1 // This file is part of Eigen, a lightweight C++ template library 2 // for linear algebra. 3 // 4 // Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud (at) inria.fr> 5 // Copyright (C) 2010 Jitse Niesen <jitse (at) maths.leeds.ac.uk> 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_HESSENBERGDECOMPOSITION_H 12 #define EIGEN_HESSENBERGDECOMPOSITION_H 13 14 namespace Eigen { 15 16 namespace internal { 17 18 template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType; 19 template<typename MatrixType> 20 struct traits<HessenbergDecompositionMatrixHReturnType<MatrixType> > 21 { 22 typedef MatrixType ReturnType; 23 }; 24 25 } 26 27 /** \eigenvalues_module \ingroup Eigenvalues_Module 28 * 29 * 30 * \class HessenbergDecomposition 31 * 32 * \brief Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation 33 * 34 * \tparam _MatrixType the type of the matrix of which we are computing the Hessenberg decomposition 35 * 36 * This class performs an Hessenberg decomposition of a matrix \f$ A \f$. In 37 * the real case, the Hessenberg decomposition consists of an orthogonal 38 * matrix \f$ Q \f$ and a Hessenberg matrix \f$ H \f$ such that \f$ A = Q H 39 * Q^T \f$. An orthogonal matrix is a matrix whose inverse equals its 40 * transpose (\f$ Q^{-1} = Q^T \f$). A Hessenberg matrix has zeros below the 41 * subdiagonal, so it is almost upper triangular. The Hessenberg decomposition 42 * of a complex matrix is \f$ A = Q H Q^* \f$ with \f$ Q \f$ unitary (that is, 43 * \f$ Q^{-1} = Q^* \f$). 44 * 45 * Call the function compute() to compute the Hessenberg decomposition of a 46 * given matrix. Alternatively, you can use the 47 * HessenbergDecomposition(const MatrixType&) constructor which computes the 48 * Hessenberg decomposition at construction time. Once the decomposition is 49 * computed, you can use the matrixH() and matrixQ() functions to construct 50 * the matrices H and Q in the decomposition. 51 * 52 * The documentation for matrixH() contains an example of the typical use of 53 * this class. 54 * 55 * \sa class ComplexSchur, class Tridiagonalization, \ref QR_Module "QR Module" 56 */ 57 template<typename _MatrixType> class HessenbergDecomposition 58 { 59 public: 60 61 /** \brief Synonym for the template parameter \p _MatrixType. */ 62 typedef _MatrixType MatrixType; 63 64 enum { 65 Size = MatrixType::RowsAtCompileTime, 66 SizeMinusOne = Size == Dynamic ? Dynamic : Size - 1, 67 Options = MatrixType::Options, 68 MaxSize = MatrixType::MaxRowsAtCompileTime, 69 MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : MaxSize - 1 70 }; 71 72 /** \brief Scalar type for matrices of type #MatrixType. */ 73 typedef typename MatrixType::Scalar Scalar; 74 typedef typename MatrixType::Index Index; 75 76 /** \brief Type for vector of Householder coefficients. 77 * 78 * This is column vector with entries of type #Scalar. The length of the 79 * vector is one less than the size of #MatrixType, if it is a fixed-side 80 * type. 81 */ 82 typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType; 83 84 /** \brief Return type of matrixQ() */ 85 typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType; 86 87 typedef internal::HessenbergDecompositionMatrixHReturnType<MatrixType> MatrixHReturnType; 88 89 /** \brief Default constructor; the decomposition will be computed later. 90 * 91 * \param [in] size The size of the matrix whose Hessenberg decomposition will be computed. 92 * 93 * The default constructor is useful in cases in which the user intends to 94 * perform decompositions via compute(). The \p size parameter is only 95 * used as a hint. It is not an error to give a wrong \p size, but it may 96 * impair performance. 97 * 98 * \sa compute() for an example. 99 */ 100 HessenbergDecomposition(Index size = Size==Dynamic ? 2 : Size) 101 : m_matrix(size,size), 102 m_temp(size), 103 m_isInitialized(false) 104 { 105 if(size>1) 106 m_hCoeffs.resize(size-1); 107 } 108 109 /** \brief Constructor; computes Hessenberg decomposition of given matrix. 110 * 111 * \param[in] matrix Square matrix whose Hessenberg decomposition is to be computed. 112 * 113 * This constructor calls compute() to compute the Hessenberg 114 * decomposition. 115 * 116 * \sa matrixH() for an example. 117 */ 118 HessenbergDecomposition(const MatrixType& matrix) 119 : m_matrix(matrix), 120 m_temp(matrix.rows()), 121 m_isInitialized(false) 122 { 123 if(matrix.rows()<2) 124 { 125 m_isInitialized = true; 126 return; 127 } 128 m_hCoeffs.resize(matrix.rows()-1,1); 129 _compute(m_matrix, m_hCoeffs, m_temp); 130 m_isInitialized = true; 131 } 132 133 /** \brief Computes Hessenberg decomposition of given matrix. 