Carlos Hernandez | 7faaa9f | 2014-08-05 17:53:32 -0700 | [diff] [blame] | 1 | // This file is part of Eigen, a lightweight C++ template library |
| 2 | // for linear algebra. |
| 3 | // |
| 4 | // Copyright (C) 2012 Gael Guennebaud <gael.guennebaud@inria.fr> |
| 5 | // Copyright (C) 2010,2012 Jitse Niesen <jitse@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_GENERALIZEDEIGENSOLVER_H |
| 12 | #define EIGEN_GENERALIZEDEIGENSOLVER_H |
| 13 | |
| 14 | #include "./RealQZ.h" |
| 15 | |
| 16 | namespace Eigen { |
| 17 | |
| 18 | /** \eigenvalues_module \ingroup Eigenvalues_Module |
| 19 | * |
| 20 | * |
| 21 | * \class GeneralizedEigenSolver |
| 22 | * |
| 23 | * \brief Computes the generalized eigenvalues and eigenvectors of a pair of general matrices |
| 24 | * |
| 25 | * \tparam _MatrixType the type of the matrices of which we are computing the |
| 26 | * eigen-decomposition; this is expected to be an instantiation of the Matrix |
| 27 | * class template. Currently, only real matrices are supported. |
| 28 | * |
| 29 | * The generalized eigenvalues and eigenvectors of a matrix pair \f$ A \f$ and \f$ B \f$ are scalars |
| 30 | * \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda Bv \f$. If |
| 31 | * \f$ D \f$ is a diagonal matrix with the eigenvalues on the diagonal, and |
| 32 | * \f$ V \f$ is a matrix with the eigenvectors as its columns, then \f$ A V = |
| 33 | * B V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we |
| 34 | * have \f$ A = B V D V^{-1} \f$. This is called the generalized eigen-decomposition. |
| 35 | * |
| 36 | * The generalized eigenvalues and eigenvectors of a matrix pair may be complex, even when the |
| 37 | * matrices are real. Moreover, the generalized eigenvalue might be infinite if the matrix B is |
| 38 | * singular. To workaround this difficulty, the eigenvalues are provided as a pair of complex \f$ \alpha \f$ |
| 39 | * and real \f$ \beta \f$ such that: \f$ \lambda_i = \alpha_i / \beta_i \f$. If \f$ \beta_i \f$ is (nearly) zero, |
| 40 | * then one can consider the well defined left eigenvalue \f$ \mu = \beta_i / \alpha_i\f$ such that: |
| 41 | * \f$ \mu_i A v_i = B v_i \f$, or even \f$ \mu_i u_i^T A = u_i^T B \f$ where \f$ u_i \f$ is |
| 42 | * called the left eigenvector. |
| 43 | * |
| 44 | * Call the function compute() to compute the generalized eigenvalues and eigenvectors of |
| 45 | * a given matrix pair. Alternatively, you can use the |
| 46 | * GeneralizedEigenSolver(const MatrixType&, const MatrixType&, bool) constructor which computes the |
| 47 | * eigenvalues and eigenvectors at construction time. Once the eigenvalue and |
| 48 | * eigenvectors are computed, they can be retrieved with the eigenvalues() and |
| 49 | * eigenvectors() functions. |
| 50 | * |
| 51 | * Here is an usage example of this class: |
| 52 | * Example: \include GeneralizedEigenSolver.cpp |
| 53 | * Output: \verbinclude GeneralizedEigenSolver.out |
| 54 | * |
| 55 | * \sa MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver |
| 56 | */ |
| 57 | template<typename _MatrixType> class GeneralizedEigenSolver |
| 58 | { |
| 59 | public: |
| 60 | |
| 61 | /** \brief Synonym for the template parameter \p _MatrixType. */ |
| 62 | typedef _MatrixType MatrixType; |
| 63 | |
| 64 | enum { |
| 65 | RowsAtCompileTime = MatrixType::RowsAtCompileTime, |
| 66 | ColsAtCompileTime = MatrixType::ColsAtCompileTime, |
| 67 | Options = MatrixType::Options, |
| 68 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, |
| 69 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime |
| 70 | }; |
| 71 | |
| 72 | /** \brief Scalar type for matrices of type #MatrixType. */ |
| 73 | typedef typename MatrixType::Scalar Scalar; |
| 74 | typedef typename NumTraits<Scalar>::Real RealScalar; |
