Eigen::ComplexEigenSolver< MatrixType_ > Class Template Reference

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

#include <ComplexEigenSolver.h>

Public Types
enum	{   RowsAtCompileTime = MatrixType::RowsAtCompileTime , ColsAtCompileTime = MatrixType::ColsAtCompileTime , Options = internal::traits<MatrixType>::Options , MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime ,   MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
typedef MatrixType_	MatrixType
	Synonym for the template parameter MatrixType_. More...
typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type MatrixType. More...
typedef NumTraits< Scalar >::Real	RealScalar
typedef Eigen::Index	Index
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...

Public Member Functions
	ComplexEigenSolver ()
	Default constructor. More...
	ComplexEigenSolver (Index size)
	Default Constructor with memory preallocation. More...
template<typename InputType >
	ComplexEigenSolver (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
	Constructor; computes eigendecomposition of given matrix. More...
const EigenvectorType &	eigenvectors () const
	Returns the eigenvectors of given matrix. More...
const EigenvalueType &	eigenvalues () const
	Returns the eigenvalues of given matrix. More...
template<typename InputType >
ComplexEigenSolver &	compute (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
	Computes eigendecomposition of given matrix. More...
ComputationInfo	info () const
	Reports whether previous computation was successful. More...
ComplexEigenSolver &	setMaxIterations (Index maxIters)
	Sets the maximum number of iterations allowed. More...
Index	getMaxIterations ()
	Returns the maximum number of iterations. More...
template<typename InputType >
ComplexEigenSolver< MatrixType > &	compute (const EigenBase< InputType > &matrix, bool computeEigenvectors)

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

Deprecated:

since Eigen 3.3

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 = internal::traits<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 matrix().

Member Function Documentation

compute() [1/2]

template<typename MatrixType_ >

template<typename InputType >

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

                                                                                                  {
  // 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::ComplexSchur< MatrixType_ >::compute(), Eigen::ComplexEigenSolver< MatrixType_ >::doComputeEigenvectors(), eigen_assert, Eigen::ComplexSchur< MatrixType_ >::info(), Eigen::ComplexEigenSolver< MatrixType_ >::m_eigenvectorsOk, Eigen::ComplexEigenSolver< MatrixType_ >::m_eivalues, Eigen::ComplexEigenSolver< MatrixType_ >::m_isInitialized, Eigen::ComplexEigenSolver< MatrixType_ >::m_schur, matrix(), Eigen::ComplexSchur< MatrixType_ >::matrixT(), Eigen::ComplexEigenSolver< MatrixType_ >::sortEigenvalues(), and Eigen::Success.

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:

Referenced by Eigen::ComplexEigenSolver< MatrixType_ >::ComplexEigenSolver(), ctms_decompositions(), eigensolver(), and eigensolver_verify_assert().

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).stableNormalize();
  }
}

References Eigen::PlainObjectBase< Derived >::coeff(), Eigen::Matrix< Scalar_, Rows_, Cols_, Options_, MaxRows_, MaxCols_ >::coeffRef(), oomph::SarahBL::epsilon, i, k, Eigen::ComplexEigenSolver< MatrixType_ >::m_eivalues, Eigen::ComplexEigenSolver< MatrixType_ >::m_eivec, Eigen::ComplexEigenSolver< MatrixType_ >::m_matX, Eigen::ComplexEigenSolver< MatrixType_ >::m_schur, Eigen::ComplexSchur< MatrixType_ >::matrixT(), Eigen::ComplexSchur< MatrixType_ >::matrixU(), Eigen::numext::maxi(), min, n, Eigen::numext::real_ref(), Eigen::value, and oomph::PseudoSolidHelper::Zero.

Referenced by Eigen::ComplexEigenSolver< MatrixType_ >::compute().

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:

                                            {
    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 eigensolver(), eigensolver_verify_assert(), and Eigen::internal::eigenvalues_selector< Derived, IsComplex >::run().

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:

                                              {
    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.

Referenced by eigensolver(), and eigensolver_verify_assert().

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.

Referenced by eigensolver().

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.

                               {
    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.

Referenced by eigensolver().

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().

Referenced by eigensolver().

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));
    }
  }
}

References i, k, Eigen::ComplexEigenSolver< MatrixType_ >::m_eivalues, Eigen::ComplexEigenSolver< MatrixType_ >::m_eivec, n, Eigen::PlainObjectBase< Derived >::swap(), and swap().

Referenced by Eigen::ComplexEigenSolver< MatrixType_ >::compute().

Member Data Documentation

m_eigenvectorsOk

template<typename MatrixType_ >

bool Eigen::ComplexEigenSolver< MatrixType_ >::m_eigenvectorsOk

protected

Referenced by Eigen::ComplexEigenSolver< MatrixType_ >::compute(), and Eigen::ComplexEigenSolver< MatrixType_ >::eigenvectors().

m_eivalues

template<typename MatrixType_ >

EigenvalueType Eigen::ComplexEigenSolver< MatrixType_ >::m_eivalues

protected

Referenced by Eigen::ComplexEigenSolver< MatrixType_ >::compute(), Eigen::ComplexEigenSolver< MatrixType_ >::doComputeEigenvectors(), Eigen::ComplexEigenSolver< MatrixType_ >::eigenvalues(), and Eigen::ComplexEigenSolver< MatrixType_ >::sortEigenvalues().

m_eivec

template<typename MatrixType_ >

EigenvectorType Eigen::ComplexEigenSolver< MatrixType_ >::m_eivec

protected

Referenced by Eigen::ComplexEigenSolver< MatrixType_ >::doComputeEigenvectors(), Eigen::ComplexEigenSolver< MatrixType_ >::eigenvectors(), and Eigen::ComplexEigenSolver< MatrixType_ >::sortEigenvalues().

m_isInitialized

template<typename MatrixType_ >

bool Eigen::ComplexEigenSolver< MatrixType_ >::m_isInitialized

protected

Referenced by Eigen::ComplexEigenSolver< MatrixType_ >::compute(), 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

Referenced by Eigen::ComplexEigenSolver< MatrixType_ >::doComputeEigenvectors().

m_schur

template<typename MatrixType_ >

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

protected

Referenced by Eigen::ComplexEigenSolver< MatrixType_ >::compute(), Eigen::ComplexEigenSolver< MatrixType_ >::doComputeEigenvectors(), Eigen::ComplexEigenSolver< MatrixType_ >::getMaxIterations(), Eigen::ComplexEigenSolver< MatrixType_ >::info(), and Eigen::ComplexEigenSolver< MatrixType_ >::setMaxIterations().

---

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

• ComplexEigenSolver.h