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

#include <src/eigen/Eigen/src/Eigenvalues/ComplexSchur.h>

Inheritance diagram for Eigen::ComplexSchur< _MatrixType >:

Collaboration diagram for Eigen::ComplexSchur< _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
	Scalar type for matrices of type `_MatrixType`.

typedef NumTraits< Scalar >::Real	RealScalar

typedef Eigen::Index	Index

typedef std::complex< RealScalar >	ComplexScalar
	Complex scalar type for `_MatrixType`.

typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime >	ComplexMatrixType
	Type for the matrices in the Schur decomposition.

Public Member Functions
	ComplexSchur (Index size=RowsAtCompileTime==Dynamic ? 1 :RowsAtCompileTime)
	Default constructor.

template<typename InputType >
	ComplexSchur (const EigenBase< InputType > &matrix, bool computeU=true)
	Constructor; computes Schur decomposition of given matrix.

const ComplexMatrixType &	matrixU () const
	Returns the unitary matrix in the Schur decomposition.

const ComplexMatrixType &	matrixT () const
	Returns the triangular matrix in the Schur decomposition.

template<typename InputType >
ComplexSchur &	compute (const EigenBase< InputType > &matrix, bool computeU=true)
	Computes Schur decomposition of given matrix.

template<typename HessMatrixType , typename OrthMatrixType >
ComplexSchur &	computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU=true)
	Compute Schur decomposition from a given Hessenberg matrix.

ComputationInfo	info () const
	Reports whether previous computation was successful.

ComplexSchur &	setMaxIterations (Index maxIters)
	Sets the maximum number of iterations allowed.

Index	getMaxIterations ()
	Returns the maximum number of iterations.

template<typename InputType >
ComplexSchur< MatrixType > &	compute (const EigenBase< InputType > &matrix, bool computeU)

template<typename HessMatrixType , typename OrthMatrixType >
ComplexSchur< MatrixType > &	computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)

Static Public Attributes
static const int	m_maxIterationsPerRow = 30
	Maximum number of iterations per row.

Protected Attributes
ComplexMatrixType	m_matT

ComplexMatrixType	m_matU

HessenbergDecomposition< MatrixType >	m_hess

ComputationInfo	m_info

bool	m_isInitialized

bool	m_matUisUptodate

Index	m_maxIters

Private Member Functions
bool	subdiagonalEntryIsNeglegible (Index i)

ComplexScalar	computeShift (Index iu, Index iter)

void	reduceToTriangularForm (bool computeU)

Friends
struct	internal::complex_schur_reduce_to_hessenberg< MatrixType, NumTraits< Scalar >::IsComplex >

Detailed Description

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

Performs a complex Schur decomposition of a real or complex square matrix.

\eigenvalues_module

Template Parameters

_MatrixType the type of the matrix of which we are computing the Schur decomposition; this is expected to be an instantiation of the Matrix class template.

Given a real or complex square matrix A, this class computes the Schur decomposition: $ A = U T U^*$ where U is a unitary complex matrix, and T is a complex upper triangular matrix. The diagonal of the matrix T corresponds to the eigenvalues of the matrix A.

Call the function compute() to compute the Schur decomposition of a given matrix. Alternatively, you can use the ComplexSchur(const MatrixType&, bool) constructor which computes the Schur decomposition at construction time. Once the decomposition is computed, you can use the matrixU() and matrixT() functions to retrieve the matrices U and V in the decomposition.

Note: This code is inspired from Jampack

See also: class RealSchur, class EigenSolver, class ComplexEigenSolver

Member Typedef Documentation

◆ ComplexMatrixType

template<typename _MatrixType >

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::ComplexSchur< _MatrixType >::ComplexMatrixType

Type for the matrices in the Schur decomposition.

This is a square matrix with entries of type ComplexScalar. The size is the same as the size of _MatrixType.

◆ ComplexScalar

template<typename _MatrixType >

typedef std::complex<RealScalar> Eigen::ComplexSchur< _MatrixType >::ComplexScalar

Complex scalar type for _MatrixType.

This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.

◆ Index

template<typename _MatrixType >

typedef Eigen::Index Eigen::ComplexSchur< _MatrixType >::Index

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::ComplexSchur< _MatrixType >::MatrixType

◆ RealScalar

template<typename _MatrixType >

typedef NumTraits<Scalar>::Real Eigen::ComplexSchur< _MatrixType >::RealScalar

◆ Scalar

template<typename _MatrixType >

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

Scalar type for matrices of type _MatrixType.

