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 (at) inria.fr> 5 // Copyright (C) 2010 Benoit Jacob <jacob.benoit.1 (at) 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 14 namespace 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 57 namespace internal { 58 59 template<typename VectorsType, typename CoeffsType, int Side> 60 struct traits<HouseholderSequence<VectorsType,CoeffsType,Side> > 61 { 62 typedef typename VectorsType::Scalar Scalar; 63 typedef typename VectorsType::StorageIndex StorageIndex; 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 76 struct HouseholderSequenceShape {}; 77 78 template<typename VectorsType, typename CoeffsType, int Side> 79 struct evaluator_traits<HouseholderSequence<VectorsType,CoeffsType,Side> > 80 : public evaluator_traits_base<HouseholderSequence<VectorsType,CoeffsType,Side> > 81 { 82 typedef HouseholderSequenceShape Shape; 83 }; 84 85 template<typename VectorsType, typename CoeffsType, int Side> 86 struct hseq_side_dependent_impl 87 { 88 typedef Block<const VectorsType, Dynamic, 1> EssentialVectorType; 89 typedef HouseholderSequence<VectorsType, CoeffsType, OnTheLeft> HouseholderSequenceType; 90 static inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k) 91 { 92 Index start = k+1+h.m_shift; 93 return Block<const VectorsType,Dynamic,1>(h.m_vectors, start, k, h.rows()-start, 1); 94 } 95 }; 96 97 template<typename VectorsType, typename CoeffsType> 98 struct hseq_side_dependent_impl<VectorsType, CoeffsType, OnTheRight> 99 { 100 typedef Transpose<Block<const VectorsType, 1, Dynamic> > EssentialVectorType; 101 typedef HouseholderSequence<VectorsType, CoeffsType, OnTheRight> HouseholderSequenceType; 102 static inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k) 103 { 104 Index start = k+1+h.m_shift; 105 return Block<const VectorsType,1,Dynamic>(h.m_vectors, k, start, 1, h.rows()-start).transpose(); 106 } 107 }; 108 109 template<typename OtherScalarType, typename MatrixType> struct matrix_type_times_scalar_type 110 { 111 typedef typename ScalarBinaryOpTraits<OtherScalarType, typename MatrixType::Scalar>::ReturnType 112 ResultScalar; 113 typedef Matrix<ResultScalar, MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime, 114 0, MatrixType::MaxRowsAtCompileTime, MatrixType::MaxColsAtCompileTime> Type; 115 }; 116 117 } // end namespace internal 118 119 template<typename VectorsType, typename CoeffsType, int Side> class HouseholderSequence 120 : public EigenBase<HouseholderSequence<VectorsType,CoeffsType,Side> > 121 { 122 typedef typename internal::hseq_side_dependent_impl<VectorsType,CoeffsType,Side>::EssentialVectorType EssentialVectorType; 123 124 public: 125 enum { 126 RowsAtCompileTime = internal::traits<HouseholderSequence>::RowsAtCompileTime, 127 ColsAtCompileTime = internal::traits<HouseholderSequence>::ColsAtCompileTime, 128 MaxRowsAtCompileTime = internal::traits<HouseholderSequence>::MaxRowsAtCompileTime, 129 MaxColsAtCompileTime = internal::traits<HouseholderSequence>::MaxColsAtCompileTime 130 }; 131 typedef typename internal::traits<HouseholderSequence>::Scalar Scalar; 132 133 typedef HouseholderSequence< 134 typename internal::conditional<NumTraits<Scalar>::IsComplex, 135 typename internal::remove_all<typename VectorsType::ConjugateReturnType>::type, 136 VectorsType>::type, 137 typename internal::conditional<NumTraits<Scalar>::IsComplex, 138 typename internal::remove_all<typename CoeffsType::ConjugateReturnType>::type, 139 CoeffsType>::type, 140 Side 141 > ConjugateReturnType; 142 143 /** \brief Constructor. 144 * \param[in] v %Matrix containing the essential parts of the Householder vectors 145 * \param[in] h Vector containing the Householder coefficients 146 * 147 * Constructs the Householder sequence with coefficients given by \p h and vectors given by \p v. The 148 * i-th Householder coefficient \f$ h_i \f$ is given by \p h(i) and the essential part of the i-th 149 * Householder vector \f$ v_i \f$ is given by \p v(k,i) with \p k > \p i (the subdiagonal part of the 150 * i-th column). If \p v has fewer columns than rows, then the Householder sequence contains as many 151 * Householder reflections as there are columns. 