Computes eigenvalues and eigenvectors of general complex matrices. More...

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

Collaboration diagram for Eigen::ComplexEigenSolver< _MatrixType >:

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

typedef _MatrixType	MatrixType
	Synonym for the template parameter `_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, ColsAtCompileTime, 1, Options &(~RowMajor), MaxColsAtCompileTime, 1 >	EigenvalueType
	Type for vector of eigenvalues as returned by eigenvalues().

typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime >	EigenvectorType
	Type for matrix of eigenvectors as returned by eigenvectors().

Public Member Functions
	ComplexEigenSolver ()
	Default constructor.

	ComplexEigenSolver (Index size)
	Default Constructor with memory preallocation.

template<typename InputType >
	ComplexEigenSolver (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
	Constructor; computes eigendecomposition of given matrix.

const EigenvectorType &	eigenvectors () const
	Returns the eigenvectors of given matrix.

const EigenvalueType &	eigenvalues () const
	Returns the eigenvalues of given matrix.

template<typename InputType >
ComplexEigenSolver &	compute (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
	Computes eigendecomposition of given matrix.

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

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

Index	getMaxIterations ()
	Returns the maximum number of iterations.

template<typename InputType >
ComplexEigenSolver< MatrixType > &	compute (const EigenBase< InputType > &matrix, bool computeEigenvectors)

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
EigenvectorType	m_eivec

EigenvalueType	m_eivalues

ComplexSchur< MatrixType >	m_schur

bool	m_isInitialized

bool	m_eigenvectorsOk

EigenvectorType	m_matX

Private Member Functions
void	doComputeEigenvectors (RealScalar matrixnorm)

void	sortEigenvalues (bool computeEigenvectors)

Detailed Description

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

Computes eigenvalues and eigenvectors of general complex matrices.

\eigenvalues_module

Template Parameters

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

The eigenvalues and eigenvectors of a matrix $ A $ are scalars $\lambda$ and vectors $ v $ such that $Av = \lambda v$ . If $ D $ is a diagonal matrix with the eigenvalues on the diagonal, and $ V $ is a matrix with the eigenvectors as its columns, then $ A V = V D $ . The matrix $ V $ is almost always invertible, in which case we have $A = V D V^{-1}$ . This is called the eigendecomposition.

The main function in this class is compute(), which computes the eigenvalues and eigenvectors of a given function. The documentation for that function contains an example showing the main features of the class.

See also: class EigenSolver, class SelfAdjointEigenSolver

Member Typedef Documentation

◆ ComplexScalar

template<typename _MatrixType >

typedef std::complex<RealScalar> Eigen::ComplexEigenSolver< _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.

◆ EigenvalueType

template<typename _MatrixType >

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

Type for vector of eigenvalues as returned by eigenvalues().

This is a column vector with entries of type ComplexScalar. The length of the vector is the size of MatrixType.

◆ EigenvectorType

template<typename _MatrixType >

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::ComplexEigenSolver< _MatrixType >::EigenvectorType

Type for matrix of eigenvectors as returned by eigenvectors().

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

◆ Index

template<typename _MatrixType >

typedef Eigen::Index Eigen::ComplexEigenSolver< _MatrixType >::Index

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::ComplexEigenSolver< _MatrixType >::MatrixType

Synonym for the template parameter _MatrixType.

◆ RealScalar

template<typename _MatrixType >

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

◆ Scalar

template<typename _MatrixType >

typedef MatrixType::Scalar Eigen::ComplexEigenSolver< _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

◆ ComplexEigenSolver() [1/3]

template<typename _MatrixType >

Eigen::ComplexEigenSolver< _MatrixType >::ComplexEigenSolver ( )

inline

Default constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via compute().

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

◆ ComplexEigenSolver() [2/3]

template<typename _MatrixType >

Eigen::ComplexEigenSolver< _MatrixType >::ComplexEigenSolver ( Index size )

inlineexplicit

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also: ComplexEigenSolver()

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

◆ ComplexEigenSolver() [3/3]

template<typename _MatrixType >

template<typename InputType >

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

inlineexplicit

Constructor; computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the eigendecomposition.

