source: pacpussensors/trunk/Vislab/lib3dv/eigen/Eigen/src/Householder/HouseholderSequence.h@ 136

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1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
5// Copyright (C) 2010 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_HOUSEHOLDER_SEQUENCE_H
12#define EIGEN_HOUSEHOLDER_SEQUENCE_H
13
14namespace Eigen {
15
16/** \ingroup Householder_Module
17 * \householder_module
18 * \class HouseholderSequence
19 * \brief Sequence of Householder reflections acting on subspaces with decreasing size
20 * \tparam VectorsType type of matrix containing the Householder vectors
21 * \tparam CoeffsType type of vector containing the Householder coefficients
22 * \tparam Side either OnTheLeft (the default) or OnTheRight
23 *
24 * This class represents a product sequence of Householder reflections where the first Householder reflection
25 * acts on the whole space, the second Householder reflection leaves the one-dimensional subspace spanned by
26 * the first unit vector invariant, the third Householder reflection leaves the two-dimensional subspace
27 * spanned by the first two unit vectors invariant, and so on up to the last reflection which leaves all but
28 * one dimensions invariant and acts only on the last dimension. Such sequences of Householder reflections
29 * are used in several algorithms to zero out certain parts of a matrix. Indeed, the methods
30 * HessenbergDecomposition::matrixQ(), Tridiagonalization::matrixQ(), HouseholderQR::householderQ(),
31 * and ColPivHouseholderQR::householderQ() all return a %HouseholderSequence.
32 *
33 * More precisely, the class %HouseholderSequence represents an \f$ n \times n \f$ matrix \f$ H \f$ of the
34 * form \f$ H = \prod_{i=0}^{n-1} H_i \f$ where the i-th Householder reflection is \f$ H_i = I - h_i v_i
35 * v_i^* \f$. The i-th Householder coefficient \f$ h_i \f$ is a scalar and the i-th Householder vector \f$
36 * v_i \f$ is a vector of the form
37 * \f[
38 * v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].
39 * \f]
40 * The last \f$ n-i \f$ entries of \f$ v_i \f$ are called the essential part of the Householder vector.
41 *
42 * Typical usages are listed below, where H is a HouseholderSequence:
43 * \code
44 * A.applyOnTheRight(H); // A = A * H
45 * A.applyOnTheLeft(H); // A = H * A
46 * A.applyOnTheRight(H.adjoint()); // A = A * H^*
47 * A.applyOnTheLeft(H.adjoint()); // A = H^* * A
48 * MatrixXd Q = H; // conversion to a dense matrix
49 * \endcode
50 * In addition to the adjoint, you can also apply the inverse (=adjoint), the transpose, and the conjugate operators.
51 *
52 * See the documentation for HouseholderSequence(const VectorsType&, const CoeffsType&) for an example.
