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

Public Types
enum	{ RowsAtCompileTime , ColsAtCompileTime , Options , MaxRowsAtCompileTime , MaxColsAtCompileTime }

typedef std::complex< RealScalar >	ComplexScalar
	Complex scalar type for MatrixType. More...

typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &(~RowMajor), MaxColsAtCompileTime, 1 >	EigenvalueType
	Type for vector of eigenvalues as returned by eigenvalues(). More...

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

typedef Eigen::Index	Index

typedef MatrixType_	MatrixType
	Synonym for the template parameter `MatrixType_`. More...

typedef NumTraits< Scalar >::Real	RealScalar

typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type MatrixType. More...

Public Member Functions
	ComplexEigenSolver ()
	Default constructor. More...

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

	ComplexEigenSolver (Index size)
	Default Constructor with memory preallocation. More...

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

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

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

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

Index	getMaxIterations ()
	Returns the maximum number of iterations. More...

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

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

Protected Attributes
bool	m_eigenvectorsOk

EigenvalueType	m_eivalues

EigenvectorType	m_eivec

bool	m_isInitialized

EigenvectorType	m_matX

ComplexSchur< MatrixType >	m_schur

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.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues>

Eigenvalues

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

Definition at line 47 of file ComplexEigenSolver.h.

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.

Definition at line 73 of file ComplexEigenSolver.h.

◆ 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.

Definition at line 80 of file ComplexEigenSolver.h.

◆ 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.

Definition at line 87 of file ComplexEigenSolver.h.

◆ Index

template<typename MatrixType_ >

typedef Eigen::Index Eigen::ComplexEigenSolver< MatrixType_ >::Index

Deprecated:: since Eigen 3.3

Definition at line 65 of file ComplexEigenSolver.h.

◆ MatrixType

template<typename MatrixType_ >

typedef MatrixType_ Eigen::ComplexEigenSolver< MatrixType_ >::MatrixType

Synonym for the template parameter MatrixType_.

Definition at line 52 of file ComplexEigenSolver.h.

◆ RealScalar

template<typename MatrixType_ >

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

Definition at line 64 of file ComplexEigenSolver.h.

◆ Scalar

template<typename MatrixType_ >

typedef MatrixType::Scalar Eigen::ComplexEigenSolver< MatrixType_ >::Scalar

Scalar type for matrices of type MatrixType.

Definition at line 63 of file ComplexEigenSolver.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 54 of file ComplexEigenSolver.h.

          {
       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().

Definition at line 94 of file ComplexEigenSolver.h.

             : 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()

Definition at line 109 of file ComplexEigenSolver.h.

             : 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.

Definition at line 128 of file ComplexEigenSolver.h.

             : 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);
     }

Member Function Documentation

◆ compute() [1/2]

template<typename MatrixType_ >

template<typename InputType >

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

Definition at line 260 of file ComplexEigenSolver.h.

 {
   // 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;
 }

◆ 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:

MatrixXcf A = MatrixXcf::Random(4,4);
cout << "Here is a random 4x4 matrix, A:" << endl << A << endl << endl;
 
ComplexEigenSolver<MatrixXcf> ces;
ces.compute(A);
cout << "The eigenvalues of A are:" << endl << ces.eigenvalues() << endl;
cout << "The matrix of eigenvectors, V, is:" << endl << ces.eigenvectors() << endl << endl;
 
complex<float> lambda = ces.eigenvalues()[0];
cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
VectorXcf v = ces.eigenvectors().col(0);
cout << "If v is the corresponding eigenvector, then lambda * v = " << endl << lambda * v << endl;
cout << "... and A * v = " << endl << A * v << endl << endl;
 
cout << "Finally, V * D * V^(-1) = " << endl
     << ces.eigenvectors() * ces.eigenvalues().asDiagonal() * ces.eigenvectors().inverse() << endl;

Output:

Here is a random 4x4 matrix, A:
  (-0.211,0.68)  (0.108,-0.444)   (0.435,0.271) (-0.198,-0.687)
  (0.597,0.566) (0.258,-0.0452)  (0.214,-0.717)  (-0.782,-0.74)
 (-0.605,0.823)  (0.0268,-0.27) (-0.514,-0.967)  (-0.563,0.998)
  (0.536,-0.33)   (0.832,0.904)  (0.608,-0.726)  (0.678,0.0259)