            : 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.derived(), computeEigenvectors);
    }

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

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Member Function Documentation

◆ check_template_parameters()

template<typename _MatrixType >

static void Eigen::ComplexEigenSolver< _MatrixType >::check_template_parameters ( )

inlinestaticprotected

    {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }

References EIGEN_STATIC_ASSERT_NON_INTEGER.

◆ compute() [1/2]

template<typename _MatrixType >

template<typename InputType >

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

{
  check_template_parameters();
  
  // this code is inspired from Jampack
  eigen_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.derived(), computeEigenvectors);
 
  if(m_schur.info() == Success)
  {
    m_eivalues = m_schur.matrixT().diagonal();
    if(computeEigenvectors)
      doComputeEigenvectors(m_schur.matrixT().norm());
    sortEigenvalues(computeEigenvectors);
  }
 
  m_isInitialized = true;
  m_eigenvectorsOk = computeEigenvectors;
  return *this;
}

References Eigen::EigenBase< Derived >::cols(), Eigen::EigenBase< Derived >::derived(), eigen_assert, Eigen::EigenBase< Derived >::rows(), and Eigen::Success.

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◆ compute() [2/2]

template<typename _MatrixType >

template<typename InputType >

ComplexEigenSolver & Eigen::ComplexEigenSolver< _MatrixType >::compute	(	const EigenBase< InputType > &	matrix,
		bool	computeEigenvectors = `true`
	)

Computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

Returns: Reference to *this

This function computes the eigenvalues of the complex matrix matrix. The eigenvalues() function can be used to retrieve them. If computeEigenvectors is true, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().

The matrix is first reduced to Schur form using the ComplexSchur class. The Schur decomposition is then used to compute the eigenvalues and eigenvectors.

The cost of the computation is dominated by the cost of the Schur decomposition, which is $ O(n^3) $ where $ n $ is the size of the matrix.

Example:

Output:

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

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◆ doComputeEigenvectors()

template<typename MatrixType >

void Eigen::ComplexEigenSolver< MatrixType >::doComputeEigenvectors ( RealScalar matrixnorm )

private

{
  const Index n = m_eivalues.size();
 
  matrixnorm = numext::maxi(matrixnorm,(std::numeric_limits<RealScalar>::min)());
 
  // 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.
        numext::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();
  }
}

References Eigen::numext::maxi(), and Eigen::numext::real_ref().

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◆ eigenvalues()

template<typename _MatrixType >

const EigenvalueType & Eigen::ComplexEigenSolver< _MatrixType >::eigenvalues ( ) const

inline

Returns the eigenvalues of given matrix.

Returns: A const reference to the column vector containing the eigenvalues.

Precondition: Either the constructor ComplexEigenSolver(const MatrixType& matrix, bool) or the member function compute(const MatrixType& matrix, bool) has been called before to compute the eigendecomposition of a matrix.

This function returns a column vector containing the eigenvalues. Eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.

Example:

Output:

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

References eigen_assert, Eigen::ComplexEigenSolver< _MatrixType >::m_eivalues, and Eigen::ComplexEigenSolver< _MatrixType >::m_isInitialized.

Referenced by Eigen::internal::eigenvalues_selector< Derived, IsComplex >::run().

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◆ eigenvectors()

template<typename _MatrixType >

const EigenvectorType & Eigen::ComplexEigenSolver< _MatrixType >::eigenvectors ( ) const

inline

Returns the eigenvectors of given matrix.

Returns: A const reference to the matrix whose columns are the eigenvectors.

Precondition: Either the constructor ComplexEigenSolver(const MatrixType& matrix, bool) or the member function compute(const MatrixType& matrix, bool) has been called before to compute the eigendecomposition of a matrix, and computeEigenvectors was set to true (the default).

This function returns a matrix whose columns are the eigenvectors. Column $ k $ is an eigenvector corresponding to eigenvalue number $ k
$ as returned by eigenvalues(). The eigenvectors are normalized to have (Euclidean) norm equal to one. The matrix returned by this function is the matrix $ V $ in the eigendecomposition $A = V D V^{-1}$ , if it exists.

