| // This file is part of Eigen, a lightweight C++ template library |
| // for linear algebra. |
| // |
| // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr> |
| // Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk> |
| // |
| // This Source Code Form is subject to the terms of the Mozilla |
| // Public License v. 2.0. If a copy of the MPL was not distributed |
| // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. |
| |
| #ifndef EIGEN_EIGENSOLVER_H |
| #define EIGEN_EIGENSOLVER_H |
| |
| #include "./RealSchur.h" |
| |
| // IWYU pragma: private |
| #include "./InternalHeaderCheck.h" |
| |
| namespace Eigen { |
| |
| /** \eigenvalues_module \ingroup Eigenvalues_Module |
| * |
| * |
| * \class EigenSolver |
| * |
| * \brief Computes eigenvalues and eigenvectors of general matrices |
| * |
| * \tparam MatrixType_ the type of the matrix of which we are computing the |
| * eigendecomposition; this is expected to be an instantiation of the Matrix |
| * class template. Currently, only real matrices are supported. |
| * |
| * The eigenvalues and eigenvectors of a matrix \f$ A \f$ are scalars |
| * \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda v \f$. If |
| * \f$ D \f$ is a diagonal matrix with the eigenvalues on the diagonal, and |
| * \f$ V \f$ is a matrix with the eigenvectors as its columns, then \f$ A V = |
| * V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we |
| * have \f$ A = V D V^{-1} \f$. This is called the eigendecomposition. |
| * |
| * The eigenvalues and eigenvectors of a matrix may be complex, even when the |
| * matrix is real. However, we can choose real matrices \f$ V \f$ and \f$ D |
| * \f$ satisfying \f$ A V = V D \f$, just like the eigendecomposition, if the |
| * matrix \f$ D \f$ is not required to be diagonal, but if it is allowed to |
| * have blocks of the form |
| * \f[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f] |
| * (where \f$ u \f$ and \f$ v \f$ are real numbers) on the diagonal. These |
| * blocks correspond to complex eigenvalue pairs \f$ u \pm iv \f$. We call |
| * this variant of the eigendecomposition the pseudo-eigendecomposition. |
| * |
| * Call the function compute() to compute the eigenvalues and eigenvectors of |
| * a given matrix. Alternatively, you can use the |
| * EigenSolver(const MatrixType&, bool) constructor which computes the |
| * eigenvalues and eigenvectors at construction time. Once the eigenvalue and |
| * eigenvectors are computed, they can be retrieved with the eigenvalues() and |
| * eigenvectors() functions. The pseudoEigenvalueMatrix() and |
| * pseudoEigenvectors() methods allow the construction of the |
| * pseudo-eigendecomposition. |
| * |
| * The documentation for EigenSolver(const MatrixType&, bool) contains an |
| * example of the typical use of this class. |
| * |
| * \note The implementation is adapted from |
| * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain). |
| * Their code is based on EISPACK. |
| * |
| * \sa MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver |
| */ |
| template <typename MatrixType_> |
| class EigenSolver { |
| public: |
| /** \brief Synonym for the template parameter \p MatrixType_. */ |
| typedef MatrixType_ MatrixType; |
| |
| enum { |
| RowsAtCompileTime = MatrixType::RowsAtCompileTime, |
| ColsAtCompileTime = MatrixType::ColsAtCompileTime, |
| Options = internal::traits<MatrixType>::Options, |
| MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, |
| MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime |
| }; |
| |
| /** \brief Scalar type for matrices of type #MatrixType. */ |
| typedef typename MatrixType::Scalar Scalar; |
