor but with preallocation of the internal data
75      * according to the specified problem \a size.
76      * \sa HouseholderQR()
77      */
78    HouseholderQR(Index rows, Index cols)
79      : m_qr(rows, cols),
80        m_hCoeffs((std::min)(rows,cols)),
81        m_temp(cols),
82        m_isInitialized(false) {}
83
84    /** \brief Constructs a QR factorization from a given matrix
85      *
86      * This constructor computes the QR factorization of the matrix \a matrix by calling
87      * the method compute(). It is a short cut for:
88      *
89      * \code
90      * HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
91      * qr.compute(matrix);
92      * \endcode
93      *
94      * \sa compute()
95      */
96    template<typename InputType>
97    explicit HouseholderQR(const EigenBase<InputType>& matrix)
98      : m_qr(matrix.rows(), matrix.cols()),
99        m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
100        m_temp(matrix.cols()),
101        m_isInitialized(false)
102    {
103      compute(matrix.derived());
104    }
105
106
107    /** \brief Constructs a QR factorization from a given matrix
108      *
109      * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when
110      * \c MatrixType is a Eigen::Ref.
111      *
112      * \sa HouseholderQR(const EigenBase&)
113      */
114    template<typename InputType>
115    explicit HouseholderQR(EigenBase<InputType>& matrix)
116      : m_qr(matrix.derived()),
117        m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
118        m_temp(matrix.cols()),
119        m_isInitialized(false)
120    {
121      computeInPlace();
122    }
123
124    /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
125      * *this is the QR decomposition, if any exists.
126      *
127      * \param b the right-hand-side of the equation to solve.
128      *
129      * \returns a solution.
130      *
131      * \note_about_checking_solutions
132      *
133      * \note_about_arbitrary_choice_of_solution
134      *
135      * Example: \include HouseholderQR_solve.cpp
136      * Output: \verbinclude HouseholderQR_solve.out
137      */
138    template<typename Rhs>
139    inline const Solve<HouseholderQR, Rhs>
140    solve(const MatrixBase<Rhs>& b) const
141    {
142      eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
143      return Solve<HouseholderQR, Rhs>(*this, b.derived());
144    }
145
146    /** This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.
147      *
148      * The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix object.
149      * Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:
150      *
151      * Example: \include HouseholderQR_householderQ.cpp
152      * Output: \verbinclude HouseholderQR_householderQ.out
153      */
154    HouseholderSequenceType householderQ() const
155    {
156      eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
157      return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
158    }
159
160    /** \returns a reference to the matrix where the Householder QR decomposition is stored
161      * in a LAPACK-compatible way.
162      */
163    const MatrixType& matrixQR() const
164    {
165        eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
166        return m_qr;
167    }
168
169    template<typename InputType>
170    HouseholderQR& compute(const EigenBase<InputType>& matrix) {
171      m_qr = matrix.derived();
172      computeInPlace();
173      return *this;
174    }
175
176    /** \returns the absolute value of the determinant of the matrix of which
177      * *this is the QR decomposition. It has only linear complexity
178      * (that is, O(n) where n is the dimension of the square matrix)
179      * as the QR decomposition has already been computed.
180      *
181      * \note This is only for square matrices.
182      *
183      * \warning a determinant can be very big or small, so for matrices
184      * of large enough dimension, there is a risk of overflow/underflow.
185      * One way to work around that is to use logAbsDeterminant() instead.
186      *
187      * \sa logAbsDeterminant(), MatrixBase::determinant()
188      */
189    typename MatrixType::RealScalar absDeterminant() const;
190
191    /** \returns the natural log of the absolute value of the determinant of the matrix of which
192      * *this is the QR decomposition. It has only linear complexity
193      * (that is, O(n) where n is the dimension of the square matrix)
194      * as the QR decomposition has already been computed.
195      *
196      * \note This is only for square matrices.
197      *
198      * \note This method is useful to work around the risk of overflow/underflow that's inherent
199      * to determinant computation.
200      *
201      * \sa absDeterminant(), MatrixBase::determinant()
202      */
203    typename MatrixType::RealScalar logAbsDeterminant() const;
204
205    inline Index rows() const { return m_qr.rows(); }
206    inline Index cols() const { return m_qr.cols(); }
207
208    /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
209      *
210      * For advanced uses only.