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

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

Constructor & Destructor Documentation

◆ ComplexSchur() [1/2]

template<typename _MatrixType >

Eigen::ComplexSchur< _MatrixType >::ComplexSchur ( Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime )

inlineexplicit

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.

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

◆ ComplexSchur() [2/2]

template<typename _MatrixType >

template<typename InputType >

Eigen::ComplexSchur< _MatrixType >::ComplexSchur	(	const EigenBase< InputType > &	matrix,
		bool	computeU = `true`
	)

inlineexplicit

Constructor; 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.

This constructor calls compute() to compute the Schur decomposition.

See also: matrixT() and matrixU() for examples.

      : m_matT(matrix.rows(),matrix.cols()),
        m_matU(matrix.rows(),matrix.cols()),
        m_hess(matrix.rows()),
        m_isInitialized(false),
        m_matUisUptodate(false),
        m_maxIters(-1)
    {
      compute(matrix.derived(), computeU);
    }

References Eigen::ComplexSchur< _MatrixType >::compute(), and Eigen::EigenBase< Derived >::derived().

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Member Function Documentation

◆ compute() [1/2]

template<typename _MatrixType >

template<typename InputType >

ComplexSchur< MatrixType > & Eigen::ComplexSchur< _MatrixType >::compute	(	const EigenBase< InputType > &	matrix,
		bool	computeU
	)

{
  m_matUisUptodate = false;
  eigen_assert(matrix.cols() == matrix.rows());
 
  if(matrix.cols() == 1)
  {
    m_matT = matrix.derived().template cast<ComplexScalar>();
    if(computeU)  m_matU = ComplexMatrixType::Identity(1,1);
    m_info = Success;
    m_isInitialized = true;
    m_matUisUptodate = computeU;
    return *this;
  }
 
  internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>::run(*this, matrix.derived(), computeU);
  computeFromHessenberg(m_matT, m_matU, computeU);
  return *this;
}

References Eigen::EigenBase< Derived >::cols(), Eigen::ComplexSchur< _MatrixType >::computeFromHessenberg(), Eigen::EigenBase< Derived >::derived(), eigen_assert, Eigen::ComplexSchur< _MatrixType >::m_info, Eigen::ComplexSchur< _MatrixType >::m_isInitialized, Eigen::ComplexSchur< _MatrixType >::m_matT, Eigen::ComplexSchur< _MatrixType >::m_matU, Eigen::ComplexSchur< _MatrixType >::m_matUisUptodate, Eigen::EigenBase< Derived >::rows(), and Eigen::Success.

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◆ compute() [2/2]

template<typename _MatrixType >

template<typename InputType >

ComplexSchur & Eigen::ComplexSchur< _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 QR iterations with a single shift. The cost of computing the Schur decomposition depends on the number of iterations; as a rough guide, it may be taken on the number of iterations; as a rough guide, it may be taken to be $25n^3$ complex flops, or $10n^3$ complex flops if computeU is false.

Example:

Output:

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

Referenced by Eigen::ComplexSchur< _MatrixType >::ComplexSchur().

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◆ computeFromHessenberg() [1/2]

template<typename _MatrixType >

template<typename HessMatrixType , typename OrthMatrixType >

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

{
  m_matT = matrixH;
  if(computeU)
    m_matU = matrixQ;
  reduceToTriangularForm(computeU);
  return *this;
}

References Eigen::ComplexSchur< _MatrixType >::m_matT, Eigen::ComplexSchur< _MatrixType >::m_matU, and Eigen::ComplexSchur< _MatrixType >::reduceToTriangularForm().

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◆ computeFromHessenberg() [2/2]

template<typename _MatrixType >

template<typename HessMatrixType , typename OrthMatrixType >

ComplexSchur & Eigen::ComplexSchur< _MatrixType >::computeFromHessenberg	(	const HessMatrixType &	matrixH,
		const OrthMatrixType &	matrixQ,
		bool	computeU = `true`
	)

Compute Schur decomposition from a given Hessenberg matrix.

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)

Referenced by Eigen::ComplexSchur< _MatrixType >::compute().