| typedef typename NumTraits<Scalar>::Real RealScalar; |
| typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3 |
| |
| /** \brief Complex scalar type for #MatrixType. |
| * |
| * This is \c std::complex<Scalar> if #Scalar is real (e.g., |
| * \c float or \c double) and just \c Scalar if #Scalar is |
| * complex. |
| */ |
| typedef std::complex<RealScalar> ComplexScalar; |
| |
| /** \brief Type for vector of eigenvalues as returned by eigenvalues(). |
| * |
| * This is a column vector with entries of type #ComplexScalar. |
| * The length of the vector is the size of #MatrixType. |
| */ |
| typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType; |
| |
| /** \brief Type for matrix of eigenvectors as returned by eigenvectors(). |
| * |
| * This is a square matrix with entries of type #ComplexScalar. |
| * The size is the same as the size of #MatrixType. |
| */ |
| typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, |
| MaxColsAtCompileTime> |
| EigenvectorsType; |
| |
| /** \brief Default constructor. |
| * |
| * The default constructor is useful in cases in which the user intends to |
| * perform decompositions via EigenSolver::compute(const MatrixType&, bool). |
| * |
| * \sa compute() for an example. |
| */ |
| EigenSolver() |
| : m_eivec(), m_eivalues(), m_isInitialized(false), m_eigenvectorsOk(false), m_realSchur(), m_matT(), m_tmp() {} |
| |
| /** \brief Default constructor with memory preallocation |
| * |
| * Like the default constructor but with preallocation of the internal data |
| * according to the specified problem \a size. |
| * \sa EigenSolver() |
| */ |
| explicit EigenSolver(Index size) |
| : m_eivec(size, size), |
| m_eivalues(size), |
| m_isInitialized(false), |
| m_eigenvectorsOk(false), |
| m_realSchur(size), |
| m_matT(size, size), |
| m_tmp(size) {} |
| |
| /** \brief Constructor; computes eigendecomposition of given matrix. |
| * |
| * \param[in] matrix Square matrix whose eigendecomposition is to be computed. |
| * \param[in] computeEigenvectors If true, both the eigenvectors and the |
| * eigenvalues are computed; if false, only the eigenvalues are |
| * computed. |
| * |
| * This constructor calls compute() to compute the eigenvalues |
| * and eigenvectors. |
| * |
| * Example: \include EigenSolver_EigenSolver_MatrixType.cpp |
| * Output: \verbinclude EigenSolver_EigenSolver_MatrixType.out |
| * |
| * \sa compute() |
| */ |
| template <typename InputType> |
| explicit EigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true) |
| : m_eivec(matrix.rows(), matrix.cols()), |
| m_eivalues(matrix.cols()), |
| m_isInitialized(false), |
| m_eigenvectorsOk(false), |
| m_realSchur(matrix.cols()), |
| m_matT(matrix.rows(), matrix.cols()), |
| m_tmp(matrix.cols()) { |
| compute(matrix.derived(), computeEigenvectors); |
| } |
| |
| /** \brief Returns the eigenvectors of given matrix. |
| * |
| * \returns %Matrix whose columns are the (possibly complex) eigenvectors. |
| * |
| * \pre Either the constructor |
| * EigenSolver(const MatrixType&,bool) or the member function |
| * compute(const MatrixType&, bool) has been called before, and |
| * \p computeEigenvectors was set to true (the default). |
| * |
| * Column \f$ k \f$ of the returned matrix is an eigenvector corresponding |
| * to eigenvalue number \f$ k \f$ as returned by eigenvalues(). The |
| * eigenvectors are normalized to have (Euclidean) norm equal to one. The |
| * matrix returned by this function is the matrix \f$ V \f$ in the |
| * eigendecomposition \f$ A = V D V^{-1} \f$, if it exists. |
| * |
| * Example: \include EigenSolver_eigenvectors.cpp |
| * Output: \verbinclude EigenSolver_eigenvectors.out |