152 * 153 * \note The %HouseholderSequence object stores \p v and \p h by reference. 154 * 155 * Example: \include HouseholderSequence_HouseholderSequence.cpp 156 * Output: \verbinclude HouseholderSequence_HouseholderSequence.out 157 * 158 * \sa setLength(), setShift() 159 */ 160 HouseholderSequence(const VectorsType& v, const CoeffsType& h) 161 : m_vectors(v), m_coeffs(h), m_trans(false), m_length(v.diagonalSize()), 162 m_shift(0) 163 { 164 } 165 166 /** \brief Copy constructor. */ 167 HouseholderSequence(const HouseholderSequence& other) 168 : m_vectors(other.m_vectors), 169 m_coeffs(other.m_coeffs), 170 m_trans(other.m_trans), 171 m_length(other.m_length), 172 m_shift(other.m_shift) 173 { 174 } 175 176 /** \brief Number of rows of transformation viewed as a matrix. 177 * \returns Number of rows 178 * \details This equals the dimension of the space that the transformation acts on. 179 */ 180 Index rows() const { return Side==OnTheLeft ? m_vectors.rows() : m_vectors.cols(); } 181 182 /** \brief Number of columns of transformation viewed as a matrix. 183 * \returns Number of columns 184 * \details This equals the dimension of the space that the transformation acts on. 185 */ 186 Index cols() const { return rows(); } 187 188 /** \brief Essential part of a Householder vector. 189 * \param[in] k Index of Householder reflection 190 * \returns Vector containing non-trivial entries of k-th Householder vector 191 * 192 * This function returns the essential part of the Householder vector \f$ v_i \f$. This is a vector of 193 * length \f$ n-i \f$ containing the last \f$ n-i \f$ entries of the vector 194 * \f[ 195 * v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ]. 196 * \f] 197 * The index \f$ i \f$ equals \p k + shift(), corresponding to the k-th column of the matrix \p v 198 * passed to the constructor. 199 * 200 * \sa setShift(), shift() 201 */ 202 const EssentialVectorType essentialVector(Index k) const 203 { 204 eigen_assert(k >= 0 && k < m_length); 205 return internal::hseq_side_dependent_impl<VectorsType,CoeffsType,Side>::essentialVector(*this, k); 206 } 207 208 /** \brief %Transpose of the Householder sequence. */ 209 HouseholderSequence transpose() const 210 { 211 return HouseholderSequence(*this).setTrans(!m_trans); 212 } 213 214 /** \brief Complex conjugate of the Householder sequence. */ 215 ConjugateReturnType conjugate() const 216 { 217 return ConjugateReturnType(m_vectors.conjugate(), m_coeffs.conjugate()) 218 .setTrans(m_trans) 219 .setLength(m_length) 220 .setShift(m_shift); 221 } 222 223 /** \brief Adjoint (conjugate transpose) of the Householder sequence. */ 224 ConjugateReturnType adjoint() const 225 { 226 return conjugate().setTrans(!m_trans); 227 } 228 229 /** \brief Inverse of the Householder sequence (equals the adjoint). */ 230 ConjugateReturnType inverse() const { return adjoint(); } 231 232 /** \internal */ 233 template<typename DestType> inline void evalTo(DestType& dst) const 234 { 235 Matrix<Scalar, DestType::RowsAtCompileTime, 1, 236 AutoAlign|ColMajor, DestType::MaxRowsAtCompileTime, 1> workspace(rows()); 237 evalTo(dst, workspace); 238 } 239 240 /** \internal */ 241 template<typename Dest, typename Workspace> 242 void evalTo(Dest& dst, Workspace& workspace) const 243 { 244 workspace.resize(rows()); 245 Index vecs = m_length; 246 if(internal::is_same_dense(dst,m_vectors)) 247 { 248 // in-place 249 dst.diagonal().setOnes(); 250 dst.template triangularView<StrictlyUpper>().setZero(); 251 for(Index k = vecs-1; k >= 0; --k) 252 { 253 Index cornerSize = rows() - k - m_shift; 254 if(m_trans) 255 dst.bottomRightCorner(cornerSize, cornerSize) 256 .applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), workspace.data()); 257 else 258 dst.bottomRightCorner(cornerSize, cornerSize) 259 .applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), workspace.data()); 260 261 // clear the off diagonal vector 262 dst.col(k).tail(rows()-k-1).setZero(); 263 } 264 // clear the remaining columns if needed 265 for(Index k = 0; k<cols()-vecs ; ++k) 266 dst.col(k).tail(rows()-k-1).setZero(); 267 } 268 else 269 { 270 dst.setIdentity(rows(), rows()); 271 for(Index k = vecs-1; k >= 0; --k) 272 { 273 Index cornerSize = rows() - k - m_shift; 274 if(m_trans) 275 dst.bottomRightCorner(cornerSize, cornerSize) 276 .applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), &workspace.coeffRef(0)); 277 else 278 dst.bottomRightCorner(cornerSize, cornerSize) 279 .applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), &workspace.coeffRef(0)); 280 } 281 } 282 } 283 284 /** \internal */ 285 template<typename Dest> inline void applyThisOnTheRight(Dest& dst) const 286 { 287 Matrix<Scalar,1,Dest::RowsAtCompileTime,RowMajor,1,Dest::MaxRowsAtCompileTime> workspace(dst.rows()); 288 applyThisOnTheRight(dst, workspace); 289 } 290 291 /** \internal */ 292 template<typename Dest, typename Workspace> 293 inline void applyThisOnTheRight(Dest& dst, Workspace& workspace) const 294 { 295 workspace.resize(dst.rows()); 296 for(Index k = 0; k < m_length; ++k) 297 { 298 Index actual_k = m_trans ? m_length-k-1 : k; 299 dst.rightCols(rows()-m_shift-actual_k) 300 .applyHouseholderOnTheRight(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data()); 301 } 302 } 303 304 /** \internal */ 305 template<typename Dest> inline void applyThisOnTheLeft(Dest& dst) const 306 { 307 Matrix<Scalar,1,Dest::ColsAtCompileTime,RowMajor,1,Dest::MaxColsAtCompileTime> workspace; 308 applyThisOnTheLeft(dst, workspace); 309 } 310 311 /** \internal */ 312 template<typename Dest, typename Workspace> 313 inline void applyThisOnTheLeft(Dest& dst, Workspace& workspace) const 314 { 315 const Index BlockSize = 48; 316 // if the entries are large enough, then apply the reflectors by block 317 if(m_length>=BlockSize && dst.cols()>1) 318 { 319 for(Index i = 0; i < m_length; i+=BlockSize) 320 { 321 Index end = m_trans ? (std::min)(m_length,i+BlockSize) : m_length-i; 322 Index k = m_trans ? i : (std::max)(Index(0),end-BlockSize); 323 Index bs = end-k; 324 Index start = k + m_shift; 325 326 typedef Block<typename internal::remove_all<VectorsType>::type,Dynamic,Dynamic> SubVectorsType; 327 SubVectorsType sub_vecs1(m_vectors.const_cast_derived(), Side==OnTheRight ? k : start, 328 Side==OnTheRight ? start : k, 329 Side==OnTheRight ? bs : m_vectors.rows()-start, 330 Side==OnTheRight ? m_vectors.cols()-start : bs); 331 typename internal::conditional<Side==OnTheRight, Transpose<SubVectorsType>, SubVectorsType&>::type sub_vecs(sub_vecs1); 332 Block<Dest,Dynamic,Dynamic> sub_dst(dst,dst.rows()-rows()+m_shift+k,0, rows()-m_shift-k,dst.cols()); 333 apply_block_householder_on_the_left(sub_dst, sub_vecs, m_coeffs.segment(k, bs), !m_trans); 334 } 335 } 336 else 337 { 338 workspace.resize(dst.cols()); 339 for(Index k = 0; k < m_length; ++k) 340 { 341 Index actual_k = m_trans ? k : m_length-k-1; 342 dst.bottomRows(rows()-m_shift-actual_k) 343 .applyHouseholderOnTheLeft(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data()); 344 } 345 } 346 } 347 348 /** \brief Computes the product of a Householder sequence with a matrix. 349 * \param[in] other %Matrix being multiplied. 350 * \returns Expression object representing the product. 351 * 352 * This function computes \f$ HM \f$ where \f$ H \f$ is the Householder sequence represented by \p *this 353 * and \f$ M \f$ is the matrix \p other. 354 */ 355 template<typename OtherDerived> 356 typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type operator*(const MatrixBase<OtherDerived>& other) const 357 { 358 typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type 359 res(other.template cast<typename internal::matrix_type_times_scalar_type<Scalar,OtherDerived>::ResultScalar>()); 360 applyThisOnTheLeft(res); 361 return res; 362 } 363 364 template<typename _VectorsType, typename _CoeffsType, int _Side> friend struct internal::hseq_side_dependent_impl; 365 366 /** \brief Sets the length of the Householder sequence. 367 * \param [in] length New value for the length. 368 * 369 * By default, the length \f$ n \f$ of the Householder sequence \f$ H = H_0 H_1 \ldots H_{n-1} \f$ is set 370 * to the number of columns of the matrix \p v passed to the constructor, or the number of rows if that 371 * is smaller. After this function is called, the length equals \p length. 