text: // 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 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_REAL_SCHUR_H
  #define EIGEN_REAL_SCHUR_H
  
  #include "./HessenbergDecomposition.h"
  
  namespace Eigen { 
  
  /** \eigenvalues_module \ingroup Eigenvalues_Module
    *
    *
    * \class RealSchur
    *
    * \brief Performs a real Schur decomposition of a square matrix
    *
    * \tparam _MatrixType the type of the matrix of which we are computing the
    * real Schur decomposition; this is expected to be an instantiation of the
    * Matrix class template.
    *
    * Given a real square matrix A, this class computes the real Schur
    * decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
    * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
    * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
    * matrix is a block-triangular matrix whose diagonal consists of 1-by-1
    * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
    * blocks on the diagonal of T are the same as the eigenvalues of the matrix
    * A, and thus the real Schur decomposition is used in EigenSolver to compute
    * the eigendecomposition of a matrix.
    *
    * Call the function compute() to compute the real Schur decomposition of a
    * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
    * constructor which computes the real Schur decomposition at construction
    * time. Once the decomposition is computed, you can use the matrixU() and
    * matrixT() functions to retrieve the matrices U and T in the decomposition.
    *
    * The documentation of RealSchur(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 class ComplexSchur, class EigenSolver, class ComplexEigenSolver
    */
  template<typename _MatrixType> class RealSchur
  {
    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 std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
      typedef typename MatrixType::Index Index;
  
      typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
      typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
  
      /** \brief Default constructor.
        *
        * \param [in] size  Positive integer, size of the matrix whose Schur decomposition will be computed.
        *
        * The default constructor is useful in cases in which the user intends to
        * perform decompositions via compute().  The \p size parameter is only
        * used as a hint. It is not an error to give a wrong \p size, but it may
        * impair performance.
        *
        * \sa compute() for an example.
        */
      RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
              : m_matT(size, size),
                m_matU(size, size),
                m_workspaceVector(size),
                m_hess(size),
                m_isInitialized(false),
                m_matUisUptodate(false)
      { }
  
      /** \brief Constructor; computes real Schur decomposition of given matrix. 
        * 
        * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
        * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
        *
        * This constructor calls compute() to compute the Schur decomposition.
        *
        * Example: \include RealSchur_RealSchur_MatrixType.cpp
        * Output: \verbinclude RealSchur_RealSchur_MatrixType.out
        */
      RealSchur(const MatrixType& matrix, bool computeU = true)
              : m_matT(matrix.rows(),matrix.cols()),
                m_matU(matrix.rows(),matrix.cols()),
                m_workspaceVector(matrix.rows()),
                m_hess(matrix.rows()),
                m_isInitialized(false),
                m_matUisUptodate(false)
      {
        compute(matrix, computeU);
      }
  
      /** \brief Returns the orthogonal matrix in the Schur decomposition. 
        *
        * \returns A const reference to the matrix U.
        *
        * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
        * member function compute(const MatrixType&, bool) has been called before
        * to compute the Schur decomposition of a matrix, and \p computeU was set
        * to true (the default value).
        *
        * \sa RealSchur(const MatrixType&, bool) for an example
        */
      const MatrixType& matrixU() const
      {
        eigen_assert(m_isInitialized && "RealSchur is not initialized.");
        eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
        return m_matU;
      }
  
      /** \brief Returns the quasi-triangular matrix in the Schur decomposition. 
        *
        * \returns A const reference to the matrix T.
        *
        * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
        * member function compute(const MatrixType&, bool) has been called before
        * to compute the Schur decomposition of a matrix.
        *
        * \sa RealSchur(const MatrixType&, bool) for an example
        */
      const MatrixType& matrixT() const
      {
        eigen_assert(m_isInitialized && "RealSchur is not initialized.");
        return m_matT;
      }
    
      /** \brief Computes Schur decomposition of given matrix. 
        * 
        * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
        * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
        * \returns    Reference to \c *this
        *
        * The Schur decomposition is computed by first reducing the matrix to
        * Hessenberg form using the class HessenbergDecomposition. The Hessenberg
        * matrix is then reduced to triangular form by performing Francis QR
        * iterations with implicit double shift. The cost of computing the Schur
        * decomposition depends on the number of iterations; as a rough guide, it
        * may be taken to be \f$25n^3\f$ flops if \a computeU is true and
        * \f$10n^3\f$ flops if \a computeU is false.
        *
        * Example: \include RealSchur_compute.cpp
        * Output: \verbinclude RealSchur_compute.out
        */
      RealSchur& compute(const MatrixType& matrix, bool computeU = true);
  