211      */
212    const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
213
214    #ifndef EIGEN_PARSED_BY_DOXYGEN
215    template<typename RhsType, typename DstType>
216    EIGEN_DEVICE_FUNC
217    void _solve_impl(const RhsType &rhs, DstType &dst) const;
218    #endif
219
220  protected:
221
222    static void check_template_parameters()
223    {
224      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
225    }
226
227    void computeInPlace();
228
229    MatrixType m_qr;
230    HCoeffsType m_hCoeffs;
231    RowVectorType m_temp;
232    bool m_isInitialized;
233};
234
235template<typename MatrixType>
236typename MatrixType::RealScalar HouseholderQR<MatrixType>::absDeterminant() const
237{
238  using std::abs;
239  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
240  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
241  return abs(m_qr.diagonal().prod());
242}
243
244template<typename MatrixType>
245typename MatrixType::RealScalar HouseholderQR<MatrixType>::logAbsDeterminant() const
246{
247  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
248  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
249  return m_qr.diagonal().cwiseAbs().array().log().sum();
250}
251
252namespace internal {
253
254/** \internal */
255template<typename MatrixQR, typename HCoeffs>
256void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0)
257{
258  typedef typename MatrixQR::Scalar Scalar;
259  typedef typename MatrixQR::RealScalar RealScalar;
260  Index rows = mat.rows();
261  Index cols = mat.cols();
262  Index size = (std::min)(rows,cols);
263
264  eigen_assert(hCoeffs.size() == size);
265
266  typedef Matrix<Scalar,MatrixQR::ColsAtCompileTime,1> TempType;
267  TempType tempVector;
268  if(tempData==0)
269  {
270    tempVector.resize(cols);
271    tempData = tempVector.data();
272  }
273
274  for(Index k = 0; k < size; ++k)
275  {
276    Index remainingRows = rows - k;
277    Index remainingCols = cols - k - 1;
278
279    RealScalar beta;
280    mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
281    mat.coeffRef(k,k) = beta;
282
283    // apply H to remaining part of m_qr from the left
284    mat.bottomRightCorner(remainingRows, remainingCols)
285        .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows-1), hCoeffs.coeffRef(k), tempData+k+1);
286  }
287}
288
289/** \internal */
290template<typename MatrixQR, typename HCoeffs,
291  typename MatrixQRScalar = typename MatrixQR::Scalar,
292  bool InnerStrideIsOne = (MatrixQR::InnerStrideAtCompileTime == 1 && HCoeffs::InnerStrideAtCompileTime == 1)>
293struct householder_qr_inplace_blocked
294{
295  // This is specialized for MKL-supported Scalar types in HouseholderQR_MKL.h
296  static void run(MatrixQR& mat, HCoeffs& hCoeffs, Index maxBlockSize=32,
297      typename MatrixQR::Scalar* tempData = 0)
298  {
299    typedef typename MatrixQR::Scalar Scalar;
300    typedef Block<MatrixQR,Dynamic,Dynamic> BlockType;
301
302    Index rows = mat.rows();
303    Index cols = mat.cols();
304    Index size = (std::min)(rows, cols);
305
306    typedef Matrix<Scalar,Dynamic,1,ColMajor,MatrixQR::MaxColsAtCompileTime,1> TempType;
307    TempType tempVector;
308    if(tempData==0)
309    {
310      tempVector.resize(cols);
311      tempData = tempVector.data();
312    }
313
314    Index blockSize = (std::min)(maxBlockSize,size);
315
316    Index k = 0;
317    for (k = 0; k < size; k += blockSize)
318    {
319      Index bs = (std::min)(size-k,blockSize);  // actual size of the block
320      Index tcols = cols - k - bs;              // trailing columns
321      Index brows = rows-k;                     // rows of the block
322
323      // partition the matrix:
324      //        A00 | A01 | A02
325      // mat  = A10 | A11 | A12
326      //        A20 | A21 | A22
327      // and performs the qr dec of [A11^T A12^T]^T
328      // and update [A21^T A22^T]^T using level 3 operations.
329      // Finally, the algorithm continue on A22
330
331      BlockType A11_21 = mat.block(k,k,brows,bs);
332      Block<HCoeffs,Dynamic,1> hCoeffsSegment = hCoeffs.segment(k,bs);
333
334      householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);
335
336      if(tcols)
337      {
338        BlockType A21_22 = mat.block(k,k+bs,brows,tcols);
339        apply_block_householder_on_the_left(A21_22,A11_21,hCoeffsSegment, false); // false == backward
340      }
341    }
342  }
343};
344
345} // end namespace internal
346
347#ifndef EIGEN_PARSED_BY_DOXYGEN
348template<typename _MatrixType>
349template<typename RhsType, typename DstType>
350void HouseholderQR<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const
351{
352  const Index rank = (std::min)(rows(), cols());
353  eigen_assert(rhs.rows() == rows());
354
355  typename RhsType::PlainObject c(rhs);
356
357  // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
358  c.applyOnTheLeft(householderSequence(
359    m_qr.leftCols(rank),
360    m_hCoeffs.head(rank)).transpose()
361  );
362
363  m_qr.topLeftCorner(rank, rank)
364      .template triangularView<Upper>()
365      .solveInPlace(c.topRows(rank));
366
367  dst.topRows(rank) = c.topRows(rank);
368  dst.bottomRows(cols()-rank).setZero();
369}
370#endif
371
372/** Performs the QR factorization of the given matrix \a matrix. The result of
373  * the factorization is stored into \c *this, and a reference to \c *this
374  * is returned.
375  *
376  * \sa class HouseholderQR, HouseholderQR(const MatrixType&)
377  */
378template<typename MatrixType>
379void HouseholderQR<MatrixType>::computeInPlace()
380{
381  check_template_parameters();
382
383  Index rows = m_qr.rows();
384  Index cols = m_qr.cols();
385  Index size = (std::min)(rows,cols);
386
387  m_hCoeffs.resize(size);
388
389  m_temp.resize(cols);
390
391  internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, 48, m_temp.data());
392
393  m_isInitialized = true;
394}
395
396/** \return the Householder QR decomposition of \c *this.
397  *
398  * \sa class HouseholderQR
399  */
400template<typename Derived>
401const HouseholderQR<typename MatrixBase<Derived>::PlainObject>
402MatrixBase<Derived>::householderQr() const
403{
404  return HouseholderQR<PlainObject>(eval());
405}
406
407} // end namespace Eigen
408
409#endif // EIGEN_QR_H
410