134 * 135 * \param[in] matrix Square matrix whose Hessenberg decomposition is to be computed. 136 * \returns Reference to \c *this 137 * 138 * The Hessenberg decomposition is computed by bringing the columns of the 139 * matrix successively in the required form using Householder reflections 140 * (see, e.g., Algorithm 7.4.2 in Golub \& Van Loan, <i>%Matrix 141 * Computations</i>). The cost is \f$ 10n^3/3 \f$ flops, where \f$ n \f$ 142 * denotes the size of the given matrix. 143 * 144 * This method reuses of the allocated data in the HessenbergDecomposition 145 * object. 146 * 147 * Example: \include HessenbergDecomposition_compute.cpp 148 * Output: \verbinclude HessenbergDecomposition_compute.out 149 */ 150 HessenbergDecomposition& compute(const MatrixType& matrix) 151 { 152 m_matrix = matrix; 153 if(matrix.rows()<2) 154 { 155 m_isInitialized = true; 156 return *this; 157 } 158 m_hCoeffs.resize(matrix.rows()-1,1); 159 _compute(m_matrix, m_hCoeffs, m_temp); 160 m_isInitialized = true; 161 return *this; 162 } 163 164 /** \brief Returns the Householder coefficients. 165 * 166 * \returns a const reference to the vector of Householder coefficients 167 * 168 * \pre Either the constructor HessenbergDecomposition(const MatrixType&) 169 * or the member function compute(const MatrixType&) has been called 170 * before to compute the Hessenberg decomposition of a matrix. 171 * 172 * The Householder coefficients allow the reconstruction of the matrix 173 * \f$ Q \f$ in the Hessenberg decomposition from the packed data. 174 * 175 * \sa packedMatrix(), \ref Householder_Module "Householder module" 176 */ 177 const CoeffVectorType& householderCoefficients() const 178 { 179 eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized."); 180 return m_hCoeffs; 181 } 182 183 /** \brief Returns the internal representation of the decomposition 184 * 185 * \returns a const reference to a matrix with the internal representation 186 * of the decomposition. 187 * 188 * \pre Either the constructor HessenbergDecomposition(const MatrixType&) 189 * or the member function compute(const MatrixType&) has been called 190 * before to compute the Hessenberg decomposition of a matrix. 191 * 192 * The returned matrix contains the following information: 193 * - the upper part and lower sub-diagonal represent the Hessenberg matrix H 194 * - the rest of the lower part contains the Householder vectors that, combined with 195 * Householder coefficients returned by householderCoefficients(), 196 * allows to reconstruct the matrix Q as 197 * \f$ Q = H_{N-1} \ldots H_1 H_0 \f$. 198 * Here, the matrices \f$ H_i \f$ are the Householder transformations 199 * \f$ H_i = (I - h_i v_i v_i^T) \f$ 200 * where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and 201 * \f$ v_i \f$ is the Householder vector defined by 202 * \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$ 203 * with M the matrix returned by this function. 204 * 205 * See LAPACK for further details on this packed storage. 206 * 207 * Example: \include HessenbergDecomposition_packedMatrix.cpp 208 * Output: \verbinclude HessenbergDecomposition_packedMatrix.out 209 * 210 * \sa householderCoefficients() 211 */ 212 const MatrixType& packedMatrix() const 213 { 214 eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized."); 215 return m_matrix; 216 } 217 218 /** \brief Reconstructs the orthogonal matrix Q in the decomposition 219 * 220 * \returns object representing the matrix Q 221 * 222 * \pre Either the constructor HessenbergDecomposition(const MatrixType&) 223 * or the member function compute(const MatrixType&) has been called 224 * before to compute the Hessenberg decomposition of a matrix. 225 * 226 * This function returns a light-weight object of template class 227 * HouseholderSequence. You can either apply it directly to a matrix or 228 * you can convert it to a matrix of type #MatrixType. 229 * 230 * \sa matrixH() for an example, class HouseholderSequence 231 */ 232 HouseholderSequenceType matrixQ() const 233 { 234 eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized."); 235 return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate()) 236 .setLength(m_matrix.rows() - 1) 237 .setShift(1); 238 } 239 240 /** \brief Constructs the Hessenberg matrix H in the decomposition 241 * 242 * \returns expression object representing the matrix H 243 * 244 * \pre Either the constructor HessenbergDecomposition(const MatrixType&) 245 * or the member function compute(const MatrixType&) has been called 246 * before to compute the Hessenberg decomposition of a matrix. 