ComplexEigenSolver.h

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// 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 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.

#ifndef EIGEN_COMPLEX_EIGEN_SOLVER_H
#define EIGEN_COMPLEX_EIGEN_SOLVER_H

#include "./EigenvaluesCommon.h"
#include "./ComplexSchur.h"

template<typename _MatrixType> class ComplexEigenSolver
{
  public:

    typedef _MatrixType MatrixType;

    enum {
      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      Options = MatrixType::Options,
      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef typename MatrixType::Index Index;

    typedef std::complex<RealScalar> ComplexScalar;

    typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options&(~RowMajor), MaxColsAtCompileTime, 1> EigenvalueType;

    typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorType;

    ComplexEigenSolver()
            : m_eivec(),
              m_eivalues(),
              m_schur(),
              m_isInitialized(false),
              m_eigenvectorsOk(false),
              m_matX()
    {}

    ComplexEigenSolver(Index size)
            : m_eivec(size, size),
              m_eivalues(size),
              m_schur(size),
              m_isInitialized(false),
              m_eigenvectorsOk(false),
              m_matX(size, size)
    {}

      ComplexEigenSolver(const MatrixType& 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, computeEigenvectors);
    }

    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;
    }

    const EigenvalueType& eigenvalues() const
    {
      eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
      return m_eivalues;
    }

    ComplexEigenSolver& compute(const MatrixType& matrix, bool computeEigenvectors = true);

    ComputationInfo info() const
    {
      eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
      return m_schur.info();
    }

  protected:
    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>
ComplexEigenSolver<MatrixType>& ComplexEigenSolver<MatrixType>::compute(const MatrixType& matrix, bool computeEigenvectors)
{
  // this code is inspired from Jampack
  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, computeEigenvectors);

  if(m_schur.info() == Success)
  {
    m_eivalues = m_schur.matrixT().diagonal();
    if(computeEigenvectors)
      doComputeEigenvectors(matrix.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();

  // 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.
        internal::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).normalize();
  }
}


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));
    }
  }
}


#endif // EIGEN_COMPLEX_EIGEN_SOLVER_H