The eigenvalues of A are:
 (0.137,0.505)
 (-0.758,1.22)
 (1.52,-0.402)
(-0.691,-1.63)
The matrix of eigenvectors, V, is:
  (-0.246,-0.106)     (0.418,0.263)   (0.0417,-0.296)    (-0.122,0.271)
  (-0.205,-0.629)    (0.466,-0.457)    (0.244,-0.456)      (0.247,0.23)
 (-0.432,-0.0359) (-0.0651,-0.0146)    (-0.191,0.334)   (0.859,-0.0877)
    (-0.301,0.46)    (-0.41,-0.397)     (0.623,0.328)    (-0.116,0.195)

Consider the first eigenvalue, lambda = (0.137,0.505)
If v is the corresponding eigenvector, then lambda * v = 
 (0.0197,-0.139)
    (0.29,-0.19)
(-0.0412,-0.223)
(-0.274,-0.0891)
... and A * v = 
 (0.0197,-0.139)
    (0.29,-0.19)
(-0.0412,-0.223)
(-0.274,-0.0891)

Finally, V * D * V^(-1) = 
  (-0.211,0.68)  (0.108,-0.444)   (0.435,0.271) (-0.198,-0.687)
  (0.597,0.566) (0.258,-0.0452)  (0.214,-0.717)  (-0.782,-0.74)
 (-0.605,0.823)  (0.0268,-0.27) (-0.514,-0.967)  (-0.563,0.998)
  (0.536,-0.33)   (0.832,0.904)  (0.608,-0.726)  (0.678,0.0259)

◆ doComputeEigenvectors()

template<typename MatrixType >

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

private

Definition at line 284 of file ComplexEigenSolver.h.

 {
   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();
   }
 }

◆ 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:

MatrixXcf ones = MatrixXcf::Ones(3,3);
ComplexEigenSolver<MatrixXcf> ces(ones, /* computeEigenvectors = */ false);
cout << "The eigenvalues of the 3x3 matrix of ones are:" 
     << endl << ces.eigenvalues() << endl;

Output:

The eigenvalues of the 3x3 matrix of ones are:
(0,-0)
 (0,0)
 (3,0)

Definition at line 184 of file ComplexEigenSolver.h.

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

◆ 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:

MatrixXcf ones = MatrixXcf::Ones(3,3);
ComplexEigenSolver<MatrixXcf> ces(ones);
cout << "The first eigenvector of the 3x3 matrix of ones is:" 
     << endl << ces.eigenvectors().col(0) << endl;

Output:

The first eigenvector of the 3x3 matrix of ones is:
(-0.816,0)
 (0.408,0)
 (0.408,0)

Definition at line 159 of file ComplexEigenSolver.h.

     {
       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;
     }

◆ getMaxIterations()

template<typename MatrixType_ >

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

inline

Returns the maximum number of iterations.

Definition at line 235 of file ComplexEigenSolver.h.

     {
       return m_schur.getMaxIterations();
     }

◆ info()

template<typename MatrixType_ >

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

inline

Reports whether previous computation was successful.

Returns: Success if computation was successful, NoConvergence otherwise.

Definition at line 221 of file ComplexEigenSolver.h.

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

◆ setMaxIterations()

template<typename MatrixType_ >

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

inline

Sets the maximum number of iterations allowed.

Definition at line 228 of file ComplexEigenSolver.h.

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

◆ sortEigenvalues()

template<typename MatrixType >

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

private

Definition at line 324 of file ComplexEigenSolver.h.

 {
   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

Definition at line 248 of file ComplexEigenSolver.h.

◆ m_eivalues

template<typename MatrixType_ >

EigenvalueType Eigen::ComplexEigenSolver< MatrixType_ >::m_eivalues

protected

Definition at line 245 of file ComplexEigenSolver.h.

◆ m_eivec

template<typename MatrixType_ >

EigenvectorType Eigen::ComplexEigenSolver< MatrixType_ >::m_eivec

protected

Definition at line 244 of file ComplexEigenSolver.h.

◆ m_isInitialized

template<typename MatrixType_ >

bool Eigen::ComplexEigenSolver< MatrixType_ >::m_isInitialized

protected

Definition at line 247 of file ComplexEigenSolver.h.

◆ m_matX

template<typename MatrixType_ >

EigenvectorType Eigen::ComplexEigenSolver< MatrixType_ >::m_matX

protected

Definition at line 249 of file ComplexEigenSolver.h.

◆ m_schur

template<typename MatrixType_ >

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

protected

Definition at line 246 of file ComplexEigenSolver.h.

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

ComplexEigenSolver.h

Public Types

Public Member Functions

Protected Attributes

Private Member Functions

Detailed Description

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

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

◆ 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

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