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◆ computeShift()

template<typename MatrixType >

ComplexSchur< MatrixType >::ComplexScalar Eigen::ComplexSchur< MatrixType >::computeShift	(	Index	iu,
		Index	iter
	)

private

Compute the shift in the current QR iteration.

{
  using std::abs;
  if (iter == 10 || iter == 20) 
  {
    // exceptional shift, taken from http://www.netlib.org/eispack/comqr.f
    return abs(numext::real(m_matT.coeff(iu,iu-1))) + abs(numext::real(m_matT.coeff(iu-1,iu-2)));
  }
 
  // compute the shift as one of the eigenvalues of t, the 2x2
  // diagonal block on the bottom of the active submatrix
  Matrix<ComplexScalar,2,2> t = m_matT.template block<2,2>(iu-1,iu-1);
  RealScalar normt = t.cwiseAbs().sum();
  t /= normt;     // the normalization by sf is to avoid under/overflow
 
  ComplexScalar b = t.coeff(0,1) * t.coeff(1,0);
  ComplexScalar c = t.coeff(0,0) - t.coeff(1,1);
  ComplexScalar disc = sqrt(c*c + RealScalar(4)*b);
  ComplexScalar det = t.coeff(0,0) * t.coeff(1,1) - b;
  ComplexScalar trace = t.coeff(0,0) + t.coeff(1,1);
  ComplexScalar eival1 = (trace + disc) / RealScalar(2);
  ComplexScalar eival2 = (trace - disc) / RealScalar(2);
 
  if(numext::norm1(eival1) > numext::norm1(eival2))
    eival2 = det / eival1;
  else
    eival1 = det / eival2;
 
  // choose the eigenvalue closest to the bottom entry of the diagonal
  if(numext::norm1(eival1-t.coeff(1,1)) < numext::norm1(eival2-t.coeff(1,1)))
    return normt * eival1;
  else
    return normt * eival2;
}

References Eigen::PlainObjectBase< Derived >::coeff(), Eigen::ComplexSchur< _MatrixType >::m_matT, and sqrt().

Referenced by Eigen::ComplexSchur< _MatrixType >::reduceToTriangularForm().

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◆ getMaxIterations()

template<typename _MatrixType >

Index Eigen::ComplexSchur< _MatrixType >::getMaxIterations ( )

inline

Returns the maximum number of iterations.

    {
      return m_maxIters;
    }

References Eigen::ComplexSchur< _MatrixType >::m_maxIters.

Referenced by Eigen::ComplexEigenSolver< _MatrixType >::getMaxIterations().

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◆ info()

template<typename _MatrixType >

ComputationInfo Eigen::ComplexSchur< _MatrixType >::info ( ) const

inline

Reports whether previous computation was successful.

Returns: Success if computation was succesful, NoConvergence otherwise.

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

References eigen_assert, Eigen::ComplexSchur< _MatrixType >::m_info, and Eigen::ComplexSchur< _MatrixType >::m_isInitialized.

Referenced by Eigen::ComplexEigenSolver< _MatrixType >::info().

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◆ matrixT()

template<typename _MatrixType >

const ComplexMatrixType & Eigen::ComplexSchur< _MatrixType >::matrixT ( ) const

inline

Returns the triangular matrix in the Schur decomposition.

Returns: A const reference to the matrix T.

It is assumed that either the constructor ComplexSchur(const MatrixType& matrix, bool computeU) or the member function compute(const MatrixType& matrix, bool computeU) has been called before to compute the Schur decomposition of a matrix.

Note that this function returns a plain square matrix. If you want to reference only the upper triangular part, use:

schur.matrixT().triangularView<Upper>()

Eigen::Upper

@ Upper

Definition Constants.h:206

Example:

Output:

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

References eigen_assert, Eigen::ComplexSchur< _MatrixType >::m_isInitialized, and Eigen::ComplexSchur< _MatrixType >::m_matT.

◆ matrixU()

template<typename _MatrixType >

const ComplexMatrixType & Eigen::ComplexSchur< _MatrixType >::matrixU ( ) const

inline

Returns the unitary matrix in the Schur decomposition.

Returns: A const reference to the matrix U.

It is assumed that either the constructor ComplexSchur(const MatrixType& matrix, bool computeU) or the member function compute(const MatrixType& matrix, bool computeU) has been called before to compute the Schur decomposition of a matrix, and that computeU was set to true (the default value).

