Performs a real Schur decomposition of a square matrix. More...

#include <RealSchur.h>

Inheritance diagram for Eigen::RealSchur< _MatrixType >:

Collaboration diagram for Eigen::RealSchur< _MatrixType >:

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
typedef _MatrixType	MatrixType
typedef MatrixType::Scalar	Scalar
typedef std::complex< typename NumTraits< Scalar >::Real >	ComplexScalar
typedef Eigen::Index	Index
typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	EigenvalueType
typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	ColumnVectorType
Public Member Functions
	RealSchur (Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)
	Default constructor.
template<typename InputType >
	RealSchur (const EigenBase< InputType > &matrix, bool computeU=true)
	Constructor; computes real Schur decomposition of given matrix.
const MatrixType &	matrixU () const
	Returns the orthogonal matrix in the Schur decomposition.
const MatrixType &	matrixT () const
	Returns the quasi-triangular matrix in the Schur decomposition.
template<typename InputType >
RealSchur &	compute (const EigenBase< InputType > &matrix, bool computeU=true)
	Computes Schur decomposition of given matrix.
template<typename HessMatrixType , typename OrthMatrixType >
RealSchur &	computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)
	Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T.
ComputationInfo	info () const
	Reports whether previous computation was successful.
RealSchur &	setMaxIterations (Index maxIters)
	Sets the maximum number of iterations allowed.
Index	getMaxIterations ()
	Returns the maximum number of iterations.
Static Public Attributes
static const int	m_maxIterationsPerRow = 40
	Maximum number of iterations per row.
Private Types
typedef Matrix< Scalar, 3, 1 >	Vector3s
Private Member Functions
Scalar	computeNormOfT ()
Index	findSmallSubdiagEntry (Index iu)
void	splitOffTwoRows (Index iu, bool computeU, const 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)
Private Attributes
MatrixType	m_matT
MatrixType	m_matU
ColumnVectorType	m_workspaceVector
HessenbergDecomposition < MatrixType >	m_hess
ComputationInfo	m_info
bool	m_isInitialized
bool	m_matUisUptodate
Index	m_maxIters

Detailed Description

template<typename _MatrixType>
class Eigen::RealSchur< _MatrixType >

Performs a real Schur decomposition of a square matrix.

Template Parameters:

_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: $ A = U T U^T $ 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, $U^{-1} = U^T$ . 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 JAMA (public domain). Their code is based on EISPACK.

See also:: class ComplexSchur, class EigenSolver, class ComplexEigenSolver

Definition at line 54 of file RealSchur.h.

Member Typedef Documentation

template<typename _MatrixType>

typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::RealSchur< _MatrixType >::ColumnVectorType

Definition at line 70 of file RealSchur.h.

template<typename _MatrixType>

typedef std::complex<typename NumTraits<Scalar>::Real> Eigen::RealSchur< _MatrixType >::ComplexScalar

Definition at line 66 of file RealSchur.h.

template<typename _MatrixType>

typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::RealSchur< _MatrixType >::EigenvalueType

Definition at line 69 of file RealSchur.h.

template<typename _MatrixType>

typedef Eigen::Index Eigen::RealSchur< _MatrixType >::Index

Deprecated:: since Eigen 3.3

Definition at line 67 of file RealSchur.h.

template<typename _MatrixType>

typedef _MatrixType Eigen::RealSchur< _MatrixType >::MatrixType

Definition at line 57 of file RealSchur.h.

template<typename _MatrixType>

typedef MatrixType::Scalar Eigen::RealSchur< _MatrixType >::Scalar

Definition at line 65 of file RealSchur.h.

template<typename _MatrixType>

typedef Matrix<Scalar,3,1> Eigen::RealSchur< _MatrixType >::Vector3s [private]

Definition at line 236 of file RealSchur.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 58 of file RealSchur.h.

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

Constructor & Destructor Documentation

template<typename _MatrixType>

Eigen::RealSchur< _MatrixType >::RealSchur ( Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime ) [inline, explicit]

Default constructor.