      /** \brief Reports whether previous computation was successful.
        *
        * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
        */
      ComputationInfo info() const
      {
        eigen_assert(m_isInitialized && "RealSchur is not initialized.");
        return m_info;
      }
  
      /** \brief Maximum number of iterations.
        *
        * Maximum number of iterations allowed for an eigenvalue to converge. 
        */
      static const int m_maxIterations = 40;
  
    private:
      
      MatrixType m_matT;
      MatrixType m_matU;
      ColumnVectorType m_workspaceVector;
      HessenbergDecomposition<MatrixType> m_hess;
      ComputationInfo m_info;
      bool m_isInitialized;
      bool m_matUisUptodate;
  
      typedef Matrix<Scalar,3,1> Vector3s;
  
      Scalar computeNormOfT();
      Index findSmallSubdiagEntry(Index iu, Scalar norm);
      void splitOffTwoRows(Index iu, bool computeU, Scalar exshift);
      void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
      void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
      void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
  };
  
  
  template<typename MatrixType>
  RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
  {
    assert(matrix.cols() == matrix.rows());
  
    // Step 1. Reduce to Hessenberg form
    m_hess.compute(matrix);
    m_matT = m_hess.matrixH();
    if (computeU)
      m_matU = m_hess.matrixQ();
  
    // Step 2. Reduce to real Schur form  
    m_workspaceVector.resize(m_matT.cols());
    Scalar* workspace = &m_workspaceVector.coeffRef(0);
  
    // The matrix m_matT is divided in three parts. 
    // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero. 
    // Rows il,...,iu is the part we are working on (the active window).
    // Rows iu+1,...,end are already brought in triangular form.
    Index iu = m_matT.cols() - 1;
    Index iter = 0;      // iteration count for current eigenvalue
    Index totalIter = 0; // iteration count for whole matrix
    Scalar exshift(0);   // sum of exceptional shifts
    Scalar norm = computeNormOfT();
  
    if(norm!=0)
    {
      while (iu >= 0)
      {
        Index il = findSmallSubdiagEntry(iu, norm);
  
        // Check for convergence
        if (il == iu) // One root found
        {
          m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
          if (iu > 0)
            m_matT.coeffRef(iu, iu-1) = Scalar(0);
          iu--;
          iter = 0;
        }
        else if (il == iu-1) // Two roots found
        {
          splitOffTwoRows(iu, computeU, exshift);
          iu -= 2;
          iter = 0;
        }
        else // No convergence yet
        {
          // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
          Vector3s firstHouseholderVector(0,0,0), shiftInfo;
          computeShift(iu, iter, exshift, shiftInfo);
          iter = iter + 1;
          totalIter = totalIter + 1;
          if (totalIter > m_maxIterations * matrix.cols()) break;
          Index im;
          initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
          performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
        }
      }
    }
    if(totalIter <= m_maxIterations * matrix.cols()) 
      m_info = Success;
    else
      m_info = NoConvergence;
  
    m_isInitialized = true;
    m_matUisUptodate = computeU;
    return *this;
  }
  
  /** \internal Computes and returns vector L1 norm of T */
  template<typename MatrixType>
  inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
  {
    const Index size = m_matT.cols();
    // FIXME to be efficient the following would requires a triangular reduxion code
    // Scalar norm = m_matT.upper().cwiseAbs().sum() 
    //               + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
    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();
    return norm;
  }
  
  /** \internal Look for single small sub-diagonal element and returns its index */
  template<typename MatrixType>
  inline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, Scalar norm)
  {
    Index res = iu;
    while (res > 0)
    {
      Scalar s = internal::abs(m_matT.coeff(res-1,res-1)) + internal::abs(m_matT.coeff(res,res));
      if (s == 0.0)
        s = norm;
      if (internal::abs(m_matT.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)
        break;
      res--;
    }
    return res;
  }
  
  /** \internal Update T given that rows iu-1 and iu decouple from the rest. */
  template<typename MatrixType>
  inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, Scalar exshift)
  {
    const Index size = m_matT.cols();
  