Example:

Output:

    {
      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;
    }

References eigen_assert, Eigen::ComplexEigenSolver< _MatrixType >::m_eigenvectorsOk, Eigen::ComplexEigenSolver< _MatrixType >::m_eivec, and Eigen::ComplexEigenSolver< _MatrixType >::m_isInitialized.

◆ getMaxIterations()

template<typename _MatrixType >

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

inline

Returns the maximum number of iterations.

    {
      return m_schur.getMaxIterations();
    }

References Eigen::ComplexSchur< _MatrixType >::getMaxIterations(), and Eigen::ComplexEigenSolver< _MatrixType >::m_schur.

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◆ info()

template<typename _MatrixType >

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

inline

Reports whether previous computation was successful.

Returns: Success if computation was succesful, NoConvergence otherwise.

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

References eigen_assert, Eigen::ComplexSchur< _MatrixType >::info(), Eigen::ComplexEigenSolver< _MatrixType >::m_isInitialized, and Eigen::ComplexEigenSolver< _MatrixType >::m_schur.

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◆ setMaxIterations()

template<typename _MatrixType >

ComplexEigenSolver & Eigen::ComplexEigenSolver< _MatrixType >::setMaxIterations ( Index maxIters )

inline

Sets the maximum number of iterations allowed.

    {
      m_schur.setMaxIterations(maxIters);
      return *this;
    }

References Eigen::ComplexEigenSolver< _MatrixType >::m_schur, and Eigen::ComplexSchur< _MatrixType >::setMaxIterations().

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◆ sortEigenvalues()

template<typename MatrixType >

void Eigen::ComplexEigenSolver< MatrixType >::sortEigenvalues ( bool computeEigenvectors )

private

{
  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));
    }
  }
}

Member Data Documentation

◆ m_eigenvectorsOk

template<typename _MatrixType >

bool Eigen::ComplexEigenSolver< _MatrixType >::m_eigenvectorsOk

protected

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

◆ m_eivalues

template<typename _MatrixType >

EigenvalueType Eigen::ComplexEigenSolver< _MatrixType >::m_eivalues

protected

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

◆ m_eivec

template<typename _MatrixType >

EigenvectorType Eigen::ComplexEigenSolver< _MatrixType >::m_eivec

protected

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

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::ComplexEigenSolver< _MatrixType >::m_isInitialized

protected

Referenced by Eigen::ComplexEigenSolver< _MatrixType >::eigenvalues(), Eigen::ComplexEigenSolver< _MatrixType >::eigenvectors(), and Eigen::ComplexEigenSolver< _MatrixType >::info().

◆ m_matX

template<typename _MatrixType >

EigenvectorType Eigen::ComplexEigenSolver< _MatrixType >::m_matX

protected

◆ m_schur

template<typename _MatrixType >

ComplexSchur<MatrixType> Eigen::ComplexEigenSolver< _MatrixType >::m_schur

protected

Referenced by Eigen::ComplexEigenSolver< _MatrixType >::getMaxIterations(), Eigen::ComplexEigenSolver< _MatrixType >::info(), and Eigen::ComplexEigenSolver< _MatrixType >::setMaxIterations().

The documentation for this class was generated from the following file:

src/eigen/Eigen/src/Eigenvalues/ComplexEigenSolver.h

Public Types

Public Member Functions

Static Protected Member Functions

Protected Attributes

Private Member Functions

Detailed Description

Member Typedef Documentation

◆ ComplexScalar

◆ EigenvalueType

◆ EigenvectorType

◆ Index

◆ MatrixType

◆ RealScalar

◆ Scalar

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ ComplexEigenSolver() [1/3]

◆ ComplexEigenSolver() [2/3]

◆ ComplexEigenSolver() [3/3]

Member Function Documentation

◆ check_template_parameters()

◆ compute() [1/2]

◆ compute() [2/2]

◆ doComputeEigenvectors()

◆ eigenvalues()

◆ eigenvectors()

◆ getMaxIterations()

◆ info()

◆ setMaxIterations()

◆ sortEigenvalues()

Member Data Documentation

◆ m_eigenvectorsOk

◆ m_eivalues

◆ m_eivec

◆ m_isInitialized

◆ m_matX

◆ m_schur