| 75 | typedef typename MatrixType::Index Index; |
| 76 | |
| 77 | /** \brief Complex scalar type for #MatrixType. |
| 78 | * |
| 79 | * This is \c std::complex<Scalar> if #Scalar is real (e.g., |
| 80 | * \c float or \c double) and just \c Scalar if #Scalar is |
| 81 | * complex. |
| 82 | */ |
| 83 | typedef std::complex<RealScalar> ComplexScalar; |
| 84 | |
| 85 | /** \brief Type for vector of real scalar values eigenvalues as returned by betas(). |
| 86 | * |
| 87 | * This is a column vector with entries of type #Scalar. |
| 88 | * The length of the vector is the size of #MatrixType. |
| 89 | */ |
| 90 | typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> VectorType; |
| 91 | |
| 92 | /** \brief Type for vector of complex scalar values eigenvalues as returned by betas(). |
| 93 | * |
| 94 | * This is a column vector with entries of type #ComplexScalar. |
| 95 | * The length of the vector is the size of #MatrixType. |
| 96 | */ |
| 97 | typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ComplexVectorType; |
| 98 | |
| 99 | /** \brief Expression type for the eigenvalues as returned by eigenvalues(). |
| 100 | */ |
| 101 | typedef CwiseBinaryOp<internal::scalar_quotient_op<ComplexScalar,Scalar>,ComplexVectorType,VectorType> EigenvalueType; |
| 102 | |
| 103 | /** \brief Type for matrix of eigenvectors as returned by eigenvectors(). |
| 104 | * |
| 105 | * This is a square matrix with entries of type #ComplexScalar. |
| 106 | * The size is the same as the size of #MatrixType. |
| 107 | */ |
| 108 | typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorsType; |
| 109 | |
| 110 | /** \brief Default constructor. |
| 111 | * |
| 112 | * The default constructor is useful in cases in which the user intends to |
| 113 | * perform decompositions via EigenSolver::compute(const MatrixType&, bool). |
| 114 | * |
| 115 | * \sa compute() for an example. |
| 116 | */ |
| 117 | GeneralizedEigenSolver() : m_eivec(), m_alphas(), m_betas(), m_isInitialized(false), m_realQZ(), m_matS(), m_tmp() {} |
| 118 | |
| 119 | /** \brief Default constructor with memory preallocation |
| 120 | * |
| 121 | * Like the default constructor but with preallocation of the internal data |
| 122 | * according to the specified problem \a size. |
| 123 | * \sa GeneralizedEigenSolver() |
| 124 | */ |
| 125 | GeneralizedEigenSolver(Index size) |
| 126 | : m_eivec(size, size), |
| 127 | m_alphas(size), |
| 128 | m_betas(size), |
| 129 | m_isInitialized(false), |
| 130 | m_eigenvectorsOk(false), |
| 131 | m_realQZ(size), |
| 132 | m_matS(size, size), |
| 133 | m_tmp(size) |
| 134 | {} |
| 135 | |
| 136 | /** \brief Constructor; computes the generalized eigendecomposition of given matrix pair. |
| 137 | * |
| 138 | * \param[in] A Square matrix whose eigendecomposition is to be computed. |
| 139 | * \param[in] B Square matrix whose eigendecomposition is to be computed. |
| 140 | * \param[in] computeEigenvectors If true, both the eigenvectors and the |
| 141 | * eigenvalues are computed; if false, only the eigenvalues are computed. |
| 142 | * |
| 143 | * This constructor calls compute() to compute the generalized eigenvalues |
| 144 | * and eigenvectors. |
| 145 | * |
| 146 | * \sa compute() |
| 147 | */ |
| 148 | GeneralizedEigenSolver(const MatrixType& A, const MatrixType& B, bool computeEigenvectors = true) |
| 149 | : m_eivec(A.rows(), A.cols()), |
| 150 | m_alphas(A.cols()), |
| 151 | m_betas(A.cols()), |
| 152 | m_isInitialized(false), |
| 153 | m_eigenvectorsOk(false), |
| 154 | m_realQZ(A.cols()), |
| 155 | m_matS(A.rows(), A.cols()), |
| 156 | m_tmp(A.cols()) |
| 157 | { |
| 158 | compute(A, B, computeEigenvectors); |
| 159 | } |
| 160 | |
| 161 | /* \brief Returns the computed generalized eigenvectors. |