Example:

Output:

    {
      eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
      eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the ComplexSchur decomposition.");
      return m_matU;
    }

References eigen_assert, Eigen::ComplexSchur< _MatrixType >::m_isInitialized, Eigen::ComplexSchur< _MatrixType >::m_matU, and Eigen::ComplexSchur< _MatrixType >::m_matUisUptodate.

◆ reduceToTriangularForm()

template<typename MatrixType >

void Eigen::ComplexSchur< MatrixType >::reduceToTriangularForm ( bool computeU )

private

{  
  Index maxIters = m_maxIters;
  if (maxIters == -1)
    maxIters = m_maxIterationsPerRow * m_matT.rows();
 
  // 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 submatrix).
  // Rows iu+1,...,end are already brought in triangular form.
  Index iu = m_matT.cols() - 1;
  Index il;
  Index iter = 0; // number of iterations we are working on the (iu,iu) element
  Index totalIter = 0; // number of iterations for whole matrix
 
  while(true)
  {
    // find iu, the bottom row of the active submatrix
    while(iu > 0)
    {
      if(!subdiagonalEntryIsNeglegible(iu-1)) break;
      iter = 0;
      --iu;
    }
 
    // if iu is zero then we are done; the whole matrix is triangularized
    if(iu==0) break;
 
    // if we spent too many iterations, we give up
    iter++;
    totalIter++;
    if(totalIter > maxIters) break;
 
    // find il, the top row of the active submatrix
    il = iu-1;
    while(il > 0 && !subdiagonalEntryIsNeglegible(il-1))
    {
      --il;
    }
 
    /* perform the QR step using Givens rotations. The first rotation
       creates a bulge; the (il+2,il) element becomes nonzero. This
       bulge is chased down to the bottom of the active submatrix. */
 
    ComplexScalar shift = computeShift(iu, iter);
    JacobiRotation<ComplexScalar> rot;
    rot.makeGivens(m_matT.coeff(il,il) - shift, m_matT.coeff(il+1,il));
    m_matT.rightCols(m_matT.cols()-il).applyOnTheLeft(il, il+1, rot.adjoint());
    m_matT.topRows((std::min)(il+2,iu)+1).applyOnTheRight(il, il+1, rot);
    if(computeU) m_matU.applyOnTheRight(il, il+1, rot);
 
    for(Index i=il+1 ; i<iu ; i++)
    {
      rot.makeGivens(m_matT.coeffRef(i,i-1), m_matT.coeffRef(i+1,i-1), &m_matT.coeffRef(i,i-1));
      m_matT.coeffRef(i+1,i-1) = ComplexScalar(0);
      m_matT.rightCols(m_matT.cols()-i).applyOnTheLeft(i, i+1, rot.adjoint());
      m_matT.topRows((std::min)(i+2,iu)+1).applyOnTheRight(i, i+1, rot);
      if(computeU) m_matU.applyOnTheRight(i, i+1, rot);
    }
  }
 
  if(totalIter <= maxIters)
    m_info = Success;
  else
    m_info = NoConvergence;
 
  m_isInitialized = true;
  m_matUisUptodate = computeU;
}

Referenced by Eigen::ComplexSchur< _MatrixType >::computeFromHessenberg().

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◆ setMaxIterations()

template<typename _MatrixType >

ComplexSchur & Eigen::ComplexSchur< _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.

    {
      m_maxIters = maxIters;
      return *this;
    }

References Eigen::ComplexSchur< _MatrixType >::m_maxIters.

Referenced by Eigen::ComplexEigenSolver< _MatrixType >::setMaxIterations().

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◆ subdiagonalEntryIsNeglegible()

template<typename MatrixType >

bool Eigen::ComplexSchur< MatrixType >::subdiagonalEntryIsNeglegible ( Index i )

inlineprivate

If m_matT(i+1,i) is neglegible in floating point arithmetic compared to m_matT(i,i) and m_matT(j,j), then set it to zero and return true, else return false.

{
  RealScalar d = numext::norm1(m_matT.coeff(i,i)) + numext::norm1(m_matT.coeff(i+1,i+1));
  RealScalar sd = numext::norm1(m_matT.coeff(i+1,i));
  if (internal::isMuchSmallerThan(sd, d, NumTraits<RealScalar>::epsilon()))
  {
    m_matT.coeffRef(i+1,i) = ComplexScalar(0);
    return true;
  }
  return false;
}