53 *
54 * \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
55 */
56
57namespace internal {
58
59template<typename VectorsType, typename CoeffsType, int Side>
60struct traits<HouseholderSequence<VectorsType,CoeffsType,Side> >
61{
62 typedef typename VectorsType::Scalar Scalar;
63 typedef typename VectorsType::Index Index;
64 typedef typename VectorsType::StorageKind StorageKind;
65 enum {
66 RowsAtCompileTime = Side==OnTheLeft ? traits<VectorsType>::RowsAtCompileTime
67 : traits<VectorsType>::ColsAtCompileTime,
68 ColsAtCompileTime = RowsAtCompileTime,
69 MaxRowsAtCompileTime = Side==OnTheLeft ? traits<VectorsType>::MaxRowsAtCompileTime
70 : traits<VectorsType>::MaxColsAtCompileTime,
71 MaxColsAtCompileTime = MaxRowsAtCompileTime,
72 Flags = 0
73 };
74};
75
76template<typename VectorsType, typename CoeffsType, int Side>
77struct hseq_side_dependent_impl
78{
79 typedef Block<const VectorsType, Dynamic, 1> EssentialVectorType;
80 typedef HouseholderSequence<VectorsType, CoeffsType, OnTheLeft> HouseholderSequenceType;
81 typedef typename VectorsType::Index Index;
82 static inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k)
83 {
84 Index start = k+1+h.m_shift;
85 return Block<const VectorsType,Dynamic,1>(h.m_vectors, start, k, h.rows()-start, 1);
86 }
87};
88
89template<typename VectorsType, typename CoeffsType>
90struct hseq_side_dependent_impl<VectorsType, CoeffsType, OnTheRight>
91{
92 typedef Transpose<Block<const VectorsType, 1, Dynamic> > EssentialVectorType;
93 typedef HouseholderSequence<VectorsType, CoeffsType, OnTheRight> HouseholderSequenceType;
94 typedef typename VectorsType::Index Index;
95 static inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k)
96 {
97 Index start = k+1+h.m_shift;
98 return Block<const VectorsType,1,Dynamic>(h.m_vectors, k, start, 1, h.rows()-start).transpose();
99 }
100};
101
102template<typename OtherScalarType, typename MatrixType> struct matrix_type_times_scalar_type
103{
104 typedef typename scalar_product_traits<OtherScalarType, typename MatrixType::Scalar>::ReturnType
105 ResultScalar;
106 typedef Matrix<ResultScalar, MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime,
107 0, MatrixType::MaxRowsAtCompileTime, MatrixType::MaxColsAtCompileTime> Type;
108};
109
110} // end namespace internal
111
112template<typename VectorsType, typename CoeffsType, int Side> class HouseholderSequence
113 : public EigenBase<HouseholderSequence<VectorsType,CoeffsType,Side> >
114{
115 typedef typename internal::hseq_side_dependent_impl<VectorsType,CoeffsType,Side>::EssentialVectorType EssentialVectorType;
116
117 public:
118 enum {
119 RowsAtCompileTime = internal::traits<HouseholderSequence>::RowsAtCompileTime,
120 ColsAtCompileTime = internal::traits<HouseholderSequence>::ColsAtCompileTime,
121 MaxRowsAtCompileTime = internal::traits<HouseholderSequence>::MaxRowsAtCompileTime,
122 MaxColsAtCompileTime = internal::traits<HouseholderSequence>::MaxColsAtCompileTime
123 };
124 typedef typename internal::traits<HouseholderSequence>::Scalar Scalar;
125 typedef typename VectorsType::Index Index;
126
127 typedef HouseholderSequence<
128 typename internal::conditional<NumTraits<Scalar>::IsComplex,
129 typename internal::remove_all<typename VectorsType::ConjugateReturnType>::type,
130 VectorsType>::type,
131 typename internal::conditional<NumTraits<Scalar>::IsComplex,
132 typename internal::remove_all<typename CoeffsType::ConjugateReturnType>::type,
133 CoeffsType>::type,
134 Side
135 > ConjugateReturnType;
136
137 /** \brief Constructor.
138 * \param[in] v %Matrix containing the essential parts of the Householder vectors
139 * \param[in] h Vector containing the Householder coefficients
140 *
141 * Constructs the Householder sequence with coefficients given by \p h and vectors given by \p v. The
142 * i-th Householder coefficient \f$ h_i \f$ is given by \p h(i) and the essential part of the i-th
143 * Householder vector \f$ v_i \f$ is given by \p v(k,i) with \p k > \p i (the subdiagonal part of the
144 * i-th column). If \p v has fewer columns than rows, then the Householder sequence contains as many
145 * Householder reflections as there are columns.
146 *
147 * \note The %HouseholderSequence object stores \p v and \p h by reference.