247 * 248 * The object returned by this function constructs the Hessenberg matrix H 249 * when it is assigned to a matrix or otherwise evaluated. The matrix H is 250 * constructed from the packed matrix as returned by packedMatrix(): The 251 * upper part (including the subdiagonal) of the packed matrix contains 252 * the matrix H. It may sometimes be better to directly use the packed 253 * matrix instead of constructing the matrix H. 254 * 255 * Example: \include HessenbergDecomposition_matrixH.cpp 256 * Output: \verbinclude HessenbergDecomposition_matrixH.out 257 * 258 * \sa matrixQ(), packedMatrix() 259 */ 260 MatrixHReturnType matrixH() const 261 { 262 eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized."); 263 return MatrixHReturnType(*this); 264 } 265 266 private: 267 268 typedef Matrix<Scalar, 1, Size, Options | RowMajor, 1, MaxSize> VectorType; 269 typedef typename NumTraits<Scalar>::Real RealScalar; 270 static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp); 271 272 protected: 273 MatrixType m_matrix; 274 CoeffVectorType m_hCoeffs; 275 VectorType m_temp; 276 bool m_isInitialized; 277 }; 278 279 /** \internal 280 * Performs a tridiagonal decomposition of \a matA in place. 281 * 282 * \param matA the input selfadjoint matrix 283 * \param hCoeffs returned Householder coefficients 284 * 285 * The result is written in the lower triangular part of \a matA. 286 * 287 * Implemented from Golub's "%Matrix Computations", algorithm 8.3.1. 288 * 289 * \sa packedMatrix() 290 */ 291 template<typename MatrixType> 292 void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp) 293 { 294 eigen_assert(matA.rows()==matA.cols()); 295 Index n = matA.rows(); 296 temp.resize(n); 297 for (Index i = 0; i<n-1; ++i) 298 { 299 // let's consider the vector v = i-th column starting at position i+1 300 Index remainingSize = n-i-1; 301 RealScalar beta; 302 Scalar h; 303 matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta); 304 matA.col(i).coeffRef(i+1) = beta; 305 hCoeffs.coeffRef(i) = h; 306 307 // Apply similarity transformation to remaining columns, 308 // i.e., compute A = H A H' 309 310 // A = H A 311 matA.bottomRightCorner(remainingSize, remainingSize) 312 .applyHouseholderOnTheLeft(matA.col(i).tail(remainingSize-1), h, &temp.coeffRef(0)); 313 314 // A = A H' 315 matA.rightCols(remainingSize) 316 .applyHouseholderOnTheRight(matA.col(i).tail(remainingSize-1).conjugate(), numext::conj(h), &temp.coeffRef(0)); 317 } 318 } 319 320 namespace internal { 321 322 /** \eigenvalues_module \ingroup Eigenvalues_Module 323 * 324 * 325 * \brief Expression type for return value of HessenbergDecomposition::matrixH() 326 * 327 * \tparam MatrixType type of matrix in the Hessenberg decomposition 328 * 329 * Objects of this type represent the Hessenberg matrix in the Hessenberg 330 * decomposition of some matrix. The object holds a reference to the 331 * HessenbergDecomposition class until the it is assigned or evaluated for 332 * some other reason (the reference should remain valid during the life time 333 * of this object). This class is the return type of 334 * HessenbergDecomposition::matrixH(); there is probably no other use for this 335 * class. 336 */ 337 template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType 338 : public ReturnByValue<HessenbergDecompositionMatrixHReturnType<MatrixType> > 339 { 340 typedef typename MatrixType::Index Index; 341 public: 342 /** \brief Constructor. 343 * 344 * \param[in] hess Hessenberg decomposition 345 */ 346 HessenbergDecompositionMatrixHReturnType(const HessenbergDecomposition<MatrixType>& hess) : m_hess(hess) { } 347 348 /** \brief Hessenberg matrix in decomposition. 349 * 350 * \param[out] result Hessenberg matrix in decomposition \p hess which 351 * was passed to the constructor 352 */ 353 template <typename ResultType> 354 inline void evalTo(ResultType& result) const 355 { 356 result = m_hess.packedMatrix(); 357 Index n = result.rows(); 358 if (n>2) 359 result.bottomLeftCorner(n-2, n-2).template triangularView<Lower>().setZero(); 360 } 361 362 Index rows() const { return m_hess.packedMatrix().rows(); } 363 Index cols() const { return m_hess.packedMatrix().cols(); } 364 365 protected: 366 const HessenbergDecomposition<MatrixType>& m_hess; 367 }; 368 369 } // end namespace internal 370 371 } // end namespace Eigen 372 373 #endif // EIGEN_HESSENBERGDECOMPOSITION_H 374