| 162 | * |
| 163 | * \returns %Matrix whose columns are the (possibly complex) eigenvectors. |
| 164 | * |
| 165 | * \pre Either the constructor |
| 166 | * GeneralizedEigenSolver(const MatrixType&,const MatrixType&, bool) or the member function |
| 167 | * compute(const MatrixType&, const MatrixType& bool) has been called before, and |
| 168 | * \p computeEigenvectors was set to true (the default). |
| 169 | * |
| 170 | * Column \f$ k \f$ of the returned matrix is an eigenvector corresponding |
| 171 | * to eigenvalue number \f$ k \f$ as returned by eigenvalues(). The |
| 172 | * eigenvectors are normalized to have (Euclidean) norm equal to one. The |
| 173 | * matrix returned by this function is the matrix \f$ V \f$ in the |
| 174 | * generalized eigendecomposition \f$ A = B V D V^{-1} \f$, if it exists. |
| 175 | * |
| 176 | * \sa eigenvalues() |
| 177 | */ |
| 178 | // EigenvectorsType eigenvectors() const; |
| 179 | |
| 180 | /** \brief Returns an expression of the computed generalized eigenvalues. |
| 181 | * |
| 182 | * \returns An expression of the column vector containing the eigenvalues. |
| 183 | * |
| 184 | * It is a shortcut for \code this->alphas().cwiseQuotient(this->betas()); \endcode |
| 185 | * Not that betas might contain zeros. It is therefore not recommended to use this function, |
| 186 | * but rather directly deal with the alphas and betas vectors. |
| 187 | * |
| 188 | * \pre Either the constructor |
| 189 | * GeneralizedEigenSolver(const MatrixType&,const MatrixType&,bool) or the member function |
| 190 | * compute(const MatrixType&,const MatrixType&,bool) has been called before. |
| 191 | * |
| 192 | * The eigenvalues are repeated according to their algebraic multiplicity, |
| 193 | * so there are as many eigenvalues as rows in the matrix. The eigenvalues |
| 194 | * are not sorted in any particular order. |
| 195 | * |
| 196 | * \sa alphas(), betas(), eigenvectors() |
| 197 | */ |
| 198 | EigenvalueType eigenvalues() const |
| 199 | { |
| 200 | eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized."); |
| 201 | return EigenvalueType(m_alphas,m_betas); |
| 202 | } |
| 203 | |
| 204 | /** \returns A const reference to the vectors containing the alpha values |
| 205 | * |
| 206 | * This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j). |
| 207 | * |
| 208 | * \sa betas(), eigenvalues() */ |
| 209 | ComplexVectorType alphas() const |
| 210 | { |
| 211 | eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized."); |
| 212 | return m_alphas; |
| 213 | } |
| 214 | |
| 215 | /** \returns A const reference to the vectors containing the beta values |
| 216 | * |
| 217 | * This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j). |
| 218 | * |
| 219 | * \sa alphas(), eigenvalues() */ |
| 220 | VectorType betas() const |
| 221 | { |
| 222 | eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized."); |
| 223 | return m_betas; |
| 224 | } |
| 225 | |
| 226 | /** \brief Computes generalized eigendecomposition of given matrix. |
| 227 | * |
| 228 | * \param[in] A Square matrix whose eigendecomposition is to be computed. |
| 229 | * \param[in] B Square matrix whose eigendecomposition is to be computed. |
| 230 | * \param[in] computeEigenvectors If true, both the eigenvectors and the |
| 231 | * eigenvalues are computed; if false, only the eigenvalues are |
| 232 | * computed. |
| 233 | * \returns Reference to \c *this |
| 234 | * |
| 235 | * This function computes the eigenvalues of the real matrix \p matrix. |
| 236 | * The eigenvalues() function can be used to retrieve them. If |
| 237 | * \p computeEigenvectors is true, then the eigenvectors are also computed |
| 238 | * and can be retrieved by calling eigenvectors(). |
| 239 | * |