148 *
149 * Example: \include HouseholderSequence_HouseholderSequence.cpp
150 * Output: \verbinclude HouseholderSequence_HouseholderSequence.out
151 *
152 * \sa setLength(), setShift()
153 */
154 HouseholderSequence(const VectorsType& v, const CoeffsType& h)
155 : m_vectors(v), m_coeffs(h), m_trans(false), m_length(v.diagonalSize()),
156 m_shift(0)
157 {
158 }
159
160 /** \brief Copy constructor. */
161 HouseholderSequence(const HouseholderSequence& other)
162 : m_vectors(other.m_vectors),
163 m_coeffs(other.m_coeffs),
164 m_trans(other.m_trans),
165 m_length(other.m_length),
166 m_shift(other.m_shift)
167 {
168 }
169
170 /** \brief Number of rows of transformation viewed as a matrix.
171 * \returns Number of rows
172 * \details This equals the dimension of the space that the transformation acts on.
173 */
174 Index rows() const { return Side==OnTheLeft ? m_vectors.rows() : m_vectors.cols(); }
175
176 /** \brief Number of columns of transformation viewed as a matrix.
177 * \returns Number of columns
178 * \details This equals the dimension of the space that the transformation acts on.
179 */
180 Index cols() const { return rows(); }
181
182 /** \brief Essential part of a Householder vector.
183 * \param[in] k Index of Householder reflection
184 * \returns Vector containing non-trivial entries of k-th Householder vector
185 *
186 * This function returns the essential part of the Householder vector \f$ v_i \f$. This is a vector of
187 * length \f$ n-i \f$ containing the last \f$ n-i \f$ entries of the vector
188 * \f[
189 * v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].
190 * \f]
191 * The index \f$ i \f$ equals \p k + shift(), corresponding to the k-th column of the matrix \p v
192 * passed to the constructor.
193 *
194 * \sa setShift(), shift()
195 */
196 const EssentialVectorType essentialVector(Index k) const
197 {
198 eigen_assert(k >= 0 && k < m_length);
199 return internal::hseq_side_dependent_impl<VectorsType,CoeffsType,Side>::essentialVector(*this, k);
200 }
201
202 /** \brief %Transpose of the Householder sequence. */
203 HouseholderSequence transpose() const
204 {
205 return HouseholderSequence(*this).setTrans(!m_trans);
206 }
207
208 /** \brief Complex conjugate of the Householder sequence. */
209 ConjugateReturnType conjugate() const
210 {
211 return ConjugateReturnType(m_vectors.conjugate(), m_coeffs.conjugate())
212 .setTrans(m_trans)
213 .setLength(m_length)
214 .setShift(m_shift);
215 }
216
217 /** \brief Adjoint (conjugate transpose) of the Householder sequence. */
218 ConjugateReturnType adjoint() const
219 {
220 return conjugate().setTrans(!m_trans);
221 }
222
223 /** \brief Inverse of the Householder sequence (equals the adjoint). */
224 ConjugateReturnType inverse() const { return adjoint(); }
225
226 /** \internal */
227 template<typename DestType> inline void evalTo(DestType& dst) const
228 {
229 Matrix<Scalar, DestType::RowsAtCompileTime, 1,
230 AutoAlign|ColMajor, DestType::MaxRowsAtCompileTime, 1> workspace(rows());
231 evalTo(dst, workspace);
232 }
233
234 /** \internal */
235 template<typename Dest, typename Workspace>
236 void evalTo(Dest& dst, Workspace& workspace) const
237 {
238 workspace.resize(rows());
239 Index vecs = m_length;
240 const typename Dest::Scalar *dst_data = internal::extract_data(dst);
241 if( internal::is_same<typename internal::remove_all<VectorsType>::type,Dest>::value
242 && dst_data!=0 && dst_data == internal::extract_data(m_vectors))
243 {
244 // in-place
245 dst.diagonal().setOnes();
246 dst.template triangularView<StrictlyUpper>().setZero();
247 for(Index k = vecs-1; k >= 0; --k)
248 {
249 Index cornerSize = rows() - k - m_shift;
250 if(m_trans)
251 dst.bottomRightCorner(cornerSize, cornerSize)
252 .applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), workspace.data());