Parameters:

[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 size parameter is only used as a hint. It is not an error to give a wrong size, but it may impair performance.

See also:: compute() for an example.

Definition at line 83 of file RealSchur.h.

                                                                   : RowsAtCompileTime)
            : m_matT(size, size),
              m_matU(size, size),
              m_workspaceVector(size),
              m_hess(size),
              m_isInitialized(false),
              m_matUisUptodate(false),
              m_maxIters(-1)
    { }

template<typename _MatrixType>

template<typename InputType >

Eigen::RealSchur< _MatrixType >::RealSchur	(	const EigenBase< InputType > &	matrix,
		bool	computeU = `true`
	)		`[inline, explicit]`

Constructor; computes real Schur decomposition of given matrix.

Parameters:

[in]	matrix	Square matrix whose Schur decomposition is to be computed.
[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:

Output:

Definition at line 104 of file RealSchur.h.

            : 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),
              m_maxIters(-1)
    {
      compute(matrix.derived(), computeU);
    }

Member Function Documentation

template<typename MatrixType >

template<typename InputType >

RealSchur< MatrixType > & Eigen::RealSchur< MatrixType >::compute	(	const EigenBase< InputType > &	matrix,
		bool	computeU = `true`
	)

Computes Schur decomposition of given matrix.

Parameters:

[in]	matrix	Square matrix whose Schur decomposition is to be computed.
[in]	computeU	If true, both T and U are computed; if false, only T is computed.

Returns:: Reference to *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 $25n^3$ flops if computeU is true and $10n^3$ flops if computeU is false.

Example:

Output:

See also:: compute(const MatrixType&, bool, Index)

Definition at line 249 of file RealSchur.h.

{
  eigen_assert(matrix.cols() == matrix.rows());
  Index maxIters = m_maxIters;
  if (maxIters == -1)
    maxIters = m_maxIterationsPerRow * matrix.rows();

  // Step 1. Reduce to Hessenberg form
  m_hess.compute(matrix.derived());

  // Step 2. Reduce to real Schur form  
  computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
  
  return *this;
}

template<typename MatrixType >

template<typename HessMatrixType , typename OrthMatrixType >

RealSchur< MatrixType > & Eigen::RealSchur< MatrixType >::computeFromHessenberg	(	const HessMatrixType &	matrixH,
		const OrthMatrixType &	matrixQ,
		bool	computeU
	)

Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T.

Parameters:

[in]	matrixH	Matrix in Hessenberg form H
[in]	matrixQ	orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
	computeU	Computes the matriX U of the Schur vectors

Returns:: Reference to *this

This routine assumes that the matrix is already reduced in Hessenberg form matrixH using either the class HessenbergDecomposition or another mean. It computes the upper quasi-triangular matrix T of the Schur decomposition of H When computeU is true, this routine computes the matrix U such that A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix

NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix is not available, the user should give an identity matrix (Q.setIdentity())

See also:: compute(const MatrixType&, bool)

Definition at line 266 of file RealSchur.h.

{  
  m_matT = matrixH; 
  if(computeU)
    m_matU = matrixQ;
  
  Index maxIters = m_maxIters;
  if (maxIters == -1)
    maxIters = m_maxIterationsPerRow * matrixH.rows();
  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);

      // 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 > maxIters) break;
        Index im;
        initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
        performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
      }
    }
  }
  if(totalIter <= maxIters)
    m_info = Success;
  else
    m_info = NoConvergence;

  m_isInitialized = true;
  m_matUisUptodate = computeU;
  return *this;
}

template<typename MatrixType >

MatrixType::Scalar Eigen::RealSchur< MatrixType >::computeNormOfT ( ) [inline, private]

Computes and returns vector L1 norm of T

Definition at line 335 of file RealSchur.h.

{
  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.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
  return norm;
}

template<typename MatrixType >

void Eigen::RealSchur< MatrixType >::computeShift	(	Index	iu,
		Index	iter,
		Scalar &	exshift,
		Vector3s &	shiftInfo
	)		`[inline, private]`

Form shift in shiftInfo, and update exshift if an exceptional shift is performed.