| 240 | * The matrix is first reduced to real generalized Schur form using the RealQZ |
| 241 | * class. The generalized Schur decomposition is then used to compute the eigenvalues |
| 242 | * and eigenvectors. |
| 243 | * |
| 244 | * The cost of the computation is dominated by the cost of the |
| 245 | * generalized Schur decomposition. |
| 246 | * |
| 247 | * This method reuses of the allocated data in the GeneralizedEigenSolver object. |
| 248 | */ |
| 249 | GeneralizedEigenSolver& compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors = true); |
| 250 | |
| 251 | ComputationInfo info() const |
| 252 | { |
| 253 | eigen_assert(m_isInitialized && "EigenSolver is not initialized."); |
| 254 | return m_realQZ.info(); |
| 255 | } |
| 256 | |
| 257 | /** Sets the maximal number of iterations allowed. |
| 258 | */ |
| 259 | GeneralizedEigenSolver& setMaxIterations(Index maxIters) |
| 260 | { |
| 261 | m_realQZ.setMaxIterations(maxIters); |
| 262 | return *this; |
| 263 | } |
| 264 | |
| 265 | protected: |
| 266 | MatrixType m_eivec; |
| 267 | ComplexVectorType m_alphas; |
| 268 | VectorType m_betas; |
| 269 | bool m_isInitialized; |
| 270 | bool m_eigenvectorsOk; |
| 271 | RealQZ<MatrixType> m_realQZ; |
| 272 | MatrixType m_matS; |
| 273 | |
| 274 | typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType; |
| 275 | ColumnVectorType m_tmp; |
| 276 | }; |
| 277 | |
| 278 | //template<typename MatrixType> |
| 279 | //typename GeneralizedEigenSolver<MatrixType>::EigenvectorsType GeneralizedEigenSolver<MatrixType>::eigenvectors() const |
| 280 | //{ |
| 281 | // eigen_assert(m_isInitialized && "EigenSolver is not initialized."); |
| 282 | // eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues."); |
| 283 | // Index n = m_eivec.cols(); |
| 284 | // EigenvectorsType matV(n,n); |
| 285 | // // TODO |
| 286 | // return matV; |
| 287 | //} |
| 288 | |
| 289 | template<typename MatrixType> |
| 290 | GeneralizedEigenSolver<MatrixType>& |
| 291 | GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors) |
| 292 | { |
| 293 | using std::sqrt; |
| 294 | using std::abs; |
| 295 | eigen_assert(A.cols() == A.rows() && B.cols() == A.rows() && B.cols() == B.rows()); |
| 296 | |
| 297 | // Reduce to generalized real Schur form: |
| 298 | // A = Q S Z and B = Q T Z |
| 299 | m_realQZ.compute(A, B, computeEigenvectors); |
| 300 | |
| 301 | if (m_realQZ.info() == Success) |
| 302 | { |
| 303 | m_matS = m_realQZ.matrixS(); |
| 304 | if (computeEigenvectors) |
| 305 | m_eivec = m_realQZ.matrixZ().transpose(); |
| 306 | |
| 307 | // Compute eigenvalues from matS |
| 308 | m_alphas.resize(A.cols()); |
| 309 | m_betas.resize(A.cols()); |
| 310 | Index i = 0; |
| 311 | while (i < A.cols()) |
| 312 | { |
| 313 | if (i == A.cols() - 1 || m_matS.coeff(i+1, i) == Scalar(0)) |
| 314 | { |
| 315 | m_alphas.coeffRef(i) = m_matS.coeff(i, i); |
| 316 | m_betas.coeffRef(i) = m_realQZ.matrixT().coeff(i,i); |
| 317 | ++i; |
| 318 | } |
| 319 | else |
| 320 | { |
| 321 | Scalar p = Scalar(0.5) * (m_matS.coeff(i, i) - m_matS.coeff(i+1, i+1)); |
| 322 | Scalar z = sqrt(abs(p * p + m_matS.coeff(i+1, i) * m_matS.coeff(i, i+1))); |
| 323 | m_alphas.coeffRef(i) = ComplexScalar(m_matS.coeff(i+1, i+1) + p, z); |
| 324 | m_alphas.coeffRef(i+1) = ComplexScalar(m_matS.coeff(i+1, i+1) + p, -z); |
| 325 | |
| 326 | m_betas.coeffRef(i) = m_realQZ.matrixT().coeff(i,i); |
| 327 | m_betas.coeffRef(i+1) = m_realQZ.matrixT().coeff(i,i); |
| 328 | i += 2; |
| 329 | } |
| 330 | } |
| 331 | } |
| 332 | |
| 333 | m_isInitialized = true; |
| 334 | m_eigenvectorsOk = false;//computeEigenvectors; |
| 335 | |
| 336 | return *this; |
| 337 | } |
| 338 | |
| 339 | } // end namespace Eigen |
| 340 | |
| 341 | #endif // EIGEN_GENERALIZEDEIGENSOLVER_H |