| * |
| * \sa eigenvalues(), pseudoEigenvectors() |
| */ |
| EigenvectorsType eigenvectors() const; |
| |
| /** \brief Returns the pseudo-eigenvectors of given matrix. |
| * |
| * \returns Const reference to matrix whose columns are the pseudo-eigenvectors. |
| * |
| * \pre Either the constructor |
| * EigenSolver(const MatrixType&,bool) or the member function |
| * compute(const MatrixType&, bool) has been called before, and |
| * \p computeEigenvectors was set to true (the default). |
| * |
| * The real matrix \f$ V \f$ returned by this function and the |
| * block-diagonal matrix \f$ D \f$ returned by pseudoEigenvalueMatrix() |
| * satisfy \f$ AV = VD \f$. |
| * |
| * Example: \include EigenSolver_pseudoEigenvectors.cpp |
| * Output: \verbinclude EigenSolver_pseudoEigenvectors.out |
| * |
| * \sa pseudoEigenvalueMatrix(), eigenvectors() |
| */ |
| const MatrixType& pseudoEigenvectors() const { |
| eigen_assert(m_isInitialized && "EigenSolver is not initialized."); |
| eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues."); |
| return m_eivec; |
| } |
| |
| /** \brief Returns the block-diagonal matrix in the pseudo-eigendecomposition. |
| * |
| * \returns A block-diagonal matrix. |
| * |
| * \pre Either the constructor |
| * EigenSolver(const MatrixType&,bool) or the member function |
| * compute(const MatrixType&, bool) has been called before. |
| * |
| * The matrix \f$ D \f$ returned by this function is real and |
| * block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2 |
| * blocks of the form |
| * \f$ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f$. |
| * These blocks are not sorted in any particular order. |
| * The matrix \f$ D \f$ and the matrix \f$ V \f$ returned by |
| * pseudoEigenvectors() satisfy \f$ AV = VD \f$. |
| * |
| * \sa pseudoEigenvectors() for an example, eigenvalues() |
| */ |
| MatrixType pseudoEigenvalueMatrix() const; |
| |
| /** \brief Returns the eigenvalues of given matrix. |
| * |
| * \returns A const reference to the column vector containing the eigenvalues. |
| * |
| * \pre Either the constructor |
| * EigenSolver(const MatrixType&,bool) or the member function |
| * compute(const MatrixType&, bool) has been called before. |
| * |
| * The eigenvalues are repeated according to their algebraic multiplicity, |
| * so there are as many eigenvalues as rows in the matrix. The eigenvalues |
| * are not sorted in any particular order. |
| * |
| * Example: \include EigenSolver_eigenvalues.cpp |
| * Output: \verbinclude EigenSolver_eigenvalues.out |
| * |
| * \sa eigenvectors(), pseudoEigenvalueMatrix(), |
| * MatrixBase::eigenvalues() |
| */ |
| const EigenvalueType& eigenvalues() const { |
| eigen_assert(m_isInitialized && "EigenSolver is not initialized."); |
| return m_eivalues; |
| } |
| |
| /** \brief Computes eigendecomposition of given matrix. |
| * |
| * \param[in] matrix Square matrix whose eigendecomposition is to be computed. |
| * \param[in] computeEigenvectors If true, both the eigenvectors and the |
| * eigenvalues are computed; if false, only the eigenvalues are |
| * computed. |
| * \returns Reference to \c *this |
| * |
| * This function computes the eigenvalues of the real matrix \p matrix. |
| * The eigenvalues() function can be used to retrieve them. If |
| * \p computeEigenvectors is true, then the eigenvectors are also computed |
| * and can be retrieved by calling eigenvectors(). |
| * |
| * The matrix is first reduced to real Schur form using the RealSchur |
| * class. The Schur decomposition is then used to compute the eigenvalues |
| * and eigenvectors. |
| * |
