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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 
15 namespace 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   */
44 template<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       */
HouseholderQR()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       */
HouseholderQR(Index rows,Index cols)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>
HouseholderQR(const EigenBase<InputType> & matrix)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>
HouseholderQR(EigenBase<InputType> & matrix)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>
solve(const MatrixBase<Rhs> & b)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       */
householderQ()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       */
matrixQR()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>
compute(const EigenBase<InputType> & matrix)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 
rows()205     inline Index rows() const { return m_qr.rows(); }
cols()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       */
hCoeffs()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 
check_template_parameters()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 
235 template<typename MatrixType>
absDeterminant()236 typename 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 
244 template<typename MatrixType>
logAbsDeterminant()245 typename 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 
252 namespace internal {
253 
254 /** \internal */
255 template<typename MatrixQR, typename HCoeffs>
256 void 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 */
290 template<typename MatrixQR, typename HCoeffs,
291   typename MatrixQRScalar = typename MatrixQR::Scalar,
292   bool InnerStrideIsOne = (MatrixQR::InnerStrideAtCompileTime == 1 && HCoeffs::InnerStrideAtCompileTime == 1)>
293 struct 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
348 template<typename _MatrixType>
349 template<typename RhsType, typename DstType>
_solve_impl(const RhsType & rhs,DstType & dst)350 void 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   */
378 template<typename MatrixType>
computeInPlace()379 void 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   */
400 template<typename Derived>
401 const HouseholderQR<typename MatrixBase<Derived>::PlainObject>
householderQr()402 MatrixBase<Derived>::householderQr() const
403 {
404   return HouseholderQR<PlainObject>(eval());
405 }
406 
407 } // end namespace Eigen
408 
409 #endif // EIGEN_QR_H
410