Definition at line 400 of file RealSchur.h.

{
  using std::sqrt;
  using std::abs;
  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 = abs(m_matT.coeff(iu,iu-1)) + 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 = 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));
    }
  }
}

template<typename MatrixType >

Index Eigen::RealSchur< MatrixType >::findSmallSubdiagEntry ( Index iu ) [inline, private]

Look for single small sub-diagonal element and returns its index

Definition at line 349 of file RealSchur.h.

{
  using std::abs;
  Index res = iu;
  while (res > 0)
  {
    Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
    if (abs(m_matT.coeff(res,res-1)) <= NumTraits<Scalar>::epsilon() * s)
      break;
    res--;
  }
  return res;
}

template<typename _MatrixType>

Index Eigen::RealSchur< _MatrixType >::getMaxIterations ( ) [inline]

Returns the maximum number of iterations.

Definition at line 213 of file RealSchur.h.

    {
      return m_maxIters;
    }

template<typename _MatrixType>

ComputationInfo Eigen::RealSchur< _MatrixType >::info ( ) const [inline]

Reports whether previous computation was successful.

Returns:: Success if computation was succesful, NoConvergence otherwise.

Definition at line 195 of file RealSchur.h.

    {
      eigen_assert(m_isInitialized && "RealSchur is not initialized.");
      return m_info;
    }

template<typename MatrixType >

void Eigen::RealSchur< MatrixType >::initFrancisQRStep	(	Index	il,
		Index	iu,
		const Vector3s &	shiftInfo,
		Index &	im,
		Vector3s &	firstHouseholderVector
	)		`[inline, private]`

Compute index im at which Francis QR step starts and the first Householder vector.

Definition at line 442 of file RealSchur.h.

{
  using std::abs;
  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) * (abs(v.coeff(1)) + abs(v.coeff(2)));
    const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
    if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
      break;
  }
}

template<typename _MatrixType>

const MatrixType& Eigen::RealSchur< _MatrixType >::matrixT ( ) const [inline]

Returns the quasi-triangular matrix in the Schur decomposition.

Returns:: A const reference to the matrix T.

Precondition:: 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.

See also:: RealSchur(const MatrixType&, bool) for an example

Definition at line 144 of file RealSchur.h.

    {
      eigen_assert(m_isInitialized && "RealSchur is not initialized.");
      return m_matT;
    }

template<typename _MatrixType>

const MatrixType& Eigen::RealSchur< _MatrixType >::matrixU ( ) const [inline]

Returns the orthogonal matrix in the Schur decomposition.

Returns:: A const reference to the matrix U.

Precondition:: 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 computeU was set to true (the default value).

See also:: RealSchur(const MatrixType&, bool) for an example

Definition at line 127 of file RealSchur.h.

    {
      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;
    }

template<typename MatrixType >

void Eigen::RealSchur< MatrixType >::performFrancisQRStep	(	Index	il,
		Index	im,
		Index	iu,
		bool	computeU,
		const Vector3s &	firstHouseholderVector,
		Scalar *	workspace
	)		`[inline, private]`

Perform a Francis QR step involving rows il:iu and columns im:iu.

Definition at line 467 of file RealSchur.h.

{
  eigen_assert(im >= il);
  eigen_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);
  }
}

template<typename _MatrixType>

RealSchur& Eigen::RealSchur< _MatrixType >::setMaxIterations ( Index maxIters ) [inline]

Sets the maximum number of iterations allowed.

If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size of the matrix.

Definition at line 206 of file RealSchur.h.

    {
      m_maxIters = maxIters;
      return *this;
    }

template<typename MatrixType >

void Eigen::RealSchur< MatrixType >::splitOffTwoRows	(	Index	iu,
		bool	computeU,
		const Scalar &	exshift
	)		`[inline, private]`

Update T given that rows iu-1 and iu decouple from the rest.

Definition at line 365 of file RealSchur.h.

{
  using std::sqrt;
  using std::abs;
  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 = sqrt(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);
}