    // The eigenvalues of the 2x2 matrix [a b; c d] are 
    // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
    Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
    Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);   // q = tr^2 / 4 - det = discr/4
    m_matT.coeffRef(iu,iu) += exshift;
    m_matT.coeffRef(iu-1,iu-1) += exshift;
  
    if (q >= Scalar(0)) // Two real eigenvalues
    {
      Scalar z = internal::sqrt(internal::abs(q));
      JacobiRotation<Scalar> rot;
      if (p >= Scalar(0))
        rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
      else
        rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
  
      m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
      m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
      m_matT.coeffRef(iu, iu-1) = Scalar(0); 
      if (computeU)
        m_matU.applyOnTheRight(iu-1, iu, rot);
    }
  
    if (iu > 1) 
      m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
  }
  
  /** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
  template<typename MatrixType>
  inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
  {
    shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
    shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
    shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
  
    // Wilkinson's original ad hoc shift
    if (iter == 10)
    {
      exshift += shiftInfo.coeff(0);
      for (Index i = 0; i <= iu; ++i)
        m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
      Scalar s = internal::abs(m_matT.coeff(iu,iu-1)) + internal::abs(m_matT.coeff(iu-1,iu-2));
      shiftInfo.coeffRef(0) = Scalar(0.75) * s;
      shiftInfo.coeffRef(1) = Scalar(0.75) * s;
      shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
    }
  
    // MATLAB's new ad hoc shift
    if (iter == 30)
    {
      Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
      s = s * s + shiftInfo.coeff(2);
      if (s > Scalar(0))
      {
        s = internal::sqrt(s);
        if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
          s = -s;
        s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
        s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
        exshift += s;
        for (Index i = 0; i <= iu; ++i)
          m_matT.coeffRef(i,i) -= s;
        shiftInfo.setConstant(Scalar(0.964));
      }
    }
  }
  
  /** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
  template<typename MatrixType>
  inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
  {
    Vector3s& v = firstHouseholderVector; // alias to save typing
  
    for (im = iu-2; im >= il; --im)
    {
      const Scalar Tmm = m_matT.coeff(im,im);
      const Scalar r = shiftInfo.coeff(0) - Tmm;
      const Scalar s = shiftInfo.coeff(1) - Tmm;
      v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
      v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
      v.coeffRef(2) = m_matT.coeff(im+2,im+1);
      if (im == il) {
        break;
      }
      const Scalar lhs = m_matT.coeff(im,im-1) * (internal::abs(v.coeff(1)) + internal::abs(v.coeff(2)));
      const Scalar rhs = v.coeff(0) * (internal::abs(m_matT.coeff(im-1,im-1)) + internal::abs(Tmm) + internal::abs(m_matT.coeff(im+1,im+1)));
      if (internal::abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
      {
        break;
      }
    }
  }
  
  /** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
  template<typename MatrixType>
  inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
  {
    assert(im >= il);
    assert(im <= iu-2);
  
    const Index size = m_matT.cols();
  
    for (Index k = im; k <= iu-2; ++k)
    {
      bool firstIteration = (k == im);
  
      Vector3s v;
      if (firstIteration)
        v = firstHouseholderVector;
      else
        v = m_matT.template block<3,1>(k,k-1);
  
      Scalar tau, beta;
      Matrix<Scalar, 2, 1> ess;
      v.makeHouseholder(ess, tau, beta);
      
      if (beta != Scalar(0)) // if v is not zero
      {
        if (firstIteration && k > il)
          m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
        else if (!firstIteration)
          m_matT.coeffRef(k,k-1) = beta;
  
        // These Householder transformations form the O(n^3) part of the algorithm
        m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
        m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
        if (computeU)
          m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
      }
    }
  
    Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
    Scalar tau, beta;
    Matrix<Scalar, 1, 1> ess;
    v.makeHouseholder(ess, tau, beta);
  
    if (beta != Scalar(0)) // if v is not zero
    {
      m_matT.coeffRef(iu-1, iu-2) = beta;
      m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
      m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
      if (computeU)
        m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
    }
  
    // clean up pollution due to round-off errors
    for (Index i = im+2; i <= iu; ++i)
    {
      m_matT.coeffRef(i,i-2) = Scalar(0);
      if (i > im+2)
        m_matT.coeffRef(i,i-3) = Scalar(0);
    }
  }
  
  } // end namespace Eigen
  
  #endif // EIGEN_REAL_SCHUR_H
  