| * The cost of the computation is dominated by the cost of the |
| * Schur decomposition, which is very approximately \f$ 25n^3 \f$ |
| * (where \f$ n \f$ is the size of the matrix) if \p computeEigenvectors |
| * is true, and \f$ 10n^3 \f$ if \p computeEigenvectors is false. |
| * |
| * This method reuses of the allocated data in the EigenSolver object. |
| * |
| * Example: \include EigenSolver_compute.cpp |
| * Output: \verbinclude EigenSolver_compute.out |
| */ |
| template <typename InputType> |
| EigenSolver& compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true); |
| |
| /** \returns NumericalIssue if the input contains INF or NaN values or overflow occurred. Returns Success otherwise. |
| */ |
| ComputationInfo info() const { |
| eigen_assert(m_isInitialized && "EigenSolver is not initialized."); |
| return m_info; |
| } |
| |
| /** \brief Sets the maximum number of iterations allowed. */ |
| EigenSolver& setMaxIterations(Index maxIters) { |
| m_realSchur.setMaxIterations(maxIters); |
| return *this; |
| } |
| |
| /** \brief Returns the maximum number of iterations. */ |
| Index getMaxIterations() { return m_realSchur.getMaxIterations(); } |
| |
| private: |
| void doComputeEigenvectors(); |
| |
| protected: |
| static void check_template_parameters() { |
| EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); |
| EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL); |
| } |
| |
| MatrixType m_eivec; |
| EigenvalueType m_eivalues; |
| bool m_isInitialized; |
| bool m_eigenvectorsOk; |
| ComputationInfo m_info; |
| RealSchur<MatrixType> m_realSchur; |
| MatrixType m_matT; |
| |
| typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType; |
| ColumnVectorType m_tmp; |
| }; |
| |
| template <typename MatrixType> |
| MatrixType EigenSolver<MatrixType>::pseudoEigenvalueMatrix() const { |
| eigen_assert(m_isInitialized && "EigenSolver is not initialized."); |
| const RealScalar precision = RealScalar(2) * NumTraits<RealScalar>::epsilon(); |
| Index n = m_eivalues.rows(); |
| MatrixType matD = MatrixType::Zero(n, n); |
| for (Index i = 0; i < n; ++i) { |
| if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i)), precision)) |
| matD.coeffRef(i, i) = numext::real(m_eivalues.coeff(i)); |
| else { |
| matD.template block<2, 2>(i, i) << numext::real(m_eivalues.coeff(i)), numext::imag(m_eivalues.coeff(i)), |
| -numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i)); |
| ++i; |
| } |
| } |
| return matD; |
| } |
| |
| template <typename MatrixType> |
| typename EigenSolver<MatrixType>::EigenvectorsType EigenSolver<MatrixType>::eigenvectors() const { |
| eigen_assert(m_isInitialized && "EigenSolver is not initialized."); |
| eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues."); |
| const RealScalar precision = RealScalar(2) * NumTraits<RealScalar>::epsilon(); |
| Index n = m_eivec.cols(); |
| EigenvectorsType matV(n, n); |
| for (Index j = 0; j < n; ++j) { |
| if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(j)), numext::real(m_eivalues.coeff(j)), precision) || |
| j + 1 == n) { |
| // we have a real eigen value |
| matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>(); |
| matV.col(j).normalize(); |
| } else { |
| // we have a pair of complex eigen values |
| for (Index i = 0; i < n; ++i) { |
| matV.coeffRef(i, j) = ComplexScalar(m_eivec.coeff(i, j), m_eivec.coeff(i, j + 1)); |
| matV.coeffRef(i, j + 1) = ComplexScalar(m_eivec.coeff(i, j), -m_eivec.coeff(i, j + 1)); |
| } |
| matV.col(j).normalize(); |
| matV.col(j + 1).normalize(); |
| ++j; |
| } |
| } |
| return matV; |
| } |
| |
| template <typename MatrixType> |
| template <typename InputType> |
| EigenSolver<MatrixType>& EigenSolver<MatrixType>::compute(const EigenBase<InputType>& matrix, |
| bool computeEigenvectors) { |
| check_template_parameters(); |
| |
| using numext::isfinite; |
| using std::abs; |
| using std::sqrt; |
| eigen_assert(matrix.cols() == matrix.rows()); |
| |
| // Reduce to real Schur form. |
| m_realSchur.compute(matrix.derived(), computeEigenvectors); |
| |
| m_info = m_realSchur.info(); |
| |
| if (m_info == Success) { |
| m_matT = m_realSchur.matrixT(); |
| if (computeEigenvectors) m_eivec = m_realSchur.matrixU(); |
| |
| // Compute eigenvalues from matT |
| m_eivalues.resize(matrix.cols()); |
| Index i = 0; |
| while (i < matrix.cols()) { |
| if (i == matrix.cols() - 1 || m_matT.coeff(i + 1, i) == Scalar(0)) { |
| m_eivalues.coeffRef(i) = m_matT.coeff(i, i); |
| if (!(isfinite)(m_eivalues.coeffRef(i))) { |
| m_isInitialized = true; |
| m_eigenvectorsOk = false; |
| m_info = NumericalIssue; |
| return *this; |
| } |
| ++i; |
| } else { |
| Scalar p = Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i + 1, i + 1)); |
| Scalar z; |
| // Compute z = sqrt(abs(p * p + m_matT.coeff(i+1, i) * m_matT.coeff(i, i+1))); |
| // without overflow |
| { |
| Scalar t0 = m_matT.coeff(i + 1, i); |
| Scalar t1 = m_matT.coeff(i, i + 1); |
| Scalar maxval = numext::maxi<Scalar>(abs(p), numext::maxi<Scalar>(abs(t0), abs(t1))); |
| t0 /= maxval; |
| t1 /= maxval; |
| Scalar p0 = p / maxval; |
| z = maxval * sqrt(abs(p0 * p0 + t0 * t1)); |
| } |
| |
| m_eivalues.coeffRef(i) = ComplexScalar(m_matT.coeff(i + 1, i + 1) + p, z); |
| m_eivalues.coeffRef(i + 1) = ComplexScalar(m_matT.coeff(i + 1, i + 1) + p, -z); |
| if (!((isfinite)(m_eivalues.coeffRef(i)) && (isfinite)(m_eivalues.coeffRef(i + 1)))) { |
| m_isInitialized = true; |
| m_eigenvectorsOk = false; |
| m_info = NumericalIssue; |
| return *this; |
| } |
| i += 2; |
| } |
| } |
| |
| // Compute eigenvectors. |
| if (computeEigenvectors) doComputeEigenvectors(); |
| } |
| |
| m_isInitialized = true; |
| m_eigenvectorsOk = computeEigenvectors; |
| |
| return *this; |
| } |
| |
| template <typename MatrixType> |
| void EigenSolver<MatrixType>::doComputeEigenvectors() { |
| using std::abs; |
| const Index size = m_eivec.cols(); |
| const Scalar eps = NumTraits<Scalar>::epsilon(); |
| |
| // inefficient! this is already computed in RealSchur |
| Scalar norm(0); |
| for (Index j = 0; j < size; ++j) { |
| norm += m_matT.row(j).segment((std::max)(j - 1, Index(0)), size - (std::max)(j - 1, Index(0))).cwiseAbs().sum(); |
| } |
| |
| // Backsubstitute to find vectors of upper triangular form |
| if (norm == Scalar(0)) { |
| return; |
| } |
| |
| for (Index n = size - 1; n >= 0; n--) { |
| Scalar p = m_eivalues.coeff(n).real(); |
| Scalar q = m_eivalues.coeff(n).imag(); |
| |
| // Scalar vector |
| if (q == Scalar(0)) { |
| Scalar lastr(0), lastw(0); |
| Index l = n; |
| |
| m_matT.coeffRef(n, n) = Scalar(1); |
| for (Index i = n - 1; i >= 0; i--) { |
| Scalar w = m_matT.coeff(i, i) - p; |
| Scalar r = m_matT.row(i).segment(l, n - l + 1).dot(m_matT.col(n).segment(l, n - l + 1)); |
| |
| if (m_eivalues.coeff(i).imag() < Scalar(0)) { |
| lastw = w; |
| lastr = r; |
| } else { |
| l = i; |
| if (m_eivalues.coeff(i).imag() == Scalar(0)) { |
| if (w != Scalar(0)) |
| m_matT.coeffRef(i, n) = -r / w; |
| else |
| m_matT.coeffRef(i, n) = -r / (eps * norm); |
| } else // Solve real equations |
| { |
| Scalar x = m_matT.coeff(i, i + 1); |
| Scalar y = m_matT.coeff(i + 1, i); |
| Scalar denom = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + |
| m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag(); |
| Scalar t = (x * lastr - lastw * r) / denom; |
| m_matT.coeffRef(i, n) = t; |
| if (abs(x) > abs(lastw)) |
| m_matT.coeffRef(i + 1, n) = (-r - w * t) / x; |
| else |
| m_matT.coeffRef(i + 1, n) = (-lastr - y * t) / lastw; |
| } |
| |
| // Overflow control |
| Scalar t = abs(m_matT.coeff(i, n)); |
| if ((eps * t) * t > Scalar(1)) m_matT.col(n).tail(size - i) /= t; |
| } |
| } |
| } else if (q < Scalar(0) && n > 0) // Complex vector |
| { |
| Scalar lastra(0), lastsa(0), lastw(0); |
| Index l = n - 1; |
| |
| // Last vector component imaginary so matrix is triangular |
| if (abs(m_matT.coeff(n, n - 1)) > abs(m_matT.coeff(n - 1, n))) { |
| m_matT.coeffRef(n - 1, n - 1) = q / m_matT.coeff(n, n - 1); |
| m_matT.coeffRef(n - 1, n) = -(m_matT.coeff(n, n) - p) / m_matT.coeff(n, n - 1); |
| } else { |
| ComplexScalar cc = |
| ComplexScalar(Scalar(0), -m_matT.coeff(n - 1, n)) / ComplexScalar(m_matT.coeff(n - 1, n - 1) - p, q); |
| m_matT.coeffRef(n - 1, n - 1) = numext::real(cc); |
| m_matT.coeffRef(n - 1, n) = numext::imag(cc); |
| } |
| m_matT.coeffRef(n, n - 1) = Scalar(0); |
| m_matT.coeffRef(n, n) = Scalar(1); |
| for (Index i = n - 2; i >= 0; i--) { |
| Scalar ra = m_matT.row(i).segment(l, n - l + 1).dot(m_matT.col(n - 1).segment(l, n - l + 1)); |
| Scalar sa = m_matT.row(i).segment(l, n - l + 1).dot(m_matT.col(n).segment(l, n - l + 1)); |
| Scalar w = m_matT.coeff(i, i) - p; |
| |
| if (m_eivalues.coeff(i).imag() < Scalar(0)) { |
| lastw = w; |
| lastra = ra; |
| lastsa = sa; |
| } else { |
| l = i; |
| if (m_eivalues.coeff(i).imag() == RealScalar(0)) { |
| ComplexScalar cc = ComplexScalar(-ra, -sa) / ComplexScalar(w, q); |
| m_matT.coeffRef(i, n - 1) = numext::real(cc); |
| m_matT.coeffRef(i, n) = numext::imag(cc); |
| } else { |
| // Solve complex equations |
| Scalar x = m_matT.coeff(i, i + 1); |
| Scalar y = m_matT.coeff(i + 1, i); |
| Scalar vr = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + |
| m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag() - q * q; |
| Scalar vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q; |
| if ((vr == Scalar(0)) && (vi == Scalar(0))) |
| vr = eps * norm * (abs(w) + abs(q) + abs(x) + abs(y) + abs(lastw)); |
| |
| ComplexScalar cc = ComplexScalar(x * lastra - lastw * ra + q * sa, x * lastsa - lastw * sa - q * ra) / |
| ComplexScalar(vr, vi); |
| m_matT.coeffRef(i, n - 1) = numext::real(cc); |
| m_matT.coeffRef(i, n) = numext::imag(cc); |
| if (abs(x) > (abs(lastw) + abs(q))) { |
| m_matT.coeffRef(i + 1, n - 1) = (-ra - w * m_matT.coeff(i, n - 1) + q * m_matT.coeff(i, n)) / x; |
| m_matT.coeffRef(i + 1, n) = (-sa - w * m_matT.coeff(i, n) - q * m_matT.coeff(i, n - 1)) / x; |
| } else { |
| cc = ComplexScalar(-lastra - y * m_matT.coeff(i, n - 1), -lastsa - y * m_matT.coeff(i, n)) / |
| ComplexScalar(lastw, q); |
| m_matT.coeffRef(i + 1, n - 1) = numext::real(cc); |
| m_matT.coeffRef(i + 1, n) = numext::imag(cc); |
| } |
| } |
| |
| // Overflow control |
| Scalar t = numext::maxi<Scalar>(abs(m_matT.coeff(i, n - 1)), abs(m_matT.coeff(i, n))); |
| if ((eps * t) * t > Scalar(1)) m_matT.block(i, n - 1, size - i, 2) /= t; |
| } |
| } |
| |
| // We handled a pair of complex conjugate eigenvalues, so need to skip them both |
| n--; |
| } else { |
| eigen_assert(0 && "Internal bug in EigenSolver (INF or NaN has not been detected)"); // this should not happen |
| } |
| } |
| |
| // Back transformation to get eigenvectors of original matrix |
| for (Index j = size - 1; j >= 0; j--) { |
| m_tmp.noalias() = m_eivec.leftCols(j + 1) * m_matT.col(j).segment(0, j + 1); |
| m_eivec.col(j) = m_tmp; |
| } |
| } |
| |
| } // end namespace Eigen |
| |
| #endif // EIGEN_EIGENSOLVER_H |
