Eigen/src/Eigenvalues/ComplexEigenSolver.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Claire Maurice
// Copyright (C) 2009 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_COMPLEX_EIGEN_SOLVER_H
#define EIGEN_COMPLEX_EIGEN_SOLVER_H

#include "./ComplexSchur.h"

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

/** \eigenvalues_module \ingroup Eigenvalues_Module
 *
 *
 * \class ComplexEigenSolver
 *
 * \brief Computes eigenvalues and eigenvectors of general complex 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.
 *
 * 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 main function in this class is compute(), which computes the
 * eigenvalues and eigenvectors of a given function. The
 * documentation for that function contains an example showing the
 * main features of the class.
 *
 * \sa class EigenSolver, class SelfAdjointEigenSolver
 */
template <typename MatrixType_>
class ComplexEigenSolver {
 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>
      EigenvectorType;

  /** \brief Default constructor.
   *
   * The default constructor is useful in cases in which the user intends to
   * perform decompositions via compute().
   */
  ComplexEigenSolver()
      : m_eivec(), m_eivalues(), m_schur(), m_isInitialized(false), m_eigenvectorsOk(false), m_matX() {}

  /** \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 ComplexEigenSolver()
   */
  explicit ComplexEigenSolver(Index size)
      : m_eivec(size, size),
        m_eivalues(size),
        m_schur(size),
        m_isInitialized(false),
        m_eigenvectorsOk(false),
        m_matX(size, 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 eigendecomposition.
   */
  template <typename InputType>
  explicit ComplexEigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true)
      : m_eivec(matrix.rows(), matrix.cols()),
        m_eivalues(matrix.cols()),
        m_schur(matrix.rows()),
        m_isInitialized(false),
        m_eigenvectorsOk(false),
        m_matX(matrix.rows(), matrix.cols()) {
    compute(matrix.derived(), computeEigenvectors);
  }

  /** \brief Returns the eigenvectors of given matrix.
   *
   * \returns  A const reference to the matrix whose columns are the eigenvectors.
   *
   * \pre Either the constructor
   * ComplexEigenSolver(const MatrixType& matrix, bool) or the member
   * function compute(const MatrixType& matrix, bool) has been called before
   * to compute the eigendecomposition of a matrix, and
   * \p computeEigenvectors was set to true (the default).
   *
   * This function returns a matrix whose columns are the eigenvectors. Column
   * \f$ k \f$ 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 ComplexEigenSolver_eigenvectors.cpp
   * Output: \verbinclude ComplexEigenSolver_eigenvectors.out
   */
  const EigenvectorType& eigenvectors() const {
    eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
    eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
    return m_eivec;
  }

  /** \brief Returns the eigenvalues of given matrix.
   *
   * \returns A const reference to the column vector containing the eigenvalues.
   *
   * \pre Either the constructor
   * ComplexEigenSolver(const MatrixType& matrix, bool) or the member
   * function compute(const MatrixType& matrix, bool) has been called before
   * to compute the eigendecomposition of a matrix.
   *
   * This function returns a column vector containing the
   * eigenvalues. 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 ComplexEigenSolver_eigenvalues.cpp
   * Output: \verbinclude ComplexEigenSolver_eigenvalues.out
   */
  const EigenvalueType& eigenvalues() const {
    eigen_assert(m_isInitialized && "ComplexEigenSolver 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 complex 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 Schur form using the
   * ComplexSchur 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 \f$ O(n^3) \f$ where \f$ n \f$
   * is the size of the matrix.
   *
   * Example: \include ComplexEigenSolver_compute.cpp
   * Output: \verbinclude ComplexEigenSolver_compute.out
   */
  template <typename InputType>
  ComplexEigenSolver& compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true);

  /** \brief Reports whether previous computation was successful.
   *
   * \returns \c Success if computation was successful, \c NoConvergence otherwise.
   */
  ComputationInfo info() const {
    eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
    return m_schur.info();
  }

  /** \brief Sets the maximum number of iterations allowed. */
  ComplexEigenSolver& setMaxIterations(Index maxIters) {
    m_schur.setMaxIterations(maxIters);
    return *this;
  }