372 * 373 * \sa length() 374 */ 375 HouseholderSequence& setLength(Index length) 376 { 377 m_length = length; 378 return *this; 379 } 380 381 /** \brief Sets the shift of the Householder sequence. 382 * \param [in] shift New value for the shift. 383 * 384 * By default, a %HouseholderSequence object represents \f$ H = H_0 H_1 \ldots H_{n-1} \f$ and the i-th 385 * column of the matrix \p v passed to the constructor corresponds to the i-th Householder 386 * reflection. After this function is called, the object represents \f$ H = H_{\mathrm{shift}} 387 * H_{\mathrm{shift}+1} \ldots H_{n-1} \f$ and the i-th column of \p v corresponds to the (shift+i)-th 388 * Householder reflection. 389 * 390 * \sa shift() 391 */ 392 HouseholderSequence& setShift(Index shift) 393 { 394 m_shift = shift; 395 return *this; 396 } 397 398 Index length() const { return m_length; } /**< \brief Returns the length of the Householder sequence. */ 399 Index shift() const { return m_shift; } /**< \brief Returns the shift of the Householder sequence. */ 400 401 /* Necessary for .adjoint() and .conjugate() */ 402 template <typename VectorsType2, typename CoeffsType2, int Side2> friend class HouseholderSequence; 403 404 protected: 405 406 /** \brief Sets the transpose flag. 407 * \param [in] trans New value of the transpose flag. 408 * 409 * By default, the transpose flag is not set. If the transpose flag is set, then this object represents 410 * \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$. 411 * 412 * \sa trans() 413 */ 414 HouseholderSequence& setTrans(bool trans) 415 { 416 m_trans = trans; 417 return *this; 418 } 419 420 bool trans() const { return m_trans; } /**< \brief Returns the transpose flag. */ 421 422 typename VectorsType::Nested m_vectors; 423 typename CoeffsType::Nested m_coeffs; 424 bool m_trans; 425 Index m_length; 426 Index m_shift; 427 }; 428 429 /** \brief Computes the product of a matrix with a Householder sequence. 430 * \param[in] other %Matrix being multiplied. 431 * \param[in] h %HouseholderSequence being multiplied. 432 * \returns Expression object representing the product. 433 * 434 * This function computes \f$ MH \f$ where \f$ M \f$ is the matrix \p other and \f$ H \f$ is the 435 * Householder sequence represented by \p h. 436 */ 437 template<typename OtherDerived, typename VectorsType, typename CoeffsType, int Side> 438 typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,OtherDerived>::Type operator*(const MatrixBase<OtherDerived>& other, const HouseholderSequence<VectorsType,CoeffsType,Side>& h) 439 { 440 typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,OtherDerived>::Type 441 res(other.template cast<typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,OtherDerived>::ResultScalar>()); 442 h.applyThisOnTheRight(res); 443 return res; 444 } 445 446 /** \ingroup Householder_Module \householder_module 447 * \brief Convenience function for constructing a Householder sequence. 448 * \returns A HouseholderSequence constructed from the specified arguments. 449 */ 450 template<typename VectorsType, typename CoeffsType> 451 HouseholderSequence<VectorsType,CoeffsType> householderSequence(const VectorsType& v, const CoeffsType& h) 452 { 453 return HouseholderSequence<VectorsType,CoeffsType,OnTheLeft>(v, h); 454 } 455 456 /** \ingroup Householder_Module \householder_module 457 * \brief Convenience function for constructing a Householder sequence. 458 * \returns A HouseholderSequence constructed from the specified arguments. 459 * \details This function differs from householderSequence() in that the template argument \p OnTheSide of 460 * the constructed HouseholderSequence is set to OnTheRight, instead of the default OnTheLeft. 461 */ 462 template<typename VectorsType, typename CoeffsType> 463 HouseholderSequence<VectorsType,CoeffsType,OnTheRight> rightHouseholderSequence(const VectorsType& v, const CoeffsType& h) 464 { 465 return HouseholderSequence<VectorsType,CoeffsType,OnTheRight>(v, h); 466 } 467 468 } // end namespace Eigen 469 470 #endif // EIGEN_HOUSEHOLDER_SEQUENCE_H 471
