RealSchur.h

// 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>
//
// 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.
//
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// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
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//
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// Eigen. If not, see <http://www.gnu.org/licenses/>.

#ifndef EIGEN_REAL_SCHUR_H
#define EIGEN_REAL_SCHUR_H

#include "./EigenvaluesCommon.h"
#include "./HessenbergDecomposition.h"

/** \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
  Scalar exshift = 0.0; // sum of exceptional shifts
  Scalar norm = computeNormOfT();

  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; 
      if (iter > m_maxIterations) break;
      Index im;
      initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
      performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
    }
  } 

  if(iter <= m_maxIterations) 
    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 = 0.0;
  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);
  }
}

#endif // EIGEN_REAL_SCHUR_H