  /** \brief Returns the maximum number of iterations. */
  Index getMaxIterations() { return m_schur.getMaxIterations(); }

 protected:
  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)

  EigenvectorType m_eivec;
  EigenvalueType m_eivalues;
  ComplexSchur<MatrixType> m_schur;
  bool m_isInitialized;
  bool m_eigenvectorsOk;
  EigenvectorType m_matX;

 private:
  void doComputeEigenvectors(RealScalar matrixnorm);
  void sortEigenvalues(bool computeEigenvectors);
};

template <typename MatrixType>
template <typename InputType>
ComplexEigenSolver<MatrixType>& ComplexEigenSolver<MatrixType>::compute(const EigenBase<InputType>& matrix,
                                                                        bool computeEigenvectors) {
  // this code is inspired from Jampack
  eigen_assert(matrix.cols() == matrix.rows());

  // Do a complex Schur decomposition, A = U T U^*
  // The eigenvalues are on the diagonal of T.
  m_schur.compute(matrix.derived(), computeEigenvectors);

  if (m_schur.info() == Success) {
    m_eivalues = m_schur.matrixT().diagonal();
    if (computeEigenvectors) doComputeEigenvectors(m_schur.matrixT().norm());
    sortEigenvalues(computeEigenvectors);
  }

  m_isInitialized = true;
  m_eigenvectorsOk = computeEigenvectors;
  return *this;
}

template <typename MatrixType>
void ComplexEigenSolver<MatrixType>::doComputeEigenvectors(RealScalar matrixnorm) {
  const Index n = m_eivalues.size();

  matrixnorm = numext::maxi(matrixnorm, (std::numeric_limits<RealScalar>::min)());

  // Compute X such that T = X D X^(-1), where D is the diagonal of T.
  // The matrix X is unit triangular.
  m_matX = EigenvectorType::Zero(n, n);
  for (Index k = n - 1; k >= 0; k--) {
    m_matX.coeffRef(k, k) = ComplexScalar(1.0, 0.0);
    // Compute X(i,k) using the (i,k) entry of the equation X T = D X
    for (Index i = k - 1; i >= 0; i--) {
      m_matX.coeffRef(i, k) = -m_schur.matrixT().coeff(i, k);
      if (k - i - 1 > 0)
        m_matX.coeffRef(i, k) -=
            (m_schur.matrixT().row(i).segment(i + 1, k - i - 1) * m_matX.col(k).segment(i + 1, k - i - 1)).value();
      ComplexScalar z = m_schur.matrixT().coeff(i, i) - m_schur.matrixT().coeff(k, k);
      if (z == ComplexScalar(0)) {
        // If the i-th and k-th eigenvalue are equal, then z equals 0.
        // Use a small value instead, to prevent division by zero.
        numext::real_ref(z) = NumTraits<RealScalar>::epsilon() * matrixnorm;
      }
      m_matX.coeffRef(i, k) = m_matX.coeff(i, k) / z;
    }
  }

  // Compute V as V = U X; now A = U T U^* = U X D X^(-1) U^* = V D V^(-1)
  m_eivec.noalias() = m_schur.matrixU() * m_matX;
  // .. and normalize the eigenvectors
  for (Index k = 0; k < n; k++) {
    m_eivec.col(k).stableNormalize();
  }
}

template <typename MatrixType>
void ComplexEigenSolver<MatrixType>::sortEigenvalues(bool computeEigenvectors) {
  const Index n = m_eivalues.size();
  for (Index i = 0; i < n; i++) {
    Index k;
    m_eivalues.cwiseAbs().tail(n - i).minCoeff(&k);
    if (k != 0) {
      k += i;
      std::swap(m_eivalues[k], m_eivalues[i]);
      if (computeEigenvectors) m_eivec.col(i).swap(m_eivec.col(k));
    }
  }
}

}  // end namespace Eigen

#endif  // EIGEN_COMPLEX_EIGEN_SOLVER_H