253 else
254 dst.bottomRightCorner(cornerSize, cornerSize)
255 .applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), workspace.data());
256
257 // clear the off diagonal vector
258 dst.col(k).tail(rows()-k-1).setZero();
259 }
260 // clear the remaining columns if needed
261 for(Index k = 0; k<cols()-vecs ; ++k)
262 dst.col(k).tail(rows()-k-1).setZero();
263 }
264 else
265 {
266 dst.setIdentity(rows(), rows());
267 for(Index k = vecs-1; k >= 0; --k)
268 {
269 Index cornerSize = rows() - k - m_shift;
270 if(m_trans)
271 dst.bottomRightCorner(cornerSize, cornerSize)
272 .applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), &workspace.coeffRef(0));
273 else
274 dst.bottomRightCorner(cornerSize, cornerSize)
275 .applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), &workspace.coeffRef(0));
276 }
277 }
278 }
279
280 /** \internal */
281 template<typename Dest> inline void applyThisOnTheRight(Dest& dst) const
282 {
283 Matrix<Scalar,1,Dest::RowsAtCompileTime,RowMajor,1,Dest::MaxRowsAtCompileTime> workspace(dst.rows());
284 applyThisOnTheRight(dst, workspace);
285 }
286
287 /** \internal */
288 template<typename Dest, typename Workspace>
289 inline void applyThisOnTheRight(Dest& dst, Workspace& workspace) const
290 {
291 workspace.resize(dst.rows());
292 for(Index k = 0; k < m_length; ++k)
293 {
294 Index actual_k = m_trans ? m_length-k-1 : k;
295 dst.rightCols(rows()-m_shift-actual_k)
296 .applyHouseholderOnTheRight(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data());
297 }
298 }
299
300 /** \internal */
301 template<typename Dest> inline void applyThisOnTheLeft(Dest& dst) const
302 {
303 Matrix<Scalar,1,Dest::ColsAtCompileTime,RowMajor,1,Dest::MaxColsAtCompileTime> workspace(dst.cols());
304 applyThisOnTheLeft(dst, workspace);
305 }
306
307 /** \internal */
308 template<typename Dest, typename Workspace>
309 inline void applyThisOnTheLeft(Dest& dst, Workspace& workspace) const
310 {
311 workspace.resize(dst.cols());
312 for(Index k = 0; k < m_length; ++k)
313 {
314 Index actual_k = m_trans ? k : m_length-k-1;
315 dst.bottomRows(rows()-m_shift-actual_k)
316 .applyHouseholderOnTheLeft(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data());
317 }
318 }
319
320 /** \brief Computes the product of a Householder sequence with a matrix.
321 * \param[in] other %Matrix being multiplied.
322 * \returns Expression object representing the product.
323 *
324 * This function computes \f$ HM \f$ where \f$ H \f$ is the Householder sequence represented by \p *this
325 * and \f$ M \f$ is the matrix \p other.
326 */
327 template<typename OtherDerived>
328 typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type operator*(const MatrixBase<OtherDerived>& other) const
329 {
330 typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type
331 res(other.template cast<typename internal::matrix_type_times_scalar_type<Scalar,OtherDerived>::ResultScalar>());
332 applyThisOnTheLeft(res);
333 return res;
334 }
335
336 template<typename _VectorsType, typename _CoeffsType, int _Side> friend struct internal::hseq_side_dependent_impl;
337
338 /** \brief Sets the length of the Householder sequence.
339 * \param [in] length New value for the length.
340 *
341 * By default, the length \f$ n \f$ of the Householder sequence \f$ H = H_0 H_1 \ldots H_{n-1} \f$ is set
342 * to the number of columns of the matrix \p v passed to the constructor, or the number of rows if that
343 * is smaller. After this function is called, the length equals \p length.
344 *
345 * \sa length()
346 */
347 HouseholderSequence& setLength(Index length)
348 {
349 m_length = length;
350 return *this;
351 }
352
353 /** \brief Sets the shift of the Householder sequence.
354 * \param [in] shift New value for the shift.
355 *
356 * By default, a %HouseholderSequence object represents \f$ H = H_0 H_1 \ldots H_{n-1} \f$ and the i-th
357 * column of the matrix \p v passed to the constructor corresponds to the i-th Householder
358 * reflection. After this function is called, the object represents \f$ H = H_{\mathrm{shift}}
359 * H_{\mathrm{shift}+1} \ldots H_{n-1} \f$ and the i-th column of \p v corresponds to the (shift+i)-th
360 * Householder reflection.
361 *
362 * \sa shift()
363 */
364 HouseholderSequence& setShift(Index shift)
365 {
366 m_shift = shift;
367 return *this;
368 }
369
370 Index length() const { return m_length; } /**< \brief Returns the length of the Householder sequence. */
371 Index shift() const { return m_shift; } /**< \brief Returns the shift of the Householder sequence. */
372
373 /* Necessary for .adjoint() and .conjugate() */
374 template <typename VectorsType2, typename CoeffsType2, int Side2> friend class HouseholderSequence;
375
376 protected:
377
378 /** \brief Sets the transpose flag.
379 * \param [in] trans New value of the transpose flag.
380 *
381 * By default, the transpose flag is not set. If the transpose flag is set, then this object represents
382 * \f$ H^T = H_{n-1}^T \ldots H_1^T H_0^T \f$ instead of \f$ H = H_0 H_1 \ldots H_{n-1} \f$.
383 *
384 * \sa trans()
385 */
386 HouseholderSequence& setTrans(bool trans)
387 {
388 m_trans = trans;
389 return *this;
390 }
391
392 bool trans() const { return m_trans; } /**< \brief Returns the transpose flag. */
393
394 typename VectorsType::Nested m_vectors;
395 typename CoeffsType::Nested m_coeffs;
396 bool m_trans;
397 Index m_length;
398 Index m_shift;
399};
400
401/** \brief Computes the product of a matrix with a Householder sequence.
402 * \param[in] other %Matrix being multiplied.
403 * \param[in] h %HouseholderSequence being multiplied.
404 * \returns Expression object representing the product.
405 *
406 * This function computes \f$ MH \f$ where \f$ M \f$ is the matrix \p other and \f$ H \f$ is the
407 * Householder sequence represented by \p h.
408 */
409template<typename OtherDerived, typename VectorsType, typename CoeffsType, int Side>
410typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,OtherDerived>::Type operator*(const MatrixBase<OtherDerived>& other, const HouseholderSequence<VectorsType,CoeffsType,Side>& h)
411{
412 typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,OtherDerived>::Type
413 res(other.template cast<typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,OtherDerived>::ResultScalar>());
414 h.applyThisOnTheRight(res);
415 return res;
416}
417
418/** \ingroup Householder_Module \householder_module
419 * \brief Convenience function for constructing a Householder sequence.
420 * \returns A HouseholderSequence constructed from the specified arguments.
421 */
422template<typename VectorsType, typename CoeffsType>
423HouseholderSequence<VectorsType,CoeffsType> householderSequence(const VectorsType& v, const CoeffsType& h)
424{
425 return HouseholderSequence<VectorsType,CoeffsType,OnTheLeft>(v, h);
426}
427
428/** \ingroup Householder_Module \householder_module
429 * \brief Convenience function for constructing a Householder sequence.
430 * \returns A HouseholderSequence constructed from the specified arguments.
431 * \details This function differs from householderSequence() in that the template argument \p OnTheSide of
432 * the constructed HouseholderSequence is set to OnTheRight, instead of the default OnTheLeft.
433 */
434template<typename VectorsType, typename CoeffsType>
435HouseholderSequence<VectorsType,CoeffsType,OnTheRight> rightHouseholderSequence(const VectorsType& v, const CoeffsType& h)
436{
437 return HouseholderSequence<VectorsType,CoeffsType,OnTheRight>(v, h);
438}
439
440} // end namespace Eigen
441
442#endif // EIGEN_HOUSEHOLDER_